CRC Handbook of chemistry and physics

PREFACE The 84th Edition of the CRC Handbook of Chemistry and Physics features a completely new version of the most hea...

Author: David R. Lide

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PREFACE The 84th Edition of the CRC Handbook of Chemistry and Physics features a completely new version of the most heavily used table, Physical Constants of Organic Compounds. This is the first revision of the table since 1994. Compounds have been selected for inclusion in the new table by a careful screening of lists of organic compounds that are important in laboratory research, industrial chemistry, environmental protection, drug development, teaching, and other active areas. In this way priorities were established for choosing the most significant compounds out of the millions of organic substances that have been reported in the literature. Property data for the selected compounds have been updated, and new structure diagrams, which show much more detail than the previous structures, have been drawn for all the compounds. This Internet version of the 84th Edition has added 17 new subsections that can be accessed as interactive tables. These include tables on Heat of Combustion, Activity Coefficients, Refrigerants, Amino Acids, Chemical Carcinogens, Laboratory Solvents, and other topics. The search screens have been modified to make them more user friendly, and there is now a subject index that permits boolean searching on the name of a physical property and the identifiers of a chemical substance (name, formula, or CAS Registry Number). An option has been added to the table displays that permits locking the left-most column, which usually contains the chemical name, when scanning a wide table. Tool-tips that explain the data in a column now appear when the cursor is held over that column heading, and it is now possible to export the results of a search directly into an Excel file. Other new features of the 84th Edition include: • An update and expansion of the table of Critical Constants of Fluids, with many new compounds and recently published data • A new version of Properties of Refrigerants, which covers fluids now used in refrigeration systems and those being considered as substitutes • A new table on Fermi Energy and Related Properties of Metals • New tables of practical laboratory data such as Flame and Bead tests, Flame Temperatures, and Density of Ethanol-Water Mixtures • An update of lists of Chemical Carcinogens and Interstellar Molecules. The Handbook of Chemistry and Physics is dependent on the efforts of many contributors throughout the world. The list of current contributors follows this Preface. The new table of Physical Constants of Organic Compounds could not have been completed without the help of Dr. Fiona Macdonald, who oversaw the structure drawing and checked names and formulas. Thanks are also due to Janice Shackleton, Trupti Desai, Nazila Kamaly, Matt Griffiths, and Lawrence Braschi, who participated in drawing the structures. David R. Lide October 27, 2003 This Edition is dedicated to my grandchildren: Mary Eleanor Lide David Alston Lide, Jr. Grace Eileen Lide David Austell Whitcomb Kate Elizabeth Whitcomb

This work contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable efforts have been made to publish reliable data and information, but the author and the publisher cannot accept responsibility for the validity of all materials or for the consequences of their use. © Copyright CRC Press LLC 2004

CURRENT CONTRIBUTORS Lev I. Berger California Institute of Electronics and Materials Science 2115 Flame Tree Way Hemet, California 92545 A. K. Covington Department of Chemistry University of Newcastle Newcastle upon Tyne NE1 7RU England Robert B. Fox 6115 Wiscassett Rd. Bethesda, Maryland 20816 H. P. R. Frederikse 9625 Dewmar Lane Kensington, Maryland 20895 J.R. Fuhr Atomic Physics Division National Institute of Standards and Technology Gaithersburg, Maryland 20899 Robert N. Goldberg Biotechnology Division National Institute of Standards and Technology Gaithersburg, Maryland 20899 Karl A. Gschneidner Ames Laboratory Energy and Mineral Resources Research Institute Iowa State University Ames, Iowa 50011

Norman E. Holden National Nuclear Data Center Brookhaven National Laboratory Upton, New York 11973 H. Donald Brooke Jenkins Department of Chemistry University of Warwick Coventry CV4 7AL England Henry V. Kehiaian ITODYS 1 rue Guy de la Brosse 75005 Paris, France J. Alistair Kerr School of Chemistry University of Birmingham Birmingham B15 2TT England Nand Kishore Department of Chemistry Indian Institute of Technology Powai, Bombay 400 076 India Rebecca Lennen Naval Surface Warfare Center Biological Sciences Group 9500 MacArthur Blvd. West Bethesda, Maryland 20817-5700 Frank J. Lovas 8616 Melwood Rd. Bethesda, Maryland 20817 William C. Martin Atomic Physics Division National Institute of Standards and Technology Gaithersburg, Maryland 20899

Joseph Reader Atomic Physics Division National Institute of Standards and Technology Gaithersburg, Maryland 20899 Lewis E. Snyder Astronomy Department University of Illinois Urbana, Illinois 61801 David W. Stocker School of Chemistry University of Leeds Leeds LS2 9JT England B. N. Taylor Physics Laboratory National Institute of Standards and Technology Gaithersburg, Maryland 20899 Thomas G. Trippe Particle Data Group Lawrence Berkeley Laboratory 1 Cyclotron Road Berkeley, California 94720 Petr Vany´sek Department of Chemistry Northern Illinois University DeKalb, Illinois 60115 Wolfgang L. Wiese Atomic Physics Division National Institute of Standards and Technology Gaithersburg, Maryland 20899

C. R. Hammond 17 Greystone Rd. West Hartford, Connecticut 06107

Joel S. Miller Department of Chemistry University of Utah Salt Lake City, Utah 84112

Edward S. Wilks E.I. du Pont de Nemours and Company Inc. Barley Mills Plaza 14/1290 Wilmington, Delaware 19880-0014

Robert F. Hampson Chemical Kinetics Division National Institute of Standards and Technology Gaithersburg, Maryland 20899

Thomas M. Miller Air Force Research Laboratory/VSBP 29 Randolph Rd. Hanscom AFB, Massachusetts 01731-3010

Christian Wohlfarth Institut für Physikalische Chemie Martin Luther University D-06217 Merseburg Germany

FUNDAMENTAL PHYSICAL CONSTANTS Peter J. Mohr and Barry N. Taylor These tables give the 1998 self-consistent set of values of the basic constants and conversion factors of physics and chemistry recommended by the Committee on Data for Science and Technology (CODATA) for international use. The 1998 set replaces the previous set of constants recommended by CODATA in 1986; assigned uncertainties have been reduced by a factor of 1/5 to 1/12 (and sometimes even greater) relative to the 1986 uncertainties. The recommended set is based on a least-squares adjustment involving all of the relevant experimental and theoretical data available through December 31, 1998. Full details of the input data and the adjustment procedure are given in Reference 1. The 1998 adjustment was carried out by P. J. Mohr and B. N. Taylor of the National Institute of Standards and Technology (NIST) under the auspices of the CODATA Task Group on Fundamental Constants. The Task Group was established in 1969 with the aim of periodically providing the scientific and technological communities with a self-consistent set of internationally recommended values of the fundamental physical constants based on all applicable information available at a given point in time. The first set was published in 1973 and was followed by a revised set first published in 1986; the current 1998 set first appeared in 1999. In the future, the CODATA Task Group plans to take advantage of the high level of automation developed for the current set in order to issue a new set of recommended values at least every four years. At the time of completion of the 1998 adjustment, the membership of the Task Group was as follows: F. Cabiati, Istituto Elettrotecnico Nazionale “Galileo Ferraris,” Italy E. R. Cohen, Science Center, Rockwell International (retired), United States of America T. Endo, Electrotechnical Laboratory, Japan R. Liu, National Institute of Metrology, China (People’s Republic of) B. A. Mamyrin, A. F. Ioffe Physical-Technical Institute, Russian Federation P. J. Mohr, National Institute of Standards and Technology, United States of America F. Nez, Laboratoire Kastler-Brossel, France B. W. Petley, National Physical Laboratory, United Kingdom T. J. Quinn, Bureau International des Poids et Mesures B. N. Taylor, National Institute of Standards and Technology, United States of America V. S. Tuninsky, D. I. Mendeleyev All-Russian Research Institute for Metrology, Russian Federation W. Wöger, Physikalisch-Technische Bundesanstalt, Germany B. M. Wood, National Research Council, Canada REFERENCES 1. Mohr, Peter J., and Taylor, Barry N., J. Phys Chem. Ref. Data 28, 1713, 1999; Rev. Mod. Phys. 72, 351, 2000. The 1998 set of recommended values is also available at the Web site of the Fundamental Constants Data Center of the NIST Physics Laboratory: http://physics.nist.gov/constants.

Fundamental Physical Constants Quantity

Symbol

Value

Unit

Relative std. uncert. u r

UNIVERSAL m s−1 N A−2 N A−2 F m−1

(exact)

ε0

299 792 458 4π × 10−7 = 12.566 370 614... × 10−7 8.854 187 817... × 10−12

Z0

376.730 313 461...

(exact)

G G/c h

6.673(10) × 10−11 6.707(10) × 10−39 6.626 068 76(52) × 10−34 4.135 667 27(16) × 10−15 1.054 571 596(82) × 10−34 6.582 118 89(26) × 10−16

m3 kg−1 s−2 (GeV/c2 )−2 Js eV s Js eV s

1.5 × 10−3 1.5 × 10−3 7.8 × 10−8 3.9 × 10−8 7.8 × 10−8 3.9 × 10−8

2.1767(16) × 10−8 1.6160(12) × 10−35 5.3906(40) × 10−44

kg m s

7.5 × 10−4 7.5 × 10−4 7.5 × 10−4

speed of light in vacuum magnetic constant

c, c0 µ0

electric constant 1/µ0 c2 characteristic√impedance of vacuum µ0 /0 = µ0 c Newtonian constant of gravitation Planck constant in eV s h/2π in eV s Planck mass (c/G)1/2 Planck length /m P c = (G/c3 )1/2 Planck time lP /c = (G/c5 )1/2

mP lP tP

(exact) (exact)

ELECTROMAGNETIC elementary charge

e e/ h

1.602 176 462(63) × 10−19 2.417 989 491(95) × 1014

C A J−1

3.9 × 10−8 3.9 × 10−8

magnetic flux quantum h/2e conductance quantum 2e2/ h inverse of conductance quantum Josephson constanta 2e/ h von Klitzing constantb h/e2 = µ0 c/2α

0 G0 G −1 0 KJ

2.067 833 636(81) × 10−15 7.748 091 696(28) × 10−5 12 906.403 786(47) 483 597.898(19) × 109

Wb S Hz V−1

3.9 × 10−8 3.7 × 10−9 3.7 × 10−9 3.9 × 10−8

RK

25 812.807 572(95)

3.7 × 10−9

Bohr magneton e/2m e in eV T−1

µB

927.400 899(37) × 10−26 5.788 381 749(43) × 10−5 13.996 246 24(56) × 109 46.686 4521(19) 0.671 7131(12)

J T−1 eV T−1 Hz T−1 m−1 T−1 K T−1

4.0 × 10−8 7.3 × 10−9 4.0 × 10−8 4.0 × 10−8 1.7 × 10−6

5.050 783 17(20) × 10−27 3.152 451 238(24) × 10−8 7.622 593 96(31) 2.542 623 66(10) × 10−2 3.658 2638(64) × 10−4

J T−1 eV T−1 MHz T−1 m−1 T−1 K T−1

4.0 × 10−8 7.6 × 10−9 4.0 × 10−8 4.0 × 10−8 1.7 × 10−6

µB / h µB / hc µB /k nuclear magneton e/2m p in eV T−1

µN µN / h µN / hc µN /k

ATOMIC AND NUCLEAR General fine-structure constant e2/4π0 c inverse fine-structure constant

α α−1

7.297 352 533(27) × 10−3 137.035 999 76(50)

3.7 × 10−9 3.7 × 10−9


Rydberg constant α2 m e c/2h

Symbol

Unit


R∞ R∞ c R∞ hc

10 973 731.568 549(83) 3.289 841 960 368(25) × 1015 2.179 871 90(17) × 10−18 13.605 691 72(53)

m−1 Hz J eV

7.6 × 10−12 7.6 × 10−12 7.8 × 10−8 3.9 × 10−8

a0

0.529 177 2083(19) × 10−10

m

3.7 × 10−9

Eh

4.359 743 81(34) × 10−18 27.211 3834(11) 3.636 947 516(27) × 10−4 7.273 895 032(53) × 10−4

J eV m2 s−1 m2 s−1

7.8 × 10−8 3.9 × 10−8 7.3 × 10−9 7.3 × 10−9

GeV−2

8.6 × 10−6

R∞ hc in eV Bohr radius α/4π R∞ = 4π0 2/m e e2 Hartree energy e2/4πε0 a0 = 2R∞ hc = α2 m e c2 in eV quantum of circulation

Value

h/2m e h/m e

Electroweak Fermi coupling constantc weak mixing angled θW (on-shell scheme) 2 ≡ 1 − (m /m )2 sin2 θW = sW W Z

G F /(c)3

1.166 39(1) × 10−5

sin2 θW

0.2224(19)

8.7 × 10−3

Electron, e− 9.109 381 88(72) × 10−31

kg

7.9 × 10−8

c2

5.485 799 110(12) × 10−4 8.187 104 14(64) × 10−14 0.510 998 902(21)

u J MeV

2.1 × 10−9 7.9 × 10−8 4.0 × 10−8

electron-muon mass ratio electron-tau mass ratio electron-proton mass ratio electron-neutron mass ratio electron-deuteron mass ratio electron to alpha particle mass ratio

m e /m µ m e /m τ m e /m p m e /m n m e /m d m e /m α

4.836 332 10(15) × 10−3 2.875 55(47) × 10−4 5.446 170 232(12) × 10−4 5.438 673 462(12) × 10−4 2.724 437 1170(58) × 10−4 1.370 933 5611(29) × 10−4

electron charge to mass quotient electron molar mass NA m e Compton wavelength h/m e c λC /2π = αa0 = α2/4π R∞ classical electron radius α2 a0 Thomson cross section (8π/3)re2

−e/m e M(e), Me λC re σe

−1.758 820 174(71) × 1011 5.485 799 110(12) × 10−7 2.426 310 215(18) × 10−12 386.159 2642(28) × 10−15 2.817 940 285(31) × 10−15 0.665 245 854(15) × 10−28

C kg−1 kg mol−1 m m m m2

4.0 × 10−8 2.1 × 10−9 7.3 × 10−9 7.3 × 10−9 1.1 × 10−8 2.2 × 10−8

µe µe /µB µe /µN

−928.476 362(37) × 10−26 −1.001 159 652 1869(41) −1 838.281 9660(39)

J T−1

4.0 × 10−8 4.1 × 10−12 2.1 × 10−9

ae ge

1.159 652 1869(41) × 10−3 −2.002 319 304 3737(82)

3.5 × 10−9 4.1 × 10−12

µe /µµ

206.766 9720(63)

3.0 × 10−8

electron mass in u, m e = Ar (e) u (electron relative atomic mass times u) energy equivalent in MeV

electron magnetic moment to Bohr magneton ratio to nuclear magneton ratio electron magnetic moment anomaly |µe |/µB − 1 electron g-factor −2(1 + ae ) electron-muon magnetic moment ratio

me

me

C

3.0 × 10−8 1.6 × 10−4 2.1 × 10−9 2.2 × 10−9 2.1 × 10−9 2.1 × 10−9

Fundamental Physical Constants Quantity electron-proton magnetic moment ratio electron to shielded proton magnetic moment ratio (H2 O, sphere, 25 ◦ C) electron-neutron magnetic moment ratio electron-deuteron magnetic moment ratio electron to shielded helione magnetic moment ratio (gas, sphere, 25 ◦ C) electron gyromagnetic ratio 2|µe |/

Symbol

Value

Unit


µe /µp

− 658.210 6875(66)

1.0 × 10−8

µe /µ p

− 658.227 5954(71)

1.1 × 10−8

µe /µn

960.920 50(23)

2.4 × 10−7

µe /µd

−2 143.923 498(23)

1.1 × 10−8

µe /µ h

864.058 255(10)

1.2 × 10−8

γe γe /2π

1.760 859 794(71) × 1011 28 024.9540(11)

s−1 T−1 MHz T−1

4.0 × 10−8 4.0 × 10−8

1.883 531 09(16) × 10−28

kg

8.4 × 10−8

0.113 428 9168(34) 1.692 833 32(14) × 10−11 105.658 3568(52)

u J MeV

3.0 × 10−8 8.4 × 10−8 4.9 × 10−8

kg mol−1

3.0 × 10−8 1.6 × 10−4 3.0 × 10−8 3.0 × 10−8 3.0 × 10−8

Muon, µ− muon mass in u, m µ = Ar (µ) u (muon relative atomic mass times u) energy equivalent in MeV

mµ

mµ

c2

muon-electron mass ratio muon-tau mass ratio muon-proton mass ratio muon-neutron mass ratio muon molar mass NA m µ

m µ /m e m µ /m τ m µ /m p m µ /m n M(µ), Mµ

206.768 2657(63) 5.945 72(97) × 10−2 0.112 609 5173(34) 0.112 454 5079(34) 0.113 428 9168(34) × 10−3

muon Compton wavelength h/m µ c λC,µ /2π muon magnetic moment to Bohr magneton ratio to nuclear magneton ratio

λC,µ

µµ µµ /µB µµ /µN

11.734 441 97(35) × 10−15 1.867 594 444(55) × 10−15 −4.490 448 13(22) × 10−26 −4.841 970 85(15) × 10−3 −8.890 597 70(27)

aµ gµ

1.165 916 02(64) × 10−3 −2.002 331 8320(13)

5.5 × 10−7 6.4 × 10−10

µµ /µp

−3.183 345 39(10)

3.2 × 10−8

muon magnetic moment anomaly |µµ |/(e/2m µ ) − 1 muon g-factor −2(1 + aµ ) muon-proton magnetic moment ratio

C,µ

m m J T−1

2.9 × 10−8 2.9 × 10−8 4.9 × 10−8 3.0 × 10−8 3.0 × 10−8

Tau, τ − tau massf in u, m τ = Ar (τ) u (tau relative atomic mass times u) energy equivalent in MeV

mτ

mτ

c2

3.167 88(52) × 10−27

kg

1.6 × 10−4

1.907 74(31) 2.847 15(46) × 10−10 1 777.05(29)

u J MeV

1.6 × 10−4 1.6 × 10−4 1.6 × 10−4


Symbol

tau-electron mass ratio tau-muon mass ratio tau-proton mass ratio tau-neutron mass ratio tau molar mass NA m τ

m τ /m e m τ /m µ m τ /m p m τ /m n M(τ), Mτ

tau Compton wavelength h/m τ c λC,τ /2π

λC,τ C,τ

Unit


3 477.60(57) 16.8188(27) 1.893 96(31) 1.891 35(31) 1.907 74(31) × 10−3

kg mol−1

1.6 × 10−4 1.6 × 10−4 1.6 × 10−4 1.6 × 10−4 1.6 × 10−4

0.697 70(11) × 10−15 0.111 042(18) × 10−15

m m

1.6 × 10−4 1.6 × 10−4

1.672 621 58(13) × 10−27

kg

7.9 × 10−8

1.007 276 466 88(13) 1.503 277 31(12) × 10−10 938.271 998(38)

u J MeV

1.3 × 10−10 7.9 × 10−8 4.0 × 10−8

C kg−1 kg mol−1

2.1 × 10−9 3.0 × 10−8 1.6 × 10−4 5.8 × 10−10 4.0 × 10−8 1.3 × 10−10

Value

Proton, p proton mass in u, m p = Ar (p) u (proton relative atomic mass times u) energy equivalent in MeV

mp

mp

c2

proton-electron mass ratio proton-muon mass ratio proton-tau mass ratio proton-neutron mass ratio proton charge to mass quotient proton molar mass NA m p

m p /m e m p /m µ m p /m τ m p /m n e/m p M(p), Mp

1 836.152 6675(39) 8.880 244 08(27) 0.527 994(86) 0.998 623 478 55(58) 9.578 834 08(38) × 107 1.007 276 466 88(13) × 10−3

proton Compton wavelength h/m p c λC,p /2π proton magnetic moment to Bohr magneton ratio to nuclear magneton ratio proton g-factor 2µp /µN

λC,p µp µp /µB µp /µN gp

1.321 409 847(10) × 10−15 0.210 308 9089(16) × 10−15 1.410 606 633(58) × 10−26 1.521 032 203(15) × 10−3 2.792 847 337(29) 5.585 694 675(57)

µp /µn µ p

−1.459 898 05(34) 1.410 570 399(59) × 10−26

µ p /µB µ p /µN

1.520 993 132(16) × 10−3 2.792 775 597(31)

1.1 × 10−8 1.1 × 10−8

σp

25.687(15) × 10−6

5.7 × 10−4

γp γp /2π

2.675 222 12(11) × 108 42.577 4825(18)


4.1 × 10−8 4.1 × 10−8

γp

2.675 153 41(11) × 108

s−1 T−1

4.2 × 10−8

γp /2π

42.576 3888(18)

MHz T−1

4.2 × 10−8

proton-neutron magnetic moment ratio shielded proton magnetic moment (H2 O, sphere, 25 ◦ C) to Bohr magneton ratio to nuclear magneton ratio proton magnetic shielding correction 1 − µ p /µp (H2 O, sphere, 25 ◦ C) proton gyromagnetic ratio 2µp / shielded proton gyromagnetic ratio 2µ p / (H2 O, sphere, 25 ◦ C)

C,p

Neutron, n

m m J T−1

J T−1

7.6 × 10−9 7.6 × 10−9 4.1 × 10−8 1.0 × 10−8 1.0 × 10−8 1.0 × 10−8 2.4 × 10−7 4.2 × 10−8


Symbol

Value

Unit


mn

1.674 927 16(13) × 10−27

kg

7.9 × 10−8

m n c2

1.008 664 915 78(55) 1.505 349 46(12) × 10−10 939.565 330(38)

u J MeV

5.4 × 10−10 7.9 × 10−8 4.0 × 10−8

neutron-electron mass ratio neutron-muon mass ratio neutron-tau mass ratio neutron-proton mass ratio neutron molar mass NA m n

m n /m e m n /m µ m n /m τ m n /m p M(n), Mn

1 838.683 6550(40) 8.892 484 78(27) 0.528 722(86) 1.001 378 418 87(58) 1.008 664 915 78(55) × 10−3

kg mol−1

2.2 × 10−9 3.0 × 10−8 1.6 × 10−4 5.8 × 10−10 5.4 × 10−10

neutron Compton wavelength h/m n c λC,n /2π neutron magnetic moment to Bohr magneton ratio to nuclear magneton ratio

λC,n µn µn /µB µn /µN

1.319 590 898(10) × 10−15 0.210 019 4142(16) × 10−15 −0.966 236 40(23) × 10−26 −1.041 875 63(25) × 10−3 −1.913 042 72(45)

gn

−3.826 085 45(90)

2.4 × 10−7

µn /µe

1.040 668 82(25) × 10−3

2.4 × 10−7

µn /µp

−0.684 979 34(16)

2.4 × 10−7

µn /µ p

−0.684 996 94(16)

2.4 × 10−7

γn γn /2π

1.832 471 88(44) × 108 29.164 6958(70)

neutron mass in u, m n = Ar (n) u (neutron relative atomic mass times u) energy equivalent in MeV

neutron g-factor 2µn /µN neutron-electron magnetic moment ratio neutron-proton magnetic moment ratio neutron to shielded proton magnetic moment ratio (H2 O, sphere, 25 ◦ C) neutron gyromagnetic ratio 2|µn |/

C,n

m m J T−1

7.6 × 10−9 7.6 × 10−9 2.4 × 10−7 2.4 × 10−7 2.4 × 10−7


2.4 × 10−7 2.4 × 10−7

3.343 583 09(26) × 10−27

kg

7.9 × 10−8

2.013 553 212 71(35) 3.005 062 62(24) × 10−10 1 875.612 762(75)

u J MeV

1.7 × 10−10 7.9 × 10−8 4.0 × 10−8

kg mol−1

2.1 × 10−9 2.0 × 10−10 1.7 × 10−10

Deuteron, d deuteron mass in u, m d = Ar (d) u (deuteron relative atomic mass times u) energy equivalent in MeV

md

md

c2

deuteron-electron mass ratio deuteron-proton mass ratio deuteron molar mass NA m d

m d /m e m d /m p M(d), Md

3 670.482 9550(78) 1.999 007 500 83(41) 2.013 553 212 71(35) × 10−3

deuteron magnetic moment to Bohr magneton ratio to nuclear magneton ratio

µd µd /µB µd /µN

0.433 073 457(18) × 10−26 0.466 975 4556(50) × 10−3 0.857 438 2284(94)

µd /µe

−4.664 345 537(50) × 10−4

1.1 × 10−8

µd /µp

0.307 012 2083(45)

1.5 × 10−8

deuteron-electron magnetic moment ratio deuteron-proton magnetic moment ratio

J T−1

4.2 × 10−8 1.1 × 10−8 1.1 × 10−8

Fundamental Physical Constants Quantity deuteron-neutron magnetic moment ratio

Symbol

Value

µd /µn

Unit


2.4 × 10−7

−0.448 206 52(11) Helion, h

helion masse in u, m h = Ar (h) u (helion relative atomic mass times u) energy equivalent in MeV helion-electron mass ratio helion-proton mass ratio helion molar mass NA m h shielded helion magnetic moment (gas, sphere, 25 ◦ C) to Bohr magneton ratio to nuclear magneton ratio shielded helion to proton magnetic moment ratio (gas, sphere, 25 ◦ C) shielded helion to shielded proton magnetic moment ratio (gas/H2 O, spheres, 25 ◦ C) shielded helion gyromagnetic ratio 2|µ h |/ (gas, sphere, 25 ◦ C)

mh

mh

c2

5.006 411 74(39) × 10−27

kg

7.9 × 10−8

3.014 932 234 69(86) 4.499 538 48(35) × 10−10 2 808.391 32(11)

u J MeV

2.8 × 10−10 7.9 × 10−8 4.0 × 10−8

kg mol−1 J T−1

2.1 × 10−9 3.1 × 10−10 2.8 × 10−10 4.2 × 10−8

m h /m e m h /m p M(h), Mh µ h

5 495.885 238(12) 2.993 152 658 50(93) 3.014 932 234 69(86) × 10−3 −1.074 552 967(45) × 10−26

µ h /µB µ h /µN

−1.158 671 474(14) × 10−3 −2.127 497 718(25)

1.2 × 10−8 1.2 × 10−8

µ h /µp

−0.761 766 563(12)

1.5 × 10−8

µ h /µ p

−0.761 786 1313(33)

4.3 × 10−9

γh

2.037 894 764(85) × 108

s−1 T−1

4.2 × 10−8

γh /2π

32.434 1025(14)

MHz T−1

4.2 × 10−8

6.644 655 98(52) × 10−27

kg

7.9 × 10−8

4.001 506 1747(10) 5.971 918 97(47) × 10−10 3 727.379 04(15)

u J MeV

2.5 × 10−10 7.9 × 10−8 4.0 × 10−8

kg mol−1

2.1 × 10−9 2.8 × 10−10 2.5 × 10−10

Alpha particle, α alpha particle mass in u, m α = Ar (α) u (alpha particle relative atomic mass times u) energy equivalent in MeV alpha particle to electron mass ratio alpha particle to proton mass ratio alpha particle molar mass NA m α

mα

mα

c2

m α /m e m α /m p M(α), Mα

7 294.299 508(16) 3.972 599 6846(11) 4.001 506 1747(10) × 10−3

PHYSICO-CHEMICAL Avogadro constant atomic mass constant 1 m u = 12 m(12 C) = 1 u = 10−3 kg mol−1/NA energy equivalent in MeV Faraday constantg NA e

NA , L

6.022 141 99(47) × 1023

mol−1

7.9 × 10−8

mu

1.660 538 73(13) × 10−27

kg

7.9 × 10−8

m u c2

1.492 417 78(12) × 10−10 931.494 013(37) 96 485.3415(39)

J MeV C mol−1

7.9 × 10−8 4.0 × 10−8 4.0 × 10−8

F


molar Planck constant molar gas constant Boltzmann constant R/NA in eV K−1

molar volume of ideal gas RT / p T = 273.15 K, p = 101.325 kPa Loschmidt constant NA /Vm T = 273.15 K, p = 100 kPa Sackur-Tetrode constant (absolute entropy constant)h 5 2 3/2 kT / p ] 1 0 2 + ln[(2πm u kT1 / h ) T1 = 1 K, p0 = 100 kPa T1 = 1 K, p0 = 101.325 kPa Stefan-Boltzmann constant (π 2 /60)k 4/3 c2 first radiation constant 2πhc2 first radiation constant for spectral radiance 2hc2 second radiation constant hc/k Wien displacement law constant b = λmax T = c2 /4.965 114 231...

Value

Unit


k/ h k/ hc

3.990 312 689(30) × 10−10 0.119 626 564 92(91) 8.314 472(15) 1.380 6503(24) × 10−23 8.617 342(15) × 10−5 2.083 6644(36) × 1010 69.503 56(12)

J s mol−1 J m mol−1 J mol−1 K−1 J K−1 eV K−1 Hz K−1 m−1 K−1

7.6 × 10−9 7.6 × 10−9 1.7 × 10−6 1.7 × 10−6 1.7 × 10−6 1.7 × 10−6 1.7 × 10−6

Vm n0 Vm

22.413 996(39) × 10−3 2.686 7775(47) × 1025 22.710 981(40) × 10−3

m3 mol−1 m−3 m3 mol−1

1.7 × 10−6 1.7 × 10−6 1.7 × 10−6

S0 /R

−1.151 7048(44) −1.164 8678(44)

σ c1 c1L c2

5.670 400(40) × 10−8 3.741 771 07(29) × 10−16 1.191 042 722(93) × 10−16 1.438 7752(25) × 10−2

W m−2 K−4 W m2 W m2 sr−1 mK

7.0 × 10−6 7.8 × 10−8 7.8 × 10−8 1.7 × 10−6

b

2.897 7686(51) × 10−3

mK

1.7 × 10−6

Symbol

NA h NA hc R k

3.8 × 10−6 3.7 × 10−6

a See the “Adopted values” table for the conventional value adopted internationally for realizing representations of the volt using the Joseph-

son effect. b See the “Adopted values” table for the conventional value adopted internationally for realizing representations of the ohm using the quantum Hall

effect. c Value recommended by the Particle Data Group, Caso et al., Eur. Phys. J. C 3(1-4), 1-794 (1998). d Based on the ratio of the masses of the W and Z bosons m /m recommended by the Particle Data Group (Caso et al., 1998). The value for W Z sin2 θW they recommend, which is based on a particular variant of the modified minimal subtraction (MS) scheme, is sin2 θˆW (MZ ) = 0.231 24(24). e The helion, symbol h, is the nucleus of the 3 He atom. f This and all other values involving m are based on the value of m c2 in MeV recommended by the Particle Data Group, Caso et al., Eur. Phys. τ τ J. C 3(1-4), 1-794 (1998), but with a standard uncertainty of 0.29 MeV rather than the quoted uncertainty of −0.26 MeV, +0.29 MeV. g The numerical value of F to be used in coulometric chemical measurements is 96 485.3432(76) [7.9×10−8 ] when the relevant current is measured in terms of representations of the volt and ohm based on the Josephson and quantum Hall effects and the internationally adopted conventional values of the Josephson and von Klitzing constants K J−90 and RK−90 given in the “Adopted values” table. h The entropy of an ideal monoatomic gas of relative atomic mass A is given by S = S + 3 R ln A − R ln( p/ p ) + 5 R ln(T/K). r r 0 0 2 2

Fundamental Physical Constants — Adopted values Relative std. uncert. u r

Quantity

Symbol

Value

Unit

molar mass of 12 C molar mass constanta M(12 C)/12 conventional value of Josephson constantb conventional value of von Klitzing constantc standard atmosphere standard acceleration of gravity

M(12 C) Mu

12 × 10−3 1 × 10−3

kg mol−1 kg mol−1

(exact) (exact)

K J−90

483 597.9

GHz V−1

(exact)

RK−90

25 812.807 101 325 9.806 65

Pa m s−2

(exact) (exact) (exact)

gn a The relative atomic mass A (X) of particle X with mass m(X) is defined by A (X) = m(X)/m , where m = m(12 C)/12 = M /N = 1 u is r r u u u A the atomic mass constant, NA is the Avogadro constant, and u is the atomic mass unit. Thus the mass of particle X in u is m(X) = A r (X) u and the molar mass of X is M(X) = Ar (X)Mu . b This is the value adopted internationally for realizing representations of the volt using the Josephson effect. c This is the value adopted internationally for realizing representations of the ohm using the quantum Hall effect.

Energy Equivalents J

kg

m−1

Hz

1J

(1 J) = 1J

(1 J)/c2 = 1.112 650 056 × 10−17 kg

(1 J)/hc = 5.034 117 62(39) × 1024 m−1

(1 J)/h = 1.509 190 50(12) × 1033 Hz

1 kg

(1 kg)c2 = 8.987 551 787 × 1016 J

(1 kg) = 1 kg

(1 kg)c/ h = 4.524 439 29(35) × 1041 m−1

(1 kg)c2 / h = 1.356 392 77(11) × 1050 Hz

1 m−1

(1 m−1 )hc = 1.986 445 44(16) × 10−25 J

(1 m−1 )h/c = 2.210 218 63(17) × 10−42 kg

(1 m−1 ) = 1 m−1

(1 m−1 )c = 299 792 458 Hz

1 Hz

(1 Hz)h = 6.626 068 76(52) × 10−34 J

(1 Hz)h/c2 = 7.372 495 78(58) × 10−51 kg

(1 Hz)/c = 3.335 640 952 × 10−9 m−1

(1 Hz) = 1 Hz

1K

(1 K)k = 1.380 6503(24) × 10−23 J

(1 K)k/c2 = 1.536 1807(27) × 10−40 kg

(1 K)k/ hc = 69.503 56(12) m−1

(1 K)k/ h = 2.083 6644(36) × 1010 Hz

1 eV

(1 eV) = 1.602 176 462(63) × 10−19 J

(1 eV)/ hc = (1 eV)/c2 = 1.782 661 731(70) × 10−36 kg 8.065 544 77(32) × 105 m−1

1u

(1 u)c2 = 1.492 417 78(12) × 10−10 J

(1 u) = 1.660 538 73(13) × 10−27 kg

(1 u)c/ h = 7.513 006 658(57) × 1014 m−1

(1 u)c2 / h = 2.252 342 733(17) × 1023 Hz

1 Eh

(1 E h ) = 4.359 743 81(34) × 10−18 J

(1 E h )/c2 = 4.850 869 19(38) × 10−35 kg

(1 E h )/ hc = 2.194 746 313 710(17) × 107 m−1

(1 E h )/ h = 6.579 683 920 735(50) × 1015 Hz

(1 eV)/ h = 2.417 989 491(95) × 1014 Hz

Derived from the relations E = mc2 = hc/λ = hν = kT , and based on the 1998 CODATA adjustment of the values of the constants; 1 m(12 C) = 10−3 kg mol−1/N , and E = 2R hc = α 2 m c2 is the Hartree energy (hartree). 1 eV = (e/C) J, 1 u = m u = 12 ∞ e A h

Energy Equivalents K

eV

u

Eh

1J

(1 J)/k = 7.242 964(13) × 1022 K

(1 J) = 6.241 509 74(24) × 1018 eV

(1 J)/c2 = 6.700 536 62(53) × 109 u

(1 J) = 2.293 712 76(18) × 1017 E h

1 kg

(1 kg)c2/k = 6.509 651(11) × 1039 K

(1 kg)c2 = 5.609 589 21(22) × 1035 eV

(1 kg) = 6.022 141 99(47) × 1026 u

(1 kg)c2 = 2.061 486 22(16) × 1034 E h

1 m−1

(1 m−1 )hc/k = 1.438 7752(25) × 10−2 K

(1 m−1 )hc = (1 m−1 )h/c = (1 m−1 )hc = −6 −15 1.239 841 857(49) × 10 eV 1.331 025 042(10) × 10 u 4.556 335 252 750(35) × 10−8 E h

1 Hz

(1 Hz)h/k = (1 Hz)h = 4.799 2374(84) × 10−11 K 4.135 667 27(16) × 10−15 eV

(1 Hz)h/c2 = (1 Hz)h = 4.439 821 637(34) × 10−24 u 1.519 829 846 003(12) × 10−16 E h

1K

(1 K) = 1K

(1 K)k = 8.617 342(15) × 10−5 eV

(1 K)k/c2 = 9.251 098(16) × 10−14 u

(1 K)k = 3.166 8153(55) × 10−6 E h

1 eV

(1 eV)/k = 1.160 4506(20) × 104 K

(1 eV) = 1 eV

(1 eV)/c2 = 1.073 544 206(43) × 10−9 u

(1 eV) = 3.674 932 60(14) × 10−2 E h

1u

(1 u)c2/k = 1.080 9528(19) × 1013 K

(1 u)c2 = 931.494 013(37) × 106 eV

(1 u) = 1u

(1 u)c2 = 3.423 177 709(26) × 107 E h

1 Eh

(1 E h )/k = 3.157 7465(55) × 105 K

(1 E h ) = 27.211 3834(11) eV

(1 E h )/c2 = 2.921 262 304(22) × 10−8 u

(1 E h ) = 1 Eh

Derived from the relations E = mc2 = hc/λ = hν = kT , and based on the 1998 CODATA adjustment of the values of the constants; 1 m(12 C) = 10−3 kg mol−1/N , and E = 2R hc = α 2 m c2 is the Hartree energy (hartree). 1 eV = (e/C) J, 1 u = m u = 12 ∞ e A h

STANDARD ATOMIC WEIGHTS (2001) This table of atomic weights includes the changes made in 1999 and 2001 by the IUPAC Commission on Atomic Weights and Isotopic Abundances. The Standard Atomic Weights apply to the elements as they exist naturally on Earth, and the uncertainties take into account the isotopic variation found in most laboratory samples. Further comments on the variability are given in the footnotes. The number in parentheses following the atomic weight value gives the uncertainty in the last digit. An atomic weight entry in brackets indicates that the element that has no stable isotopes; the value given is the atomic mass in u (or the mass number, if the mass is not accurately known) for the isotope of longest half-life. Thorium, protactinium, and uranium have no stable isotopes, but the terrestrial isotopic composition is sufficiently uniform to permit a standard atomic weight to be specified.

REFERENCES 1. 2. 3. 4.

Vocke, R. D., Pure Appl. Chem. 71, 1593, 1999. Coplen, T. D., Pure Appl. Chem. 73, 667, 2001. Coplen, T. D., J. Phys. Chem. Ref. Data, 30, 701, 2001. Loss, R. D., Report of the IUPAC Commission on Atomic Weights and Isotopic Abundances, Chemistry International, 23, 179, 2001.

Name Actinium Aluminum Americium Antimony Argon Arsenic Astatine Barium Berkelium Beryllium Bismuth Bohrium Boron Bromine Cadmium Calcium Californium Carbon Cerium Cesium Chlorine Chromium Cobalt Copper Curium Dubnium Dysprosium Einsteinium Erbium Europium Fermium Fluorine Francium Gadolinium Gallium Germanium Gold Hafnium

Symbol

Atomic No.

Ac Al Am Sb Ar As At Ba Bk Be Bi Bh B Br Cd Ca Cf C Ce Cs Cl Cr Co Cu Cm Db Dy Es Er Eu Fm F Fr Gd Ga Ge Au Hf

89 13 95 51 18 33 85 56 97 4 83 107 5 35 48 20 98 6 58 55 17 24 27 29 96 105 66 99 68 63 100 9 87 64 31 32 79 72

Atomic Weight [227.0277] 26.981538(2) [243.0614] 121.760(1) 39.948(1) 74.92160(2) [209.9871] 137.327(7) [247.0703] 9.012182(3) 208.98038(2) [264.12] 10.811(7) 79.904(1) 112.411(8) 40.078(4) [251.0796] 12.0107(8) 140.116(1) 132.90545(2) 35.453(2) 51.9961(6) 58.933200(9) 63.546(3) [247.0704] [262.1141] 162.500(1) [252.0830] 167.259(3) 151.964(1) [257.0951] 18.9984032(5) [223.0197] 157.25(3) 69.723(1) 72.64(1) 196.96655(2) 178.49(2)

1-12

Footnotes a a g gr a a

a gmr g g a gr g gmr

r a a g a g g a a g

STANDARD ATOMIC WEIGHTS (2001) (continued) Name

Symbol

Atomic No.

Hassium Helium Holmium Hydrogen Indium Iodine Iridium Iron Krypton Lanthanum Lawrencium Lead Lithium Lutetium Magnesium Manganese Meitnerium Mendelevium Mercury Molybdenum Neodymium Neon Neptunium Nickel Niobium Nitrogen Nobelium Osmium Oxygen Palladium Phosphorus Platinum Plutonium Polonium Potassium Praseodymium Promethium Protactinium Radium Radon Rhenium Rhodium Rubidium Ruthenium Rutherfordium Samarium Scandium Seaborgium Selenium Silicon Silver Sodium Strontium Sulfur Tantalum Technetium Tellurium Terbium

Hs He Ho H In I Ir Fe Kr La Lr Pb Li Lu Mg Mn Mt Md Hg Mo Nd Ne Np Ni Nb N No Os O Pd P Pt Pu Po K Pr Pm Pa Ra Rn Re Rh Rb Ru Rf Sm Sc Sg Se Si Ag Na Sr S Ta Tc Te Tb

108 2 67 1 49 53 77 26 36 57 103 82 3 71 12 25 109 101 80 42 60 10 93 28 41 7 102 76 8 46 15 78 94 84 19 59 61 91 88 86 75 45 37 44 104 62 21 106 34 14 47 11 38 16 73 43 52 65

Atomic Weight [277] 4.002602(2) 164.93032(2) 1.00794(7) 114.818(3) 126.90447(3) 192.217(3) 55.845(2) 83.798(2) 138.9055(2) [262.1097] 207.2(1) 6.941(2) 174.967(1) 24.3050(6) 54.938049(9) [268.1388] [258.0984] 200.59(2) 95.94(2) 144.24(3) 20.1797(6) [237.0482] 58.6934(2) 92.90638(2) 14.0067(2) [259.1010] 190.23(3) 15.9994(3) 106.42(1) 30.973761(2) 195.078(2) [244.0642] [208.9824] 39.0983(1) 140.90765(2) [144.9127] 231.03588(2) [226.0254] [222.0176] 186.207(1) 102.90550(2) 85.4678(3) 101.07(2) [261.1088] 150.36(3) 44.955910(8) [266.1219] 78.96(3) 28.0855(3) 107.8682(2) 22.989770(2) 87.62(1) 32.065(5) 180.9479(1) [97.9072] 127.60(3) 158.92534(2)

1-13

Footnotes a gr gmr

gm g a gr bgmr g

a a g g gm a

gr a g gr g

a a g a a a

g g a g a r r g gr gr a g

STANDARD ATOMIC WEIGHTS (2001) (continued)

a

b

g

m

r

Name

Symbol

Atomic No.

Thallium Thorium Thulium Tin Titanium Tungsten Ununbium Ununhexium Ununnilium Ununquadium Unununium Uranium Vanadium Xenon Ytterbium Yttrium Zinc Zirconium

Tl Th Tm Sn Ti W Uub Uuh Uun Uuq Uuu U V Xe Yb Y Zn Zr

81 90 69 50 22 74 112 116 110 114 111 92 23 54 70 39 30 40

Atomic Weight

204.3833(2) 232.0381(1) 168.93421(2) 118.710(7) 47.867(1) 183.84(1) [285] [289] [281] [289] [272.1535] 238.02891(3) 50.9415(1) 131.293(6) 173.04(3) 88.90585(2) 65.409(4) 91.224(2)

Footnotes

g g

a a a a a gm gm g

g

No stable isotope exists. The atomic mass in u (or the mass number, if the mass is not accurately known) is given in brackets for the isotope of longest half-life. Commercially available Li materials have atomic weights that range between 6.939 and 6.996; if a more accurate value is required, it must be determined for the specific material. Geological specimens are known in which the element has an isotopic composition outside the limits for the normal material. The difference between the atomic weight of the element in such specimens and that given in the table may exceed the stated uncertainty. Modified isotopic compositions may be found in commercially available material because it has been subject to an undisclosed or inadvertent isotopic fractionation. Substantial deviations in atomic weight of the element from that given in the table can occur. Range in isotopic composition of normal terrestrial material prevents a more precise atomic weight being given; the tabulated value should be applicable to any normal material.

1-14

ATOMIC MASSES AND ABUNDANCES This table lists the mass (in atomic mass units, symbol u) and the natural abundance (in percent) of the stable nuclides and a few important radioactive nuclides. A complete table of all nuclides may be found in Section 11 (“Table of the Isotopes”). The atomic masses are based on the 1995 evaluation of Audi and Wapstra (Reference 2). The number in parentheses following the mass value is the uncertainty in the last digit(s) given. Natural abundance values are also followed by uncertainties in the last digit(s) of the stated values. This uncertainty includes both the estimated measurement uncertainty and the reported range of variation in different terrestrial sources of the element (see Reference 3 and 4 for more details). The absence of an entry in the Abundance column indicates a radioactive nuclide not present in nature or an element whose isotopic composition varies so widely that a meaningful natural abundance cannot be defined. An electronic version of these data is available on the Web site of the NIST Physics Laboratory (Reference 5). REFERENCES 1. Holden, N. E., “Table of the Isotopes”, in Lide, D. R., Ed., CRC Handbook of Chemistry and Physics, 82nd Ed., CRC Press, Boca Raton FL, 2001. 2. Audi, G., and Wapstra, A. H., Nucl. Phys., A595, 409, 1995. 3. Rosman, K. J. R., and Taylor, P. D. P., J. Phys. Chem. Ref. Data, 27, 1275, 1998. 4. R. D. Vocke (for IUPAC Commission on Atomic Weights and Isotopic Abundances), Pure Appl. Chem., 71, 1593, 1999. 5. Coursey, J. S., and Dragoset, R. A., Atomic Weights and Isotopic Compositions (version 2.1). Available: http://physics.nist.gov/Compositions/ National Institute of Standards and Technology, Gaithersburg, MD. Z 1

Isotope 1H 2D 3T

2

3He 4He

3

6Li 7Li

4 5

9Be 10B 11B

6

12C 13C

7

14N 15N

8

16O 17O 18O

9 10

19F 20Ne 21Ne 22Ne

11 12

23Na 24Mg 25Mg 26Mg

13 14

27Al 28Si 29Si 30Si

15 16

31P 32S 33S 34S 36S

17

35Cl 37Cl

18

36Ar 38Ar

Mass in u 1.0078250321(4) 2.0141017780(4) 3.0160492675(11) 3.0160293097(9) 4.0026032497(10) 6.0151223(5) 7.0160040(5) 9.0121821(4) 10.0129370(4) 11.0093055(5) 12.0000000(0) 13.0033548378(10) 14.0030740052(9) 15.0001088984(9) 15.9949146221(15) 16.99913150(22) 17.9991604(9) 18.99840320(7) 19.9924401759(20) 20.99384674(4) 21.99138551(23) 22.98976967(23) 23.98504190(20) 24.98583702(20) 25.98259304(21) 26.98153844(14) 27.9769265327(20) 28.97649472(3) 29.97377022(5) 30.97376151(20) 31.97207069(12) 32.97145850(12) 33.96786683(11) 35.96708088(25) 34.96885271(4) 36.96590260(5) 35.96754628(27) 37.9627322(5)

Abundance in %

Z

Isotope 40Ar

99.9850(70) 0.0115(70)

19

39K 40K 41K

0.000137(3) 99.999863(3) 7.59(4) 92.41(4) 100 19.9(7) 80.1(7) 98.93(8) 1.07(8) 99.632(7) 0.368(7) 99.757(16) 0.038(1) 0.205(14) 100 90.48(3) 0.27(1) 9.25(3) 100 78.99(4) 10.00(1) 11.01(3) 100 92.2297(7) 4.6832(5) 3.0872(5) 100 94.93(31) 0.76(2) 4.29(28) 0.02(1) 75.78(4) 24.22(4) 0.3365(30) 0.0632(5)

20

40Ca 42Ca 43Ca 44Ca 46Ca 48Ca

21 22

45Sc 46Ti 47Ti 48Ti 49Ti 50Ti

23

50V 51V

24

50Cr 52Cr 53Cr 54Cr

25 26

55Mn 54Fe 56Fe 57Fe 58Fe

27 28

59Co 58Ni 60Ni 61Ni 62Ni 64Ni

29

63Cu 65Cu

30

64Zn 66Zn 67Zn

1-15

Mass in u 39.962383123(3) 38.9637069(3) 39.96399867(29) 40.96182597(28) 39.9625912(3) 41.9586183(4) 42.9587668(5) 43.9554811(9) 45.9536928(25) 47.952534(4) 44.9559102(12) 45.9526295(12) 46.9517638(10) 47.9479471(10) 48.9478708(10) 49.9447921(11) 49.9471628(14) 50.9439637(14) 49.9460496(14) 51.9405119(15) 52.9406538(15) 53.9388849(15) 54.9380496(14) 53.9396148(14) 55.9349421(15) 56.9353987(15) 57.9332805(15) 58.9332002(15) 57.9353479(15) 59.9307906(15) 60.9310604(15) 61.9283488(15) 63.9279696(16) 62.9296011(15) 64.9277937(19) 63.9291466(18) 65.9260368(16) 66.9271309(17)

Abundance in % 99.6003(30) 93.2581(44) 0.0117(1) 6.7302(44) 96.941(156) 0.647(23) 0.135(10) 2.086(110) 0.004(3) 0.187(21) 100 8.25(3) 7.44(2) 73.72(3) 5.41(2) 5.18(2) 0.250(4) 99.750(4) 4.345(13) 83.789(18) 9.501(17) 2.365(7) 100 5.845(35) 91.754(36) 2.119(10) 0.282(4) 100 68.0769(89) 26.2231(77) 1.1399(6) 3.6345(17) 0.9256(9) 69.17(3) 30.83(3) 48.63(60) 27.90(27) 4.10(13)

ATOMIC MASSES AND ABUNDANCES (continued) Z

Isotope 68Zn 70Zn

31

69Ga 71Ga

32

70Ge 72Ge 73Ge 74Ge 76Ge

33 34

75As 74Se 76Se 77Se 78Se 80Se 82Se

35

79Br 81Br

36

78Kr 80Kr 82Kr 83Kr 84Kr 86Kr

37

85Rb 87Rb

38

84Sr 86Sr 87Sr 88Sr

39 40

89Y 90Zr 91Zr 92Zr 94Zr 96Zr

41 42

93Nb 92Mo 94Mo 95Mo 96Mo 97Mo 98Mo 100Mo

43

97Tc 98Tc 99Tc

44

96Ru 98Ru 99Ru 100Ru 101Ru 102Ru 104Ru

45 46

103Rh 102Pd 104Pd 105Pd

Mass in u 67.9248476(17) 69.925325(4) 68.925581(3) 70.9247050(19) 69.9242504(19) 71.9220762(16) 72.9234594(16) 73.9211782(16) 75.9214027(16) 74.9215964(18) 73.9224766(16) 75.9192141(16) 76.9199146(16) 77.9173095(16) 79.9165218(20) 81.9167000(22) 78.9183376(20) 80.916291(3) 77.920386(7) 79.916378(4) 81.9134846(28) 82.914136(3) 83.911507(3) 85.9106103(12) 84.9117893(25) 86.9091835(27) 83.913425(4) 85.9092624(24) 86.9088793(24) 87.9056143(24) 88.9058479(25) 89.9047037(23) 90.9056450(23) 91.9050401(23) 93.9063158(25) 95.908276(3) 92.9063775(24) 91.906810(4) 93.9050876(20) 94.9058415(20) 95.9046789(20) 96.9060210(20) 97.9054078(20) 99.907477(6) 96.906365(5) 97.907216(4) 98.9062546(21) 95.907598(8) 97.905287(7) 98.9059393(21) 99.9042197(22) 100.9055822(22) 101.9043495(22) 103.905430(4) 102.905504(3) 101.905608(3) 103.904035(5) 104.905084(5)

Abundance in %

Z

Isotope 106Pd

18.75(51) 0.62(3) 60.108(9) 39.892(9) 20.84(87) 27.54(34) 7.73(5) 36.28(73) 7.61(38) 100 0.89(4) 9.37(29) 7.63(16) 23.77(28) 49.61(41) 8.73(22) 50.69(7) 49.31(7) 0.35(1) 2.28(6) 11.58(14) 11.49(6) 57.00(4) 17.30(22) 72.17(2) 27.83(2) 0.56(1) 9.86(1) 7.00(1) 82.58(1) 100 51.45(40) 11.22(5) 17.15(8) 17.38(28) 2.80(9) 100 14.84(35) 9.25(12) 15.92(13) 16.68(2) 9.55(8) 24.13(31) 9.63(23)

108Pd 110Pd

47

107Ag 109Ag

48

106Cd 108Cd 110Cd 111Cd 112Cd 113Cd 114Cd 116Cd

49

113In 115In

50

112Sn 114Sn 115Sn 116Sn 117Sn 118Sn 119Sn 120Sn 122Sn 124Sn

51

121Sb 123Sb

52

120Te 122Te 123Te 124Te 125Te 126Te 128Te 130Te

53 54

127I 124Xe 126Xe 128Xe 129Xe 130Xe 131Xe 132Xe 134Xe 136Xe

55 56

133Cs 130Ba 132Ba

5.54(14) 1.87(3) 12.76(14) 12.60(7) 17.06(2) 31.55(14) 18.62(27) 100 1.02(1) 11.14(8) 22.33(8)

134Ba 135Ba 136Ba 137Ba 138Ba

57

138La 139La

58

136Ce 138Ce 140Ce

1-16

Mass in u 105.903483(5) 107.903894(4) 109.905152(12) 106.905093(6) 108.904756(3) 105.906458(6) 107.904183(6) 109.903006(3) 110.904182(3) 111.9027572(30) 112.9044009(30) 113.9033581(30) 115.904755(3) 112.904061(4) 114.903878(5) 111.904821(5) 113.902782(3) 114.903346(3) 115.901744(3) 116.902954(3) 117.901606(3) 118.903309(3) 119.9021966(27) 121.9034401(29) 123.9052746(15) 120.9038180(24) 122.9042157(22) 119.904020(11) 121.9030471(20) 122.9042730(19) 123.9028195(16) 124.9044247(20) 125.9033055(20) 127.9044614(19) 129.9062228(21) 126.904468(4) 123.9058958(21) 125.904269(7) 127.9035304(15) 128.9047795(9) 129.9035079(10) 130.9050819(10) 131.9041545(12) 133.9053945(9) 135.907220(8) 132.905447(3) 129.906310(7) 131.905056(3) 133.904503(3) 134.905683(3) 135.904570(3) 136.905821(3) 137.905241(3) 137.907107(4) 138.906348(3) 135.907140(50) 137.905986(11) 139.905434(3)

Abundance in % 27.33(3) 26.46(9) 11.72(9) 51.839(8) 48.161(8) 1.25(6) 0.89(3) 12.49(18) 12.80(12) 24.13(21) 12.22(12) 28.73(42) 7.49(18) 4.29(5) 95.71(5) 0.97(1) 0.66(1) 0.34(1) 14.54(9) 7.68(7) 24.22(9) 8.59(4) 32.58(9) 4.63(3) 5.79(5) 57.21(5) 42.79(5) 0.09(1) 2.55(12) 0.89(3) 4.74(14) 7.07(15) 18.84(25) 31.74(8) 34.08(62) 100 0.09(1) 0.09(1) 1.92(3) 26.44(24) 4.08(2) 21.18(3) 26.89(6) 10.44(10) 8.87(16) 100 0.106(1) 0.101(1) 2.417(18) 6.592(12) 7.854(24) 11.232(24) 71.698(42) 0.090(1) 99.910(1) 0.185(2) 0.251(2) 88.450(51)


Isotope 142Ce

59 60

141Pr 142Nd 143Nd 144Nd 145Nd 146Nd 148Nd 150Nd

61

145Pm 147Pm

62

144Sm 147Sm 148Sm 149Sm 150Sm 152Sm 154Sm

63

151Eu 153Eu

64

152Gd 154Gd 155Gd 156Gd 157Gd 158Gd 160Gd

65 66

159Tb 156Dy 158Dy 160Dy 161Dy 162Dy 163Dy 164Dy

67 68

165Ho 162Er 164Er 166Er 167Er 168Er 170Er

69 70

169Tm 168Yb 170Yb 171Yb 172Yb 173Yb 174Yb 176Yb

71

175Lu 176Lu

72

174Hf 176Hf 177Hf 178Hf 179Hf 180Hf

Mass in u 141.909240(4) 140.907648(3) 141.907719(3) 142.909810(3) 143.910083(3) 144.912569(3) 145.913112(3) 147.916889(3) 149.920887(4) 144.912744(4) 146.915134(3) 143.911995(4) 146.914893(3) 147.914818(3) 148.917180(3) 149.917271(3) 151.919728(3) 153.922205(3) 150.919846(3) 152.921226(3) 151.919788(3) 153.920862(3) 154.922619(3) 155.922120(3) 156.923957(3) 157.924101(3) 159.927051(3) 158.925343(3) 155.924278(7) 157.924405(4) 159.925194(3) 160.926930(3) 161.926795(3) 162.928728(3) 163.929171(3) 164.930319(3) 161.928775(4) 163.929197(4) 165.930290(3) 166.932045(3) 167.932368(3) 169.935460(3) 168.934211(3) 167.933894(5) 169.934759(3) 170.936322(3) 171.9363777(30) 172.9382068(30) 173.9388581(30) 175.942568(3) 174.9407679(28) 175.9426824(28) 173.940040(3) 175.9414018(29) 176.9432200(27) 177.9436977(27) 178.9458151(27) 179.9465488(27)

Abundance in %

Z

11.114(51) 100 27.2(5) 12.2(2) 23.8(3) 8.3(1) 17.2(3) 5.7(1) 5.6(2)

73

Isotope 180Ta 181Ta

74

180W 182W 183W 184W 186W

75

185Re 187Re

76

184Os 186Os 187Os

3.07(7) 14.99(18) 11.24(10) 13.82(7) 7.38(1) 26.75(16) 22.75(29) 47.81(3) 52.19(3) 0.20(1) 2.18(3) 14.80(12) 20.47(9) 15.65(2) 24.84(7) 21.86(19) 100 0.06(1) 0.10(1) 2.34(8) 18.91(24) 25.51(26) 24.90(16) 28.18(37) 100 0.14(1) 1.61(3) 33.61(35) 22.93(17) 26.78(26) 14.93(27) 100 0.13(1) 3.04(15) 14.28(57) 21.83(67) 16.13(27) 31.83(92) 12.76(41) 97.41(2) 2.59(2) 0.16(1) 5.26(7) 18.60(9) 27.28(7) 13.62(2) 35.08(16)

188Os 189Os 190Os 192Os

77

191Ir 193Ir

78

190Pt 192Pt 194Pt 195Pt 196Pt 198Pt

79 80

197Au 196Hg 198Hg 199Hg 200Hg 201Hg 202Hg 204Hg

81

203Tl 205Tl

82

204Pb 206Pb 207Pb 208Pb

83 84

209Bi 209Po 210Po

85

210At 211At

86

211Rn 220Rn 222Rn

87 88

223Fr 223Ra 224Ra 226Ra 228Ra

89 90

227Ac 230Th 232Th

91 92

231Pa 233U 234U 235U

1-17

Mass in u 179.947466(3) 180.947996(3) 179.946706(5) 181.948206(3) 182.9502245(29) 183.9509326(29) 185.954362(3) 184.9529557(30) 186.9557508(30) 183.952491(3) 185.953838(3) 186.9557479(30) 187.9558360(30) 188.9581449(30) 189.958445(3) 191.961479(4) 190.960591(3) 192.962924(3) 189.959930(7) 191.961035(4) 193.962664(3) 194.964774(3) 195.964935(3) 197.967876(4) 196.966552(3) 195.965815(4) 197.966752(3) 198.968262(3) 199.968309(3) 200.970285(3) 201.970626(3) 203.973476(3) 202.972329(3) 204.974412(3) 203.973029(3) 205.974449(3) 206.975881(3) 207.976636(3) 208.980383(3) 208.982416(3) 209.982857(3) 209.987131(9) 210.987481(4) 210.990585(8) 220.0113841(29) 222.0175705(27) 223.0197307(29) 223.018497(3) 224.0202020(29) 226.0254026(27) 228.0310641(27) 227.0277470(29) 230.0331266(22) 232.0380504(22) 231.0358789(28) 233.039628(3) 234.0409456(21) 235.0439231(21)

Abundance in % 0.012(2) 99.988(2) 0.12(1) 26.50(16) 14.31(4) 30.64(2) 28.43(19) 37.40(2) 62.60(2) 0.02(1) 1.59(3) 1.96(2) 13.24(8) 16.15(5) 26.26(2) 40.78(19) 37.3(2) 62.7(2) 0.014(1) 0.782(7) 32.967(99) 33.832(10) 25.242(41) 7.163(55) 100 0.15(1) 9.97(20) 16.87(22) 23.10(19) 13.18(9) 29.86(26) 6.87(15) 29.524(14) 70.476(14) 1.4(1) 24.1(1) 22.1(1) 52.4(1) 100

100 100 0.0055(2) 0.7200(51)


Isotope

236U 238U

93

237Np 239Np

94

238Pu 239Pu 240Pu 241Pu 242Pu 244Pu

95

241Am 243Am

96

243Cm 244Cm 245Cm 246Cm 247Cm 248Cm

97

247Bk

Mass in u

236.0455619(21) 238.0507826(21) 237.0481673(21) 239.0529314(23) 238.0495534(21) 239.0521565(21) 240.0538075(21) 241.0568453(21) 242.0587368(21) 244.064198(5) 241.0568229(21) 243.0613727(23) 243.0613822(24) 244.0627463(21) 245.0654856(29) 246.0672176(24) 247.070347(5) 248.072342(5) 247.070299(6)

Abundance in %

Z

Isotope

249Bk

99.2745(106)

98

249Cf 250Cf 251Cf 252Cf

99 100 101

252Es 257Fm 256Md 258Md

102 103 104 105 106 107 108 109 110 111

259No 262Lr 261Rf 262Db 263Sg 264Bh 265Hs 268Mt 269Uun 272Uuu

*Mass values derived not purely from experimental data, but at least partly from systematic trends.

1-18

Mass in u

249.074980(3) 249.074847(3) 250.0764000(24) 251.079580(5) 252.081620(5) 252.082970(50) 257.095099(7) 256.094050(60) 258.098425(5) 259.101020(110)* 262.109690(320)* 261.108750(110)* 262.114150(200)* 263.118310(130)* 264.124730(300)* 265.130000(320)* 268.138820(340)* 269.145140(310)* 272.153480(360)*

Abundance in %

ELECTRON CONFIGURATION OF NEUTRAL ATOMS IN THE GROUND STATE Atomic no.

n= Element

K 1 s

L 2 s

p

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56

H He Li Be B C N O F Ne Na Mg Al Si P S Cl Ar K Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn Ga Ge As Se Br Kr Rb Sr Y Zr Nb Mo Tc Ru Rh Pd Ag Cd In Sn Sb Te I Xe Cs Ba

1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

1 2 3 4 5 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6

s

M 3 p d

s

N 4 p d

1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

1 2 3 4 5 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6

1 2 2 2 2 1 2 2 2 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

1 2 3 4 5 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6

1 2 3 5 5 6 7 8 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10

1-19

1 2 4 5 5 7 8 10 10 10 10 10 10 10 10 10 10 10

O 5 f

s

p

d

f

s

1 2 2 2 1 1 2 1 1 1 2 2 2 2 2 2 2 2 2

1 2 3 4 5 6 6 6

1 2

P 6 p d

Q 7 s p

ELECTRON CONFIGURATION OF NEUTRAL ATOMS IN THE GROUND STATE (continued) Atomic no. 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104

n= Element

K 1 s

L 2 s

p

s

M 3 p d

La Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb Lu Hf Ta W Re Os Ir Pt Au Hg Tl Pb Bi Po At Rn Fr Ra Ac Th Pa U Np Pu Am Cm Bk Cf Es Fm Md No Lr Rf

2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6

2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6

10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10

s 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

N 4 p d 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6

10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10

O 5 f

s

1 3 4 5 6 7 7 9 10 11 12 13 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14

2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

p 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6

d

f

s

2 3 4 6 7 7 9 10 11 12 13 14 14 14

2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

1 1

1

1 2 3 4 5 6 7 9 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10

P 6 p d

1 2 3 4 5 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6

1 2 1 1 1

1

2

Q 7 s

1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

p

1

REFERENCE Martin, W. C., Musgrove, A., and Kotochigova, S., Ground Levels and Ionization Energies for Neutral Atoms, Web Version 1.2.2, http:// physics.nist.gov/IonEnergy, National Institute of Standards and Technology, Gaithersburg, MD, December 2002.

1-20

INTERNATIONAL TEMPERATURE SCALE OF 1990 (ITS-90) B. W. Mangum A new temperature scale, the International Temperature Scale of 1990 (ITS-90), was officially adopted by the Comité International des Poids et Mesures (CIPM), meeting 26—28 September 1989 at the Bureau International des Poids et Mesures (BIPM). The ITS-90 was recommended to the CIPM for its adoption following the completion of the final details of the new scale by the Comité Consultatif de Thermométrie (CCT), meeting 12—14 September 1989 at the BIPM in its 17th Session. The ITS-90 became the official international temperature scale on 1 January 1990. The ITS90 supersedes the present scales, the International Practical Temperature Scale of 1968 (IPTS-68) and the 1976 Provisional 0.5 to 30 K Temperature Scale (EPT-76). The ITS-90 extends upward from 0.65 K, and temperatures on this scale are in much better agreement with thermodynamic values that are those on the IPTS-68 and the EPT-76. The new scale has subranges and alternative definitions in certain ranges that greatly facilitate its use. Furthermore, its continuity, precision, and reproducibility throughout its ranges are much improved over that of the present scales. The replacement of the thermocouple with the platinum resistance thermometer at temperatures below 961.78°C resulted in the biggest improvement in reproducibility. The ITS-90 is divided into four primary ranges: 1. Between 0.65 and 3.2 K, the ITS-90 is defined by the vapor pressure-temperature relation of 3He, and between 1.25 and 2.1768 K (the λ point) and between 2.1768 and 5.0 K by the vapor pressure-temperature relations of 4He. T90 is defined by the vapor pressure equations of the form: 9

[(

)

T90 / K = A0 + ∑ Ai ln( p/ Pa ) – B / C i =1

]

i

The values of the coefficients Ai, and of the constants Ao, B, and C of the equations are given below. 2. Between 3.0 and 24.5561 K, the ITS-90 is defined in terms of a 3He or 4He constant volume gas thermometer (CVGT). The thermometer is calibrated at three temperatures — at the triple point of neon (24.5561 K), at the triple point of equilibrium hydrogen (13.8033 K), and at a temperature between 3.0 and 5.0 K, the value of which is determined by using either 3He or 4He vapor pressure thermometry. 3. Between 13.8033 K (–259.3467°C) and 1234.93 K (961.78°C), the ITS-90 is defined in terms of the specified fixed points given below, by resistance ratios of platinum resistance thermometers obtained by calibration at specified sets of the fixed points, and by reference functions and deviation functions of resistance ratios which relate to T90 between the fixed points. 4. Above 1234.93 K, the ITS-90 is defined in terms of Planck’s radiation law, using the freezing-point temperature of either silver, gold, or copper as the reference temperature. Full details of the calibration procedures and reference functions for various subranges are given in: The International Temperature Scale of 1990, Metrologia, 27, 3, 1990; errata in Metrologia, 27, 107, 1990.

Defining Fixed Points of the ITS-90 Materiala

Equilibrium stateb

Temperature T90 (K)

He e-H2 e-H2 (or He) e-H2 (or He) Nec O2 Ar Hgc H2O Gac Inc Sn Zn Alc Ag Au

VP TP VP (or CVGT) VP (or CVGT) TP TP TP TP TP MP FP FP FP FP FP FP

Cuc

FP

3 to 5 13.8033 ≈17 ≈20.3 24.5561 54.3584 83.8058 234.3156 273.16 302.9146 429.7485 505.078 692.677 933.473 1234.93 1337.33 1357.77

1-15

t90 (°C)

–270.15 to –268.15 –259.3467 ≈ –256.15 ≈ –252.85 –248.5939 –218.7916 –189.3442 –38.8344 0.01 29.7646 156.5985 231.928 419.527 660.323 961.78 1064.18 1084.62

INTERNATIONAL TEMPERATURE SCALE OF 1990 (ITS-90) (continued) Defining Fixed Points of the ITS-90 (continued) a

b

c

e-H2 indicates equilibrium hydrogen, that is, hydrogen with the equilibrium distribution of its ortho and para states. Normal hydrogen at room temperature contains 25% para hydrogen and 75% ortho hydrogen. VP indicates vapor pressure point; CVGT indicates constant volume gas thermometer point; TP indicates triple point (equilibrium temperature at which the solid, liquid, and vapor phases coexist); FP indicates freezing point, and MP indicates melting point (the equilibrium temperatures at which the solid and liquid phases coexist under a pressure of 101 325 Pa, one standard atmosphere). The isotopic composition is that naturally occurring. Previously, these were secondary fixed points.

Values of Coefficients in the Vapor Pressure Equations for Helium Coef.or constant A0 A1 A2 A3 A4 A5 A6 A7 A8 A9 B C

3He 0.65—3.2 K

4He 1.25—2.1768 K

1.053 447 0.980 106 0.676 380 0.372 692 0.151 656 –0.002 263 0.006 596 0.088 966 –0.004 770 –0.054 943 7.3 4.3

1.392 408 0.527 153 0.166 756 0.050 988 0.026 514 0.001 975 –0.017 976 0.005 409 0.013 259 0 5.6 2.9

1-16

4He 2.1768—5.0 K

3.146 631 1.357 655 0.413 923 0.091 159 0.016 349 0.001 826 –0.004 325 –0.004 973 0 0 10.3 1.9

CONVERSION OF TEMPERATURES FROM THE 1948 AND 1968 SCALES TO ITS-90 This table gives temperature corrections from older scales to the current International Temperature Scale of 1990 (see the preceding table for details on ITS-90). The first part of the table may be used for converting Celsius temperatures in the range -180 to 4000°C from IPTS-68 or IPTS-48 to ITS90. Within the accuracy of the corrections, the temperature in the first column may be identified with either t68, t48, or t90. The second part of the table is designed for use at lower temperatures to convert values expressed in kelvins from EPT-76 or IPTS-68 to ITS-90. The references give analytical equations for expressing these relations. Note that Reference 1 supersedes Reference 2 with respect to corrections in the 630 to 1064°C range. REFERENCES 1. Burns, G. W. et al., in Temperature: Its Measurement and Control in Science and Industry, Vol. 6, Schooley, J. F., Ed., American Institute of Physics, New York, 1993. 2. Goldberg, R. N. and Weir, R. D., Pure and Appl. Chem., 1545, 1992. t/°C

t90-t68

t90-t48

t/°C

t90-t68

t90-t48

t/°C

t90-t68

t90-t48

-180 -170 -160 -150 -140 -130 -120 -110 -100 -90 -80 -70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 210 220 230 240 250 260

0.008 0.010 0.012 0.013 0.014 0.014 0.014 0.013 0.013 0.012 0.012 0.011 0.010 0.009 0.008 0.006 0.004 0.002 0.000 -0.002 -0.005 -0.007 -0.010 -0.013 -0.016 -0.018 -0.021 -0.024 -0.026 -0.028 -0.030 -0.032 -0.034 -0.036 -0.037 -0.038 -0.039 -0.039 -0.040 -0.040 -0.040 -0.040 -0.040 -0.040 -0.040

0.020 0.017 0.007 0.000 0.001 0.008 0.017 0.026 0.035 0.041 0.045 0.045 0.042 0.038 0.032 0.024 0.016 0.008 0.000 -0.006 -0.012 -0.016 -0.020 -0.023 -0.026 -0.026 -0.027 -0.027 -0.026 -0.024 -0.023 -0.020 -0.018 -0.016 -0.012 -0.009 -0.005 -0.001 0.003 0.007 0.011 0.014 0.018 0.021 0.024

270 280 290 300 310 320 330 340 350 360 370 380 390 400 410 420 430 440 450 460 470 480 490 500 510 520 530 540 550 560 570 580 590 600 610 620 630 640 650 660 670 680 690 700 710

-0.039 -0.039 -0.039 -0.039 -0.039 -0.039 -0.040 -0.040 -0.041 -0.042 -0.043 -0.045 -0.046 -0.048 -0.051 -0.053 -0.056 -0.059 -0.062 -0.065 -0.068 -0.072 -0.075 -0.079 -0.083 -0.087 -0.090 -0.094 -0.098 -0.101 -0.105 -0.108 -0.112 -0.115 -0.118 -0.122 -0.125 -0.11 -0.10 -0.09 -0.07 -0.05 -0.04 -0.02 -0.01

0.028 0.030 0.032 0.034 0.035 0.036 0.036 0.037 0.036 0.035 0.034 0.032 0.030 0.028 0.024 0.022 0.019 0.015 0.012 0.009 0.007 0.004 0.002 0.000 -0.001 -0.002 -0.001 0.000 0.002 0.007 0.011 0.018 0.025 0.035 0.047 0.060 0.075 0.12 0.15 0.19 0.24 0.29 0.32 0.37 0.41

720 730 740 750 760 770 780 790 800 810 820 830 840 850 860 870 880 890 900 910 920 930 940 950 960 970 980 990 1000 1010 1020 1030 1040 1050 1060 1070 1080 1090 1100 1200 1300 1400 1500 1600 1700

0.00 0.02 0.03 0.03 0.04 0.05 0.05 0.05 0.05 0.05 0.04 0.04 0.03 0.02 0.01 0.00 -0.02 -0.03 -0.05 -0.06 -0.08 -0.10 -0.11 -0.13 -0.15 -0.16 -0.18 -0.19 -0.20 -0.22 -0.23 -0.23 -0.24 -0.25 -0.25 -0.25 -0.26 -0.26 -0.26 -0.30 -0.35 -0.39 -0.44 -0.49 -0.54

0.45 0.49 0.53 0.56 0.60 0.63 0.66 0.69 0.72 0.75 0.76 0.79 0.81 0.83 0.85 0.87 0.87 0.89 0.90 0.92 0.93 0.94 0.96 0.97 0.97 0.99 1.00 1.02 1.04 1.05 1.07 1.10 1.12 1.14 1.17 1.19 1.20 1.20 1.2 1.4 1.5 1.6 1.8 1.9 2.1

1-17

CONVERSION OF TEMPERATURES FROM THE 1948 AND 1968 SCALES TO ITS-90 (continued) t/°C

t90-t68

t90-t48

T/K

1800 1900 2000 2100 2200 2300 2400 2500 2600 2700 2800 2900 3000 3100 3200 3300 3400 3500 3600 3700 3800 3900 4000

-0.60 -0.66 -0.72 -0.79 -0.85 -0.93 -1.00 -1.07 -1.15 -1.24 -1.32 -1.41 -1.50 -1.59 -1.69 -1.78 -1.89 -1.99 -2.10 -2.21 -2.32 -2.43 -2.55

2.2 2.3 2.5 2.7 2.9 3.1 3.2 3.4 3.7 3.8 4.0 4.2 4.4 4.6 4.8 5.1 5.3 5.5 5.8 6.0 6.3 6.6 6.8

T/K

T90-T76

T90-T68

-0.0001 -0.0002 -0.0003 -0.0004 -0.0005 -0.0006 -0.0007 -0.0008 -0.0010 -0.0011 -0.0013 -0.0014 -0.0016 -0.0018 -0.0020 -0.0022 -0.0025 -0.0027 -0.0030 -0.0032 -0.0035 -0.0038 -0.0041

-0.006 -0.003 -0.004 -0.006 -0.008 -0.009 -0.009 -0.008 -0.007 -0.007 -0.006 -0.005 -0.004 -0.004

28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76

5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27

T90-T76

1-18

T90-T68

T/K

-0.005 -0.006 -0.006 -0.007 -0.008 -0.008 -0.008 -0.007 -0.007 -0.007 -0.006 -0.006 -0.006 -0.006 -0.006 -0.006 -0.006 -0.007 -0.007 -0.007 -0.006 -0.006 -0.006 -0.005 -0.005 -0.004 -0.003 -0.002 -0.001 0.000 0.001 0.002 0.003 0.003 0.004 0.004 0.005 0.005 0.006 0.006 0.007 0.007 0.007 0.007 0.007 0.007 0.007 0.008 0.008

77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 110 120 130 140 150 160 170 180 190 200 210 220 230 240 250 260 270 273.16 300 400 500 600 700 800 900

T90-T76

T90-T68 0.008 0.008 0.008 0.008 0.008 0.008 0.008 0.008 0.008 0.008 0.008 0.008 0.008 0.008 0.008 0.008 0.008 0.008 0.008 0.008 0.009 0.009 0.009 0.009 0.011 0.013 0.014 0.014 0.014 0.014 0.013 0.012 0.012 0.011 0.010 0.009 0.008 0.007 0.005 0.003 0.001 0.000 -0.006 -0.031 -0.040 -0.040 -0.055 -0.089 -0.124

INTERNATIONAL SYSTEM OF UNITS (SI)

1 SI base units Table 1 gives the seven base quantities, assumed to be mutually independent, on which the SI is founded; and the names and symbols of their respective units, called ``SI base units.' ' Definitions of the SI base units are given in Appendix A. The kelvin and its symbol K are also used to express the value of a temperature interval or a temperature difference.

Table 1. SI base units SI base unit Base quantity length mass time electric current thermodynamic temperature amount of substance luminous intensity

Name

Symbol

meter kilogram second ampere kelvin mole candela

m kg s A K mol cd

2 SI deriived units Derived units are expressed algebraically in terms of base units or other derived units (including the radian and steradian which are the two supplementary units – see Sec. 3). The symbols for derived units are obtained by means of the mathematical operations of multiplication and division. For example, the derived unit for the derived quantity molar mass (mass divided by amount of substance) is the kilogram per mole, symbol kg/mol. Additional examples of derived units expressed in terms of SI base units are given in Table 2.

Table 2. Examples of SI derived units expressed in terms of SI base units SI derived unit Derived quantity area volume speed, velocity acceleration wave number mass density (density) specific volume current density magnetic field strength amount-of-substance concentration (concentration) luminance

2.1

Name

Symbol

square meter cubic meter meter per second meter per second squared reciprocal meter kilogram per cubic meter cubic meter per kilogram ampere per square meter ampere per meter

m2 m3 m/s m/s 2 m21 kg/m 3 m 3 /kg A/m 2 A/m

mole per cubic meter candela per square meter

mol/m 3 cd/m 2

SI derived units with special names and symbols

Certain SI derived units have special names and symbols; these are given in Tables 3a and 3b. As discussed in Sec. 3, the radian and steradian, which are the two supplementary units, are included in Table 3a.

INTERNATIONAL SYSTEM OF UNITS (SI) (continued) Table 3a. SI derived units with special names and symbols, including the radian and steradian SI derived unit Derived quantity

Special name

plane angle solid angle frequency force pressure, stress energy, work, quantity of heat power, radiant flux electric charge, quantity of electricity electric potential, potential difference, electromotive force capacitance electric resistance electric conductance magnetic flux magnetic flux density inductance Celsius temperature(a) luminous flux illuminance (a) (b)

Special symbol

Expression in terms of other SI units

Expression in terms of SI base units

radian steradian hertz newton pascal

rad sr Hz N Pa

N/m2

m ? m21 = 1 m 2 ? m22 = 1 s21 m ? kg ? s22 m21 ? kg ? s22

joule watt

J W

N?m J/s

m2 ? kg ? s22 m2 ? kg ? s23

coulomb

C

volt farad ohm siemens weber tesla henry degree Celsius lumen lux

V F V S Wb T H 8C lm lx

s?A W/A C/V V/A A/V V?s Wb/m2 Wb/A cd ? sr lm/m2

m2 ? kg ? s23 ? A21 m22 ? kg21 ? s4 ? A2 m2 ? kg ? s23 ? A22 m22 ? kg21 ? s3 ? A2 m2 ? kg ? s22 ? A21 kg ? s22 ? A21 m2 ? kg ? s22 ? A22 K cd ? sr (b) m22 ? cd ? sr (b)

See Sec. 2.1.1. The steradian (sr) is not an SI base unit. However, in photometry the steradian (sr) is maintained in expressions for units (see Sec. 3).

Table 3b. SI derived units with special names and symbols admitted for reasons of safeguarding human health (a) SI derived unit Derived quantity

Special name

Special symbol

Expression in terms of other SI units


activity (of a radionuclide)

becquerel

Bq

absorbed dose, specific energy (imparted), kerma

gray

Gy

J/kg

m2 ? s22

dose equivalent, ambient dose equivalent, directional dose equivalent, personal dose equivalent, equivalent dose

sievert

Sv

J/kg

m2 ? s22

(a)

s21

The derived quantities to be expressed in the gray and the sievert have been revised in accordance with the recommendations of the International Commission on Radiation Units and Measurements (ICRU).

2.1.1 Degree Celsius In addition to the quantity thermodynamic temperature (symbol T ), expressed in the unit kelvin, use is also made of the quantity Celsius temperature (symbol t ) defined by the equation t = T2T 0 , where T 0 = 273.15 K by definition. To express Celsius temperature, the unit degree Celsius, symbol 8C, which is equal in magnitude to the unit kelvin, is used; in this case, ``degree Celsius' ' is a special name used in place of ``kelvin.' ' An interval or difference of Celsius temperature can, however, be expressed in the unit kelvin as well as in the unit degree Celsius. (Note that the thermodynamic temperature T0 is exactly 0.01 K below the thermodynamic temperature of the triple point of water.)

INTERNATIONAL SYSTEM OF UNITS (SI) (continued)

2.2

Use of SI derived units with special names and symbols

Examples of SI derived units that can be expressed with the aid of SI derived units having special names and symbols (including the radian and steradian) are given in Table 4. Table 4. Examples of SI derived units expressed with the aid of SI derived units having special names and symbols SI derived unit Derived quantity angular velocity angular acceleration dynamic viscosity moment of force surface tension heat flux density, irradiance radiant intensity radiance heat capacity, entropy specific heat capacity, specific entropy specific energy thermal conductivity energy density electric field strength electric charge density electric flux density permittivity permeability molar energy molar entropy, molar heat capacity exposure (x and g rays) absorbed dose rate (a)

Name

Symbol


radian per second radian per second squared pascal second newton meter newton per meter

rad/s rad/s 2 Pa ? s N?m N/m

m ? m 21 ? s 21 = s 21 m ? m 21 ? s 22 = s 22 m21 ? kg ? s21 m 2 ? kg ? s22 kg ? s22

watt per square meter watt per steradian watt per square meter steradian joule per kelvin joule per kilogram kelvin joule per kilogram watt per meter kelvin joule per cubic meter volt per meter coulomb per cubic meter coulomb per square meter farad per meter henry per meter joule per mole

W/m 2 W/sr

kg ? s23 m 2 ? kg ? s 23 ? sr 21 (a)

W/(m 2 ? sr) J/K

kg ? s 23 ? sr 21 (a) m 2 ? kg ? s22 ? K21

J/(kg ? K) J/kg W/(m ? K) J/m 3 V/m C/m 3 C/m 2 F/m H/m J/mol

m 2 ? s22 ? K21 m 2 ? s22 m ? kg ? s23 ? K21 m21 ? kg ? s22 m ? kg ? s23 ? A21 m23 ? s ? A m22 ? s ? A m23 ? kg21 ? s4 ? A2 m ? kg ? s22 ? A22 m 2 ? kg ? s22 ? mol21

joule per mole kelvin coulomb per kilogram gray per second

J/(mol ? K) C/kg Gy/s

m 2 ? kg ? s22 ? K21 ? mol21 kg21 ? s ? A m 2 ? s23

The steradian (sr) is not an SI base unit. However, in radiometry the steradian (sr) is maintained in expressions for units (see Sec. 3).

The advantages of using the special names and symbols of SI derived units are apparent in Table 4. Consider, for example, the quantity molar entropy: the unit J/(mol ? K) is obviously more easily understood than its SI base-unit equivalent, m 2 ? kg ? s 22 ? K 21 ? mol 21. Nevertheless, it should always be recognized that the special names and symbols exist for convenience; either the form in which special names or symbols are used for certain combinations of units or the form in which they are not used is correct. For example, because of the descriptive value implicit in the compound-unit form, communication is sometimes facilitated if magnetic flux (see Table 3a) is expressed in terms of the volt second (V ? s) instead of the weber (Wb). Tables 3a, 3b, and 4 also show that the values of several different quantities are expressed in the same SI unit. For example, the joule per kelvin (J/K) is the SI unit for heat capacity as well as for entropy. Thus the name of the unit is not sufficient to define the quantity measured. A derived unit can often be expressed in several different ways through the use of base units and derived units with special names. In practice, with certain quantities, preference is given to using certain units with special names, or combinations of units, to facilitate the distinction between quantities whose values have identical expressions in terms of SI base units. For example, the SI unit of frequency is specified as the hertz (Hz) rather than the reciprocal second (s 21 ), and the SI unit of moment of force is specified as the newton meter (N ? m) rather than the joule (J).


Similarly, in the field of ionizing radiation, the SI unit of activity is designated as the becquerel (Bq) rather than the reciprocal second (s 21 ), and the SI units of absorbed dose and dose equivalent are designated as the gray (Gy) and the sievert (Sv), respectively, rather than the joule per kilogram (J/kg). 3 SI supplementary units As previously stated, there are two units in this class: the radian, symbol rad, the SI unit of the quantity plane angle; and the steradian, symbol sr, the SI unit of the quantity solid angle. Definitions of these units are given in Appendix A. The SI supplementary units are now interpreted as so-called dimensionless derived units for which the CGPM allows the freedom of using or not using them in expressions for SI derived units.3 Thus the radian and steradian are not given in a separate table but have been included in Table 3a together with other derived units with special names and symbols (seeSec.2.1). This interpretation of the supplementary units implies that plane angle and solid angle are considered derived quantities of dimension one (so-called dimensionless quantities), each of which has the which has the unit one, symbol 1, as its coherent SI unit. However, in practice, when one expresses the values of derived quantities involving plane angle or solid angle, it often aids understanding if the special names (or symbols) ``radian' ' (rad) or ``steradian' ' (sr) are used in place of the number 1. For example, although values of the derived quantity angular velocity (plane angle divided by time) may be expressed in the unit s21, such values are usually expressed in the unit rad/s. Because the radian and steradian are now viewed as so-called dimensionless derived units, the Consultative Committee for Units (CCU, Comité Consultatif des Unités) of the CIPM as result of a 1993 request it received from ISO/TC12, recommended to the CIPM that it request the CGPM to abolish the class of supplementary units as a separate class in the SI. The CIPM accepted the CCU recommendation, and if the abolishment is approved by the CGPM as is likely (the question will be on the agenda of the 20th CGPM, October 1995), the SI will consist of only two classes of units: base units and derived units, with the radian and steradian subsumed into the class of derived units of the SI. (The option of using or not using them in expressions for SI derived units, as is convenient, would remain unchanged.) 4 Decimal multiples and submultiples of SI units: SI prefixes Table 5 gives the SI prefixes that are used to form decimal multiples and submultiples of SI units. They allow very large or very small numerical values to be avoided. A prefix attaches directly to the name of a unit, and a prefix symbol attaches directly to the symbol for a unit. For example, one kilometer, symbol 1 km, is equal to one thousand meters, symbol 1000 m or 103 m. When prefixes are attached to SI units, the units so formed are called ``multiples and submultiples of SI units' ' in order to distinguish them from the coherent system of SI units.

Note:

3

Alternative definitions of the SI prefixes and their symbols are not permitted. For example, it is unacceptable to use kilo (k) to represent 2 10 = 1024, mega (M) to represent 2 20 = 1 048 576, or giga (G) to represent 2 30 = 1 073 741 824.

This interpretation was given in 1980 by the CIPM . It was deemed necessary because Resolution 12 of the 11th CGPM, which established the SI in 1960 , did not specify the nature of the supplementary units. The interpretation is based on two principal considerations: that plane angle is generally expressed as the ratio of two lengths and solid angle as the ratio of an area and the square of a length, and are thus quantities of dimension one (so-called dimensionless quantities); and that treating the radian and steradian as SI base units – a possibility not disallowed by Resolution 12 – could compromise the internal coherence of the SI based on only seven base units. (See ISO 31-0 for a discussion of the concept of dimension.)


Table 5. SI prefixes Factor

Prefix

10 24 10 21 10 18 10 15 10 12 10 9 10 6 10 3 10 2 10 1

= = = = = = = =

(10 3 ) 8 (10 3 ) 7 (10 3 ) 6 (10 3 ) 5 (10 3 ) 4 (10 3 ) 3 (10 3 ) 2 (10 3 ) 1

5

Units Outside the SI

Symbol

yotta zetta exa peta tera giga mega kilo hecto deka

Y Z E P T G M k h da

Factor 1021 1022 1023 1026 1029 10212 10215 10218 10221 10224

= = = = = = = =

Prefix

Symbol

deci centi milli micro nano pico femto atto zepto yocto

(10 3 ) 21 (10 3 ) 22 (10 3 ) 23 (10 3 ) 24 (10 3 ) 25 (10 3 ) 26 (10 3 ) 27 (10 3 ) 28

d c m m n p f a z y

Units that are outside the SI may be divided into three categories: –

those units that are accepted for use with the SI;

– those units that are temporarily accepted for use with the SI; and – those units that are not accepted for use with the SI and thus must strictly be avoided. 5.1

Units accepted for use with the SI The following sections discuss in detail the units that are acceptable for use with the SI.

5.1.1

Hour, degree, liter, and the like

Certain units that are not part of the SI are essential and used so widely that they are accepted by the CIPM for use with the SI. These units are given in Table 6. The combination of units of this table with SI units to form derived units should be restricted to special cases in order not to lose the advantages of the coherence of SI units. Additionally, it is recognized that it may be necessary on occasion to use time-related units other than those given in Table 6; in particular, circumstances may require that intervals of time be expressed in weeks, months, or years. In such cases, if a standardized symbol for the unit is not available, the name of the unit should be written out in full.

Table 6. Units accepted for use with the SI Name

Symbol

6 6

minute hour time day degree minute plane angle second liter metric ton (c)

min h d 8 ' " l, L (b) t

Value in SI units 1 min 1h 1d 18 1' 1" 1L 1t

= = = = = = = =

60 s 60 min = 3600 s 24 h = 86 400 s (p/180) rad (1/60)8=(p/10 800) rad (1/60)' =(p/648 000) rad 1 dm3 = 1023 m3 10 3 kg

(b)

The alternative symbol for the liter, L, was adopted by the CGPM in order to avoid the risk of confusion between the letter l and the number 1 . Thus, although both l and L are internationally accepted symbols for the liter, to avoid this risk the symbol to be used in the United States is L . The script letter , is not an approved symbol for the liter.

(c)

This is the name to be used for this unit in the United States; it is also used in some other English-speaking countries. However, ``tonne' ' is used in many countries.


5.1.2

Neper, bel, shannon, and the like

There are a few highly specialized units not listed in Table 6 that are given by the International Organization for Standardization (ISO) or the International Electrotechnical Commission (IEC) and which are also acceptable for use with the SI. They include the neper (Np), bel (B), octave, phon, and sone, and units used in information technology, including the baud (Bd), bit (bit), erlang (E), hartley (Hart), and shannon (Sh).4 It is the position of NIST that the only such additional units that may be used with the SI are those given in either the International Standards on quantities and units of ISO or of IEC . 5.1.3

Electronvolt and unified atomic mass unit

The CIPM also finds it necessary to accept for use with the SI the two units given in Table 7. These units are used in specialized fields; their values in SI units must be obtained from experiment and, therefore, are not known exactly.

Note :

In some fields the unified atomic mass unit is called the dalton, symbol Da; however, this name and symbol are not accepted by the CGPM, CIPM, ISO, or IEC for use with the SI. Similarly, AMU is not an acceptable unit symbol for the unified atomic mass unit. The only allowed name is `ùnified atomic mass unit' ' and the only allowed symbol is u.

Table 7. Units accepted for use with the SI whose values in SI units are obtained experimentally Name electronvolt unified atomic mass unit (a)

Symbol eV u

Definition (a) (b)

The electronvolt is the kinetic energy acquired by an electron in passing through a potential difference of 1 V in vacuum; 1 eV = 1.602 177 33310219 J with a combined standard uncertainty of 0.000 000 49310219 J . The unified atomic mass unit is equal to 1/12 of the mass of an atom of the nuclide 12 C; 1 u = 1.660 540 23 10227 kg with a combined standard uncertainty of 0.000 001 0310227 kg .

(b)

5.1.4

Natural and atomic units

In some cases, particularly in basic science, the values of quantities are expressed in terms of fundamental constants of nature or so-called natural units.The use of these units with the SI is permissible when it is necessary for the most effective communication of information. In such cases, the specific natural units that are used must be identified. This requirement applies even to the system of units customarily called `àtomicunits' ' used in theoretical atomic physics and chemistry, inasmuch as there are several different systems that have the appellation `àtomic units.'' Examples of physical quantities used as natural units are given in Table 8. NIST also takes the position that while theoretical results intended primarily for other theorists may be left in natural units, if they are also intended for experimentalists, they must also be given in acceptable units.

4

The symbol in parentheses following the name of the unit is its internationally accepted unit symbol, but the octave, phon, and sone have no such unit symbols. For additional information on the neper and bel, see Sec. 0.5 of ISO 31-2. The question of the byte (B) is under international consideration.

INTERNATIONAL SYSTEM OF UNITS (SI) (continued) Table 8. Examples of physical quantities sometimes used as natural units Kind of quantity

Physical quantity used as a unit

action electric charge energy length length magnetic flux magnetic moment magnetic moment mass mass speed

Planck constant divided by 2p elementary charge Hartree energy Bohr radius Compton wavelength (electron) magnetic flux quantum Bohr magneton nuclear magneton electron rest mass proton rest mass speed of electromagnetic waves in vacuum

5.2

Symbol

h e Eh a0 lC F0 mB mN me mp c

Units temporarily accepted for use with the SI

Because of existing practice in certain fields or countries, in 1978 the CIPM considered that it was permissible for the units given in Table 9 to continue to be used with the SI until the CIPM considers that their use is no longer necessary. However, these units must not be introduced where they are not presently used. Further, NIST strongly discourages the continued use of these units except for the nautical mile, knot, are, and hectare; and except for the curie, roentgen, rad, and rem until the year 2000 (the cessation date suggested by the Committee for Ineragency Radiation Research and Policy Coordination or CIRRPC, a United States Government interagency group). 5

Table 9. Units temporarily accepted for use with the SI (a) Name nautical mile knot ångström are(b) hectare(b) barn bar gal curie roentgen rad rem (a) (b) (c)

Symbol

Å a ha b bar Gal Ci R rad (c) rem

Value in SI units 1 nautical mile = 1852 m 1 nautical mile per hour = (1852/3600) m/s 1 Å = 0.1 nm = 10210 m 1 a = 1 dam2 = 10 2 m2 1 ha = 1 hm2 = 10 4 m2 1 b = 100 fm2 = 10228 m2 1 bar=0.1 MPa=100 kPa=1000 hPa=10 5 Pa 1 Gal = 1 cm/s 2 = 1022 m/s 2 1 Ci = 3.731010 Bq 1 R = 2.5831024 C/kg 1 rad = 1 cGy = 1022 Gy 1 rem = 1 cSv = 1022 Sv

See Sec. 5.2 regarding the continued use of these units. This unit and its symbol are used to express agrarian areas. When there is risk of confusion with the symbol for the radian, rd may be used as the symbol for rad.

5 In 1993 the CCU (see Sec. 3) was requested by ISO/TC 12 to consider asking the CIPM to deprecate the use of the units of Table 9 except for the nautical mile and knot, and possibly the are and hectare. The CCU discussed this request at its February 1995 meeting.


Appendix A. A.1

Definitions of the SI Base Units and the Radian and Steradian

Introduction

The following definitions of the SI base units are taken from NIST SP 330; the definitions of the SI supplementary units, the radian and steradian, which are now interpreted as SI derived units (see Sec. 3), are those generally accepted and are the same as those given in ANSI/IEEE Std 268-1992. SI derived units are uniquely defined only in terms of SI base units; for example, 1 V = 1 m 2 ? kg ? s 23 ? A 21. A.2

Meter (17th CGPM, 1983)

The meter is the length of the path travelled by light in vacuum during a time interval of 1/299 792 458 of a second. A.3

Kilogram (3d CGPM, 1901)

The kilogram is the unit of mass; it is equal to the mass of the international prototype of the kilogram. A.4

Second (13th CGPM, 1967)

The second is the duration of 9 192 631 770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the cesium- 133 atom. A.5

Ampere (9th CGPM, 1948)

The ampere is that constant current which, if maintained in two straight parallel conductors of infinite length, of negligible circular cross section, and placed 1 meter apart in vacuum, would produce between these conductors a force equal to 2 3 10 27 newton per meter of length. A.6

Kelvin (13th CGPM, 1967)

The kelvin, unit of thermodynamic temperature, is the fraction 1/273.16 of the thermodynamic temperature of the triple point of water. A.7

Mole (14th CGPM, 1971)

1. The mole is the amount of substance of a system which contains as many elementary entities as there are atoms in 0.012 kilogram of carbon 12. 2. When the mole is used, the elementary entities must be specified and may be atoms, molecules, ions, electrons, other particles, or specified groups of such particles. In the definition of the mole, it is understood that unbound atoms of carbon 12, at rest and in their ground state, are referred to. Note that this definition specifies at the same time the nature of the quantity whose unit is the mole. A.8

Candela (16th CGPM, 1979)

The candela is the luminous intensity, in a given direction, of a source that emits monochromatic radiation of frequency 540 3 10 12 hertz and that has a radiant intensity in that direction of ( 1/ 683) watt per steradian. A.9

Radian

The radian is the plane angle between two radii of a circle that cut off on the circumference an arc equal in length to the radius. A.10

Steradian

The steradian is the solid angle that, having its vertex in the center of a sphere, cuts off an area of the surface of the sphere equal to that of a square with sides of length equal to the radius of the sphere.

CONVERSION FACTORS The following table gives conversion factors from various units of measure to SI units. It is reproduced from NIST Special Publication 811, Guide for the Use of the International System of Units (SI). The table gives the factor by which a quantity expressed in a non-SI unit should be multiplied in order to calculate its value in the SI. The SI values are expressed in terms of the base, supplementary, and derived units of SI in order to provide a coherent presentation of the conversion factors and facilitate computations (see the table “International System of Units” in this Section). If desired, powers of ten can be avoided by using SI Prefixes and shifting the decimal point if necessary. Conversion from a non-SI unit to a different non-SI unit may be carried out by using this table in two stages, e.g., 1 calth = 4.184 J 1 BtuIT = 1.055056 E+03 J Thus, 1 BtuIT = (1.055056 E+03 ÷ 4.184) calth = 252.164 calth Conversion factors are presented for ready adaptation to computer readout and electronic data transmission. The factors are written as a number equal to or greater than one and less than ten with six or fewer decimal places. This number is followed by the letter E (for exponent), a plus or a minus sign, and two digits which indicate the power of 10 by which the number must be multiplied to obtain the correct value. For example: 3.523 907 E-02 is 3.523 907 × 10-2 or 0.035 239 07 Similarly: 3.386 389 E+03 is 3.386 389 × 103 or 3 386.389 A factor in boldface is exact; i.e., all subsequent digits are zero. All other conversion factors have been rounded to the figures given in accordance with accepted practice. Where less than six digits after the decimal point are shown, more precision is not warranted. It is often desirable to round a number obtained from a conversion of units in order to retain information on the precision of the value. The following rounding rules may be followed: (1) If the digits to be discarded begin with a digit less than 5, the digit preceding the first discarded digit is not changed. Example: 6.974 951 5 rounded to 3 digits is 6.97 (2) If the digits to be discarded begin with a digit greater than 5, the digit preceding the first discarded digit is increased by one. Example: 6.974 951 5 rounded to 4 digits is 6.975 (3) If the digits to be discarded begin with a 5 and at least one of the following digits is greater than 0, the digit preceding the 5 is increased by 1. Example: 6.974 851 rounded to 5 digits is 6.974 9 (4) If the digits to be discarded begin with a 5 and all of the following digits are 0, the digit preceding the 5 is unchanged if it is even and increased by one if it is odd. (Note that this means that the final digit is always even.) Examples: 6.974 951 5 rounded to 7 digits is 6.974 952 6.974 950 5 rounded to 7 digits is 6.974 950

REFERENCE Taylor, B. N., Guide for the Use of the International System of Units (SI), NIST Special Publication 811, 1995 Edition, Superintendent of Documents, U.S. Government Printing Office, Washington, DC 20402, 1995.

© 2000 by CRC PRESS LLC

Factors in boldface are exact To convert from

to

Multiply by

abampere. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ampere (A) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.0 abcoulomb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . coulomb (C) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.0

E+01 E+01

abfarad . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . farad (F) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.0 abhenry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . henry (H) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.0 abmho. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . siemens (S) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.0

E+09 Eⴚ09 E+09

abohm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ohm (⍀) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.0 abvolt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . volt (V) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.0 acceleration of free fall, standard (g n ). . . . . . . . . . . . . . . meter per second squared (m/s 2 ) . . . . . . . . . . . . . . 9.806 65 acre (based on U.S. survey foot)9 . . . . . . . . . . . . . . . . . . . square meter (m2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.046 873

Eⴚ09 Eⴚ08

acre foot (based on U.S. survey foot)9 . . . . . . . . . . . . . . cubic meter (m3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.233 489 ampere hour (A ⭈ h) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . coulomb (C) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6

E+03 E+03

ångstro¨ m (Å) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . meter (m) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ångstro¨ m (Å) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . nanometer (nm). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . are (a) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . square meter (m2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . astronomical unit (AU). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . meter (m) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . atmosphere, standard (atm). . . . . . . . . . . . . . . . . . . . . . . . . . pascal (Pa) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . atmosphere, standard (atm). . . . . . . . . . . . . . . . . . . . . . . . . . kilopascal (kPa). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . atmosphere, technical (at) 10 . . . . . . . . . . . . . . . . . . . . . . . . . pascal (Pa) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Eⴚ10 Eⴚ01 E+02 E+11 E+05 E+02

1.0 1.0 1.0 1.495 979 1.013 25 1.013 25

9.806 65 atmosphere, technical (at) 10 . . . . . . . . . . . . . . . . . . . . . . . . . kilopascal (kPa). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.806 65

bar (bar). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . pascal (Pa) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . bar (bar). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . kilopascal (kPa). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . barn (b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . square meter (m2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . barrel [for petroleum, 42 gallons (U.S.)](bbl) . . . . . . . cubic meter (m3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . barrel [for petroleum, 42 gallons (U.S.)](bbl) . . . . . . . liter (L) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . biot (Bi) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ampere (A) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . British thermal unit IT (BtuIT )11 . . . . . . . . . . . . . . . . . . . . . . joule (J). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . British thermal unit th (Btuth )11 . . . . . . . . . . . . . . . . . . . . . . joule (J). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . British thermal unit (mean) (Btu) . . . . . . . . . . . . . . . . . . . joule (J). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . British thermal unit (39 ⬚F) (Btu) . . . . . . . . . . . . . . . . . . . joule (J). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . British thermal unit (59 ⬚F) (Btu) . . . . . . . . . . . . . . . . . . . joule (J). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . British thermal unit (60 ⬚F) (Btu) . . . . . . . . . . . . . . . . . . . joule (J). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . British thermal unitIT foot per hour square foot degree Fahrenheit [BtuIT ⭈ ft/(h ⭈ ft2 ⭈ ⬚F)] . . . . . . . . . . . . . . . . . . . . . . . . . . . watt per meter kelvin [W/(m ⭈ K)]. . . . . . . . . . . . . British thermal unitth foot per hour square foot degree Fahrenheit [Btuth ⭈ ft/(h ⭈ ft2 ⭈ ⬚F)]. . . . . . . . . . . . . . . . . . . . . . . . . . . . watt per meter kelvin [W/(m ⭈ K)]. . . . . . . . . . . . .

E+03

E+04 E+01

1.0 1.0 1.0 1.589 873 1.589 873 1.0 1.055 056

E+05 E+02 Eⴚ28 E⫺01 E+02 E+01 E+03

1.054 350 1.055 87 1.059 67 1.054 80 1.054 68

E+03 E+03 E+03 E+03 E+03

1.730 735

E+00

1.729 577

E+00

British thermal unitIT inch per hour square foot degree Fahrenheit [BtuIT ⭈ in/(h ⭈ ft2 ⭈ ⬚F)] . . . . . . . . . . . . . . . . . . . . . . . . . . . watt per meter kelvin [W/(m ⭈ K)]. . . . . . . . . . . . . 1.442 279 British thermal unitth inch per hour square foot degree Fahrenheit [Btuth ⭈ in/(h ⭈ ft2 ⭈ ⬚F)] . . . . . . . . . . . . . . . . . . . . . . . . . . . watt per meter kelvin [W/(m ⭈ K)]. . . . . . . . . . . . . 1.441 314 British thermal unitIT inch per second square foot degree Fahrenheit [BtuIT ⭈ in/(s ⭈ ft2 ⭈ ⬚F)] . . . . . . . . . . . . . . . . . . . . . . . . . . . watt per meter kelvin [W/(m ⭈ K)]. . . . . . . . . . . . . 5.192 204

9

E+00

E⫺01 E⫺01 E+02

The U.S. survey foot equals (1200/3937) m. 1 international foot = 0.999998 survey foot. One technical atmosphere equals one kilogram-force per square centimeter (1 at = 1 kgf/cm 2 ). 11 The Fifth International Conference on the Properties of Steam (London, July 1956) defined the International Table calorie as 4.1868 J. Therefore the exact conversion factor for the International Table Btu is 1.055 055 852 62 kJ. Note that the notation for International Table used in this listing is subscript ‘‘IT’’. Similarily, the notation for thermochemical is subscript ‘‘th.’’ Further, the thermochemical Btu, Btu th , is based on the thermochemical calorie, cal th , where cal th = 4.184 J exactly. 10


To convert from

to

British thermal unitth inch per second square foot degree Fahrenheit [Btuth ⭈ in/(s ⭈ ft2 ⭈ ⬚F)] . . . . . . . . . . . . . . . . . . . . . . . . . . . watt per meter kelvin [W/(m ⭈ K)]. . . . . . . . . . . . . British thermal unitIT per cubic foot (BtuIT /ft3). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . joule per cubic meter (J/m3) . . . . . . . . . . . . . . . . . . . British thermal unitth per cubic foot (Btuth /ft3). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . joule per cubic meter (J/m3) . . . . . . . . . . . . . . . . . . . British thermal unitIT per degree Fahrenheit (BtuIT / ⬚F) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . joule per kelvin (J/ k) . . . . . . . . . . . . . . . . . . . . . . . . . . British thermal unitth per degree Fahrenheit (Btuth /⬚F) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . joule per kelvin (J/ k) . . . . . . . . . . . . . . . . . . . . . . . . . . British thermal unitIT per degree Rankine (BtuIT /⬚R) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . joule per kelvin (J/ k) . . . . . . . . . . . . . . . . . . . . . . . . . . British thermal unitth per degree Rankine (Btuth /⬚R) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . joule per kelvin (J/ k) . . . . . . . . . . . . . . . . . . . . . . . . . . British thermal unitIT per hour (BtuIT /h) . . . . . . . . . . . . watt (W) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . British thermal unitth per hour (Btuth /h). . . . . . . . . . . . . watt (W) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . British thermal unitIT per hour square foot degree Fahrenheit [BtuIT /(h ⭈ ft2 ⭈ ⬚F)] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . watt per square meter kelvin [W/(m2 ⭈ K)] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . British thermal unitth per hour square foot degree Fahrenheit [Btuth /(h ⭈ ft2 ⭈ ⬚F)] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . watt per square meter kelvin [W/(m2 ⭈ K)] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . British thermal unitth per minute (Btuth /min) . . . . . . . watt (W) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . British thermal unitIT per pound (BtuIT /lb). . . . . . . . . . joule per kilogram (J/kg) . . . . . . . . . . . . . . . . . . . . . . British thermal unitth per pound (Btuth /lb) . . . . . . . . . . joule per kilogram (J/kg) . . . . . . . . . . . . . . . . . . . . . . British thermal unitIT per pound degree Fahrenheit [BtuIT /(lb ⭈ ⬚F)]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . joule per kilogram kelvin (J/(kg ⭈ K)]. . . . . . . . . . British thermal unitth per pound degree Fahrenheit [Btuth /(lb ⭈ ⬚F)] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . joule per kilogram kelvin [J/(kg ⭈ K)]. . . . . . . . . . British thermal unitIT per pound degree Rankine [BtuIT /(lb ⭈ ⬚R)] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . joule per kilogram kelvin [J/(kg ⭈ K)]. . . . . . . . . . British thermal unitth per pound degree Rankine [Btuth /(lb ⭈ ⬚R)]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . joule per kilogram kelvin [J/(kg ⭈ K)]. . . . . . . . . . British thermal unitIT per second (BtuIT /s) . . . . . . . . . . watt (W) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . British thermal unitth per second (Btuth /s) . . . . . . . . . . watt (W) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . British thermal unitIT per second square foot degree Fahrenheit [BtuIT /(s ⭈ ft2 ⭈ ⬚F)] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . watt per square meter kelvin [W/(m2 ⭈ K)] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . British thermal unitth per second square foot degree Fahrenheit [Btuth /(s ⭈ ft2 ⭈ ⬚F)] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . watt per square meter kelvin [W/(m2 ⭈ K)] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . British thermal unitIT per square foot (BtuIT /ft2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . joule per square meter (J/m2) . . . . . . . . . . . . . . . . . . British thermal unitth per square foot (Btuth /ft2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . joule per square meter (J/m2) . . . . . . . . . . . . . . . . . . British thermal unitIT per square foot hour [(BtuIT /(ft2 ⭈ h)] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . watt per square meter (W/m2) . . . . . . . . . . . . . . . . . British thermal unitth per square foot hour [Btuth /(ft2 ⭈ h)]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . watt per square meter (W/m2) . . . . . . . . . . . . . . . . . British thermal unitth per square foot minute [Btuth /(ft2 ⭈ min)] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . watt per square meter (W/m2) . . . . . . . . . . . . . . . . . British thermal unitIT per square foot second [(BtuIT /(ft2 ⭈ s)] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . watt per square meter (W/m2) . . . . . . . . . . . . . . . . . British thermal unitth per square foot second [Btuth /(ft2 ⭈ s)] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . watt per square meter (W/m2) . . . . . . . . . . . . . . . . . British thermal unitth per square inch second [Btuth /(in2 ⭈ s)] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . watt per square meter (W/m2) . . . . . . . . . . . . . . . . .


Multiply by 5.188 732

E+02

3.725 895

E+04

3.723 403

E+04

1.899 101

E+03

1.897 830

E+03

1.899 101

E+03

1.897 830 2.930 711

E+03 E⫺01

2.928 751

E⫺01

5.678 263

E+00

5.674 466 1.757 250

E+00 E+01

2.326 2.324 444

E+03 E+03

4.1868

E+03

4.184

E+03

4.1868

E+03

4.184 1.055 056 1.054 350

E+03 E+03 E+03

2.044 175

E+04

2.042 808

E+04

1.135 653

E+04

1.134 893

E+04

3.154 591

E+00

3.152 481

E+00

1.891 489

E+02

1.135 653

E+04

1.134 893

E+04

1.634 246

E+06

To convert from

to

Multiply by 3

bushel (U.S.) (bu) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . cubic meter (m ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.523 907

E⫺02

bushel (U.S.) (bu) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . liter (L) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.523 907

E+01

calorie IT (calIT) 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . joule (J). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . calorie th (calth) 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . joule (J). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . calorie (cal) (mean) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . joule (J). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . calorie (15 ⬚C) (cal 15) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . joule (J). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . calorie (20 ⬚C) (cal 20) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . joule (J). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.1868 4.184

E+00 E+00

4.190 02 4.185 80 4.181 90

E+00 E+00 E+00

4.1868 4.184 4.190 02

E+03 E+03 E+03

4.184 4.1868 4.184

E+02 E+03 E+03

4.1868

E+03

4.184 4.1868 4.184 6.973 333 4.184 4.184

E+03 E+03 E+03 E⫺02 E+00 E+04

6.973 333

E+02

4.184 1.550 003 2.0 2.0 1.333 22 1.333 22 1.333 224 1.333 224 9.806 38

E+04 E+03 Eⴚ04 Eⴚ01 E+03 E+00 E+03 E+00 E+01

9.806 65 1.0 1.0 2.011 684 5.067 075 5.067 075 1.55 3.624 556 2.831 685 4.719 474 4.719 474 2.831 685

E+01 Eⴚ03 Eⴚ06 E+01 E⫺10 E⫺04 E⫺01 E+00 E⫺02 E⫺04 E⫺01 E⫺02

calorieIT , kilogram (nutrition) 12 . . . . . . . . . . . . . . . . . . . . . joule (J). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . calorieth , kilogram (nutrition) 12 . . . . . . . . . . . . . . . . . . . . . joule (J). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . calorie (mean), kilogram (nutrition) 12 . . . . . . . . . . . . . . . joule (J). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . calorieth per centimeter second degree Celsius [calth /(cm ⭈ s ⭈ ⬚C)] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . watt per meter kelvin [W/(m ⭈ K)]. . . . . . . . . . . . . calorieIT per gram (calIT /g). . . . . . . . . . . . . . . . . . . . . . . . . . joule per kilogram (J/kg) . . . . . . . . . . . . . . . . . . . . . . calorieth per gram (calth /g) . . . . . . . . . . . . . . . . . . . . . . . . . . joule per kilogram (J/kg) . . . . . . . . . . . . . . . . . . . . . . calorieIT per gram degree Celsius [calIT /(g ⭈ ⬚C)] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . joule per kilogram kelvin [J/(kg ⭈ K)]. . . . . . . . . . calorieth per gram degree Celsius [calth /(g ⭈ ⬚C)] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . joule per kilogram kelvin [J/(kg ⭈ K)]. . . . . . . . . . calorieIT per gram kelvin [cal IT /(g ⭈ K)] . . . . . . . . . . . . . joule per kilogram kelvin [J / (kg ⭈ K)] . . . . . . . . . calorieth per gram kelvin [cal th / (g ⭈ K)] . . . . . . . . . . . . . joule per kilogram kelvin [J / (kg ⭈ K)] . . . . . . . . . calorieth per minute (calth /min). . . . . . . . . . . . . . . . . . . . . . watt (W) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . calorieth per second (calth /s). . . . . . . . . . . . . . . . . . . . . . . . . watt (W) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . calorieth per square centimeter (calth /cm2). . . . . . . . . . . joule per square meter (J/m2) . . . . . . . . . . . . . . . . . . calorieth per square centimeter minute [calth /(cm2 ⭈ min)] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . watt per square meter (W/m2) . . . . . . . . . . . . . . . . . calorieth per square centimeter second [calth /(cm2 ⭈ s)] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . watt per square meter (W/m2) . . . . . . . . . . . . . . . . . candela per square inch (cd/in2) . . . . . . . . . . . . . . . . . . . . candela per square meter (cd/m2). . . . . . . . . . . . . . carat, metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . kilogram (kg) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . carat, metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . gram (g) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . centimeter of mercury (0 ⬚C) 13 . . . . . . . . . . . . . . . . . . . . . . pascal (Pa) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . centimeter of mercury (0 ⬚C) 13 . . . . . . . . . . . . . . . . . . . . . . kilopascal (kPa). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . centimeter of mercury, conventional (cmHg) 13 . . . . . . . pascal (Pa) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . centimeter of mercury, conventional (cmHg) 13 . . . . . . . kilopascal (kPa). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . centimeter of water (4 ⬚C) 13 . . . . . . . . . . . . . . . . . . . . . . . . . pascal (Pa) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . centimeter of water, conventional (cmH 2 O) 13 . . . . . . . . pascal (Pa) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . centipoise (cP) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . pascal second (Pa ⭈ s) . . . . . . . . . . . . . . . . . . . . . . . . . . centistokes (cSt) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . meter squared per second (m2 /s). . . . . . . . . . . . . . . chain (based on U.S. survey foot) (ch)9 . . . . . . . . . . . . . meter (m) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . circular mil. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . square meter (m2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . circular mil. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . square millimeter (mm2) . . . . . . . . . . . . . . . . . . . . . . . clo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . square meter kelvin per watt (m2 ⭈ K /W). . . . . . . cord (128 ft 3 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . cubic meter (m 3 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . cubic foot (ft3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . cubic meter (m3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . cubic foot per minute (ft3 /min) . . . . . . . . . . . . . . . . . . . . . cubic meter per second (m3 /s) . . . . . . . . . . . . . . . . . cubic foot per minute (ft3 /min) . . . . . . . . . . . . . . . . . . . . . liter per second (L / s) . . . . . . . . . . . . . . . . . . . . . . . . . . cubic foot per second (ft3 /s) . . . . . . . . . . . . . . . . . . . . . . . . cubic meter per second (m3 /s) . . . . . . . . . . . . . . . . . 12

The kilogram calorie or ‘‘large calorie’’ is an obsolete term used for the kilocalorie, which is the calorie used to express the energy content of foods. However, in practice, the prefix ‘‘kilo’’ is usually omitted. 13 Conversion factors for mercury manometer pressure units are calculated using the standard value for the acceleration of gravity and the density of mercury at the stated temperature. Additional digits are not justified because the definitions of the units do not take into account the compressibility of mercury or the change in density caused by the revised practical temperature scale, ITS-90. Similar comments also apply to water manometer pressure units. Conversion factors for conventional mercury and water manometer pressure units are based on ISO 31-3.


To convert from

to

Multiply by

cubic inch (in3) 14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . cubic meter (m3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.638 706 cubic inch per minute (in3 /min). . . . . . . . . . . . . . . . . . . . . cubic meter per second (m3 /s) . . . . . . . . . . . . . . . . . 2.731 177 cubic mile (mi3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . cubic meter (m3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.168 182 cubic yard (yd3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . cubic meter (m3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.645 549

E⫺05 E⫺07 E+09 E⫺01

cubic yard per minute (yd3 /min) . . . . . . . . . . . . . . . . . . . . cubic meter per second (m3 /s) . . . . . . . . . . . . . . . . . 1.274 258 cup (U.S.) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . cubic meter (m 3). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.365 882 cup (U.S.) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . liter (L) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.365 882

E⫺02 E⫺04 E⫺01

cup (U.S.) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . milliliter (mL) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.365 882 curie (Ci) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . becquerel (Bq) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7

E+02 E+10

darcy15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . meter squared (m2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.869 233 day (d) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . second (s). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.64

E⫺13 E+04

day (sidereal). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . second (s). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.616 409 debye (D) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . coulomb meter (C ⭈ m) . . . . . . . . . . . . . . . . . . . . . . . . 3.335 641

E+04 E⫺30

degree (angle) (⬚). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . radian (rad) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.745 329 E⫺02 degree Celsius (temperature) (⬚C) . . . . . . . . . . . . . . . . . . . kelvin (K). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . T / K = t / ⬚C+273.15 degree Celsius (temperature interval) (⬚C) . . . . . . . . . . kelvin (K). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.0 E+00 degree centigrade (temperature) 16 . . . . . . . . . . . . . . . . . . . degree Celsius (⬚C) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . t / ⬚C ≈ t / deg. cent. degree centigrade (temperature interval) 16 . . . . . . . . . . degree Celsius (⬚C) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.0 E+00 degree Fahrenheit (temperature) (⬚F). . . . . . . . . . . . . . . . degree Celsius (⬚C) . . . . . . . . . . . . . . . . . . . . . . . . . . .t / ⬚C = (t / ⬚F ⫺ 32) / 1.8 degree Fahrenheit (temperature) (⬚F). . . . . . . . . . . . . . . . kelvin (K). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . T / K = (t / ⬚F + 459.67)/1.8 degree Fahrenheit (temperature interval)(⬚F) . . . . . . . . degree Celsius (⬚C) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.555 556 E⫺01 degree Fahrenheit (temperature interval)(⬚F) . . . . . . . . kelvin (K). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.555 556 E⫺01 degree Fahrenheit hour per British thermal unit IT (⬚F ⭈ h / Btu IT) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . kelvin per watt (K / W) . . . . . . . . . . . . . . . . . . . . . . . . . 1.895 634 E+00 degree Fahrenheit hour per British thermal unit th (⬚F ⭈ h / Btu th). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . kelvin per watt (K / W) . . . . . . . . . . . . . . . . . . . . . . . . . 1.896 903 E+00 degree Fahrenheit hour square foot per British thermal unitIT (⬚F ⭈ h ⭈ ft2 /BtuIT). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . square meter kelvin per watt (m2 ⭈ K /W) . . . . . . 1.761 102 E⫺01 degree Fahrenheit hour square foot per British thermal unitth (⬚F ⭈ h ⭈ ft2 /Btuth) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . square meter kelvin per watt (m2 ⭈ K /W) . . . . . . 1.762 280 E⫺01 degree Fahrenheit hour square foot per British thermal unitIT inch [⬚F ⭈ h ⭈ ft2 /(BtuIT ⭈ in)] . . . . . . . . . . . . . . . . . . . . . . . . . . . meter kelvin per watt (m ⭈ K /W) . . . . . . . . . . . . . . 6.933 472 E+00 degree Fahrenheit hour square foot per British thermal unitth inch [⬚F ⭈ h ⭈ ft2 /(Btuth ⭈ in)] . . . . . . . . . . . . . . . . . . . . . . . . . . . meter kelvin per watt (m ⭈ K /W) . . . . . . . . . . . . . . 6.938 112 E+00 degree Fahrenheit second per British thermal unit IT (⬚F ⭈ s / Btu IT). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . kelvin per watt (K / W) . . . . . . . . . . . . . . . . . . . . . . . . . 5.265 651 E⫺04 degree Fahrenheit second per British thermal unit th (⬚F ⭈ s / Btu th) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . kelvin per watt (K / W) . . . . . . . . . . . . . . . . . . . . . . . . . 5.269 175 E⫺04 degree Rankine (⬚R) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . kelvin (K). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .T / K = (T / ⬚R) / 1.8 degree Rankine (temperature interval) (⬚R) . . . . . . . . . kelvin (K). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.555 556 E⫺01 denier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . kilogram per meter (kg/m) . . . . . . . . . . . . . . . . . . . . 1.111 111 E⫺07 denier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . gram per meter (g/m) . . . . . . . . . . . . . . . . . . . . . . . . . 1.111 111 E⫺04 dyne (dyn) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . newton (N) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.0

Eⴚ05

dyne centimeter (dyn ⭈ cm). . . . . . . . . . . . . . . . . . . . . . . . . . newton meter (N ⭈ m). . . . . . . . . . . . . . . . . . . . . . . . . . 1.0 dyne per square centimeter (dyn/cm2) . . . . . . . . . . . . . . pascal (Pa) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.0

Eⴚ07 Eⴚ01

electronvolt (eV) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . joule (J). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.602 177

E⫺19

EMU of capacitance (abfarad) . . . . . . . . . . . . . . . . . . . . . . farad (F) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.0 EMU of current (abampere). . . . . . . . . . . . . . . . . . . . . . . . . ampere (A) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.0 EMU of electric potential (abvolt) . . . . . . . . . . . . . . . . . . volt (V) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.0

E+09 E+01 Eⴚ08

EMU of inductance (abhenry) . . . . . . . . . . . . . . . . . . . . . . henry (H) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.0

Eⴚ09

14

The exact conversion factor is 1.638 706 4 E⫺05. The darcy is a unit for expressing the permeability of porous solids, not area. 16 The centigrade temperature scale is obsolete; the degree centigrade is only approximately equal to the degree Celsius. 15


To convert from

to

Multiply by

EMU of resistance (abohm) . . . . . . . . . . . . . . . . . . . . . . . . . ohm (⍀) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.0

Eⴚ09

erg (erg). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . joule (J). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.0

Eⴚ07

erg per second (erg/s). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . watt (W) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.0

Eⴚ07

erg per square centimeter second |1obrkt㜸1ru|/ (cm2 ⭈ s)]. . . . . . . . . . . . . . . . . . . . . . . . . . . . watt per square meter (W/m2) . . . . . . . . . . . . . . . . . ESU of capacitance (statfarad) . . . . . . . . . . . . . . . . . . . . . . farad (F) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ESU of current (statampere) . . . . . . . . . . . . . . . . . . . . . . . . ampere (A) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ESU of electric potential (statvolt) . . . . . . . . . . . . . . . . . . volt (V) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ESU of inductance (stathenry) . . . . . . . . . . . . . . . . . . . . . . henry (H) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ESU of resistance (statohm) . . . . . . . . . . . . . . . . . . . . . . . . . ohm (⍀) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.0

Eⴚ03

1.112 650 3.335 641 2.997 925 8.987 552 8.987 552

E⫺12 E⫺10 E+02 E+11 E+11

faraday (based on carbon 12) . . . . . . . . . . . . . . . . . . . . . . . coulomb (C) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.648 531 fathom (based on U.S. survey foot)9 . . . . . . . . . . . . . . . . meter (m) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.828 804 fermi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . meter (m) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.0 fermi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . femtometer (fm) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.0 fluid ounce (U.S.) (fl oz). . . . . . . . . . . . . . . . . . . . . . . . . . . . cubic meter (m3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.957 353 fluid ounce (U.S.) (fl oz). . . . . . . . . . . . . . . . . . . . . . . . . . . . milliliter (mL) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.957 353 foot (ft) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . meter (m) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.048 foot (U.S. survey) (ft)9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . meter (m) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.048 006

E+04

footcandle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . lux (lx) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.076 391 footlambert. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . candela per square meter (cd/m2). . . . . . . . . . . . . . 3.426 259 foot of mercury, conventional (ftHg) 13 . . . . . . . . . . . . . . pascal (Pa) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.063 666

E+00 Eⴚ15 E+00 E⫺05 E+01 Eⴚ01 E⫺01 E+01 E+00

4.063 666 2.988 98

E+04 E+01 E+03

2.988 98 2.989 067 2.989 067 8.466 667 5.08 3.048

E+00 E+03 E+00 E⫺05 Eⴚ03 Eⴚ01

3.048 4.214 011 1.355 818 3.766 161 2.259 697 1.355 818

Eⴚ01 E⫺02 E+00 E⫺04 E⫺02 E+00

8.630 975 franklin (Fr) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . coulomb (C) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.335 641

E⫺03 E⫺10

gal (Gal) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . meter per second squared (m/s2). . . . . . . . . . . . . . . gallon [Canadian and U.K. (Imperial)] (gal) . . . . . . . . cubic meter (m3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . gallon [Canadian and U.K. (Imperial)] (gal) . . . . . . . . liter (L) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . gallon (U.S.) (gal) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . cubic meter (m3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.0 4.546 09 4.546 09

Eⴚ02 Eⴚ03 E+00

3.785 412 3.785 412 4.381 264 4.381 264

E⫺03 E+00 E⫺08 E⫺05

1.410 089 gallon (U.S.) per horsepower hour [gal / (hp ⭈ h)] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . liter per joule (L / J) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.410 089 gallon (U.S.) per minute (gpm)(gal / min) . . . . . . . . . . . cubic meter per second (m3 /s) . . . . . . . . . . . . . . . . . 6.309 020 gallon (U.S.) per minute (gpm) (gal / min) . . . . . . . . . . . liter per second (L / s) . . . . . . . . . . . . . . . . . . . . . . . . . . 6.309 020

E⫺09

foot of mercury, conventional (ftHg) 13 . . . . . . . . . . . . . . kilopascal (kPa). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . foot of water (39.2 ⬚F) 13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . pascal (Pa) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . foot of water (39.2 ⬚F) 13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . kilopascal (kPa). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . foot of water, conventional (ftH 2 O) 13 . . . . . . . . . . . . . . . pascal (Pa) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . foot of water, conventional (ftH 2 O) 13 . . . . . . . . . . . . . . . kilopascal (kPa). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . foot per hour (ft/h). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . meter per second (m/s) . . . . . . . . . . . . . . . . . . . . . . . . foot per minute (ft/min) . . . . . . . . . . . . . . . . . . . . . . . . . . . . meter per second (m/s) . . . . . . . . . . . . . . . . . . . . . . . . foot per second (ft/s). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . meter per second (m/s) . . . . . . . . . . . . . . . . . . . . . . . . foot per second squared (ft/s2) . . . . . . . . . . . . . . . . . . . . . . meter per second squared (m/s2). . . . . . . . . . . . . . . foot foot foot foot foot foot

poundal. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . joule (J). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . pound-force (ft ⭈ lbf) . . . . . . . . . . . . . . . . . . . . . . . . . . . joule (J). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . pound-force per hour (ft ⭈ lbf/h). . . . . . . . . . . . . . . . watt (W) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . pound-force per minute (ft ⭈ lbf/min) . . . . . . . . . . watt (W) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . pound-force per second (ft ⭈ lbf/s) . . . . . . . . . . . . . watt (W) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . to the fourth power (ft 4) 17 . . . . . . . . . . . . . . . . . . . . . . meter to the fourth power (m4) . . . . . . . . . . . . . . . .

gallon (U.S.) (gal) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . liter (L) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . gallon (U.S.) per day (gal / d) . . . . . . . . . . . . . . . . . . . . . . . . cubic meter per second (m3 /s) . . . . . . . . . . . . . . . . . gallon (U.S.) per day (gal / d) . . . . . . . . . . . . . . . . . . . . . . . . liter per second (L / s) . . . . . . . . . . . . . . . . . . . . . . . . . . gallon (U.S.) per horsepower hour [gal / (hp ⭈ h)] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . cubic meter per joule (m3 /J) . . . . . . . . . . . . . . . . . . .

17 This is a unit for the quantity second moment of area, which is sometimes called the ‘‘moment of section’’ or ‘‘area moment of inertia’’ of a plane section about a specified axis.


E⫺06 E⫺05 E⫺02

To convert from

to

Multiply by

gamma (␥) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . tesla (T) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.0

Eⴚ09

gauss (Gs, G) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . tesla (T) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.0

Eⴚ04

gilbert (Gi) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ampere (A) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . gill [Canadian and U.K. (Imperial)] (gi) . . . . . . . . . . . . cubic meter (m3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . gill [Canadian and U.K. (Imperial)] (gi) . . . . . . . . . . . . liter (L) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . gill (U.S.) (gi). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . cubic meter (m3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . gill (U.S.) (gi). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . liter (L) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . gon (also called grade) (gon) . . . . . . . . . . . . . . . . . . . . . . . . radian (rad) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . gon (also called grade) (gon) . . . . . . . . . . . . . . . . . . . . . . . . degree (angle) (⬚) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . grain (gr) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . kilogram (kg) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . grain (gr) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . milligram (mg) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . grain per gallon (U.S.) (gr / gal) . . . . . . . . . . . . . . . . . . . . . kilogram per cubic meter (kg/m3) . . . . . . . . . . . . .

7.957 747 1.420 653 1.420 653 1.182 941 1.182 941 1.570 796 9.0 6.479 891 6.479 891

E⫺01 E⫺04 E⫺01 E⫺04 E⫺01 E⫺02 Eⴚ01 Eⴚ05 E+01

1.711 806 grain per gallon (U.S.) (gr / gal) . . . . . . . . . . . . . . . . . . . . . milligram per liter (mg / L). . . . . . . . . . . . . . . . . . . . . 1.711 806 gram-force per square centimeter (gf / cm 2). . . . . . . . . . pascal (Pa) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.806 65 gram per cubic centimeter (g / cm 3) . . . . . . . . . . . . . . . . . kilogram per cubic meter (kg/m3) . . . . . . . . . . . . . 1.0

E⫺02 E+01 E+01

hectare (ha) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . square meter (m2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . horsepower (550 ft ⭈ lbf/s) (hp) . . . . . . . . . . . . . . . . . . . . . watt (W) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . horsepower (boiler) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . watt (W) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . horsepower (electric). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . watt (W) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . horsepower (metric). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . watt (W) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . horsepower (U.K.) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . watt (W) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . horsepower (water). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . watt (W) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . hour (h) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . second (s). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . hour (sidereal) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . second (s). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . hundredweight (long, 112 lb). . . . . . . . . . . . . . . . . . . . . . . . kilogram (kg) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . hundredweight (short, 100 lb) . . . . . . . . . . . . . . . . . . . . . . . kilogram (kg) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.0 7.456 999 9.809 50 7.46 7.354 988 7.4570 7.460 43 3.6 3.590 170 5.080 235 4.535 924

E+04 E+02 E+03 E+02 E+02 E+02 E+02 E+03 E+03 E+01 E+01

inch (in) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . meter (m) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . inch (in) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . centimeter (cm) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . inch of mercury (32 ⬚F) 13 . . . . . . . . . . . . . . . . . . . . . . . . . . . pascal (Pa) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . inch of mercury (32 ⬚F) 13 . . . . . . . . . . . . . . . . . . . . . . . . . . . kilopascal (kPa). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . inch of mercury (60 ⬚F) 13 . . . . . . . . . . . . . . . . . . . . . . . . . . . pascal (Pa) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . inch of mercury (60 ⬚F) 13 . . . . . . . . . . . . . . . . . . . . . . . . . . . kilopascal (kPa). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . inch of mercury, conventional (inHg) 13 . . . . . . . . . . . . . pascal (Pa) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . inch of mercury, conventional (inHg) 13 . . . . . . . . . . . . . kilopascal (kPa). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . inch of water (39.2 ⬚F) 13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . pascal (Pa) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . inch of water (60 ⬚F) 13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . pascal (Pa) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . inch of water, conventional (inH 2 O) 13 . . . . . . . . . . . . . . pascal (Pa) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.54 2.54 3.386 38 3.386 38 3.376 85

Eⴚ02 E+00 E+03 E+00 E+03

3.376 85 3.386 389 3.386 389 2.490 82 2.4884

E+00 E+03 E+00 E+02 E+02

2.490 889 inch per second (in/s). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . meter per second (m/s) . . . . . . . . . . . . . . . . . . . . . . . . 2.54 inch per second squared (in/s2) . . . . . . . . . . . . . . . . . . . . . meter per second squared (m/s2). . . . . . . . . . . . . . . 2.54 inch to the fourth power (in4) 17 . . . . . . . . . . . . . . . . . . . . . meter to the fourth power (m4) . . . . . . . . . . . . . . . . 4.162 314

E+03

E+02 Eⴚ02 Eⴚ02 E⫺07

E+02 kayser (K) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . reciprocal meter (m ⫺1 ) . . . . . . . . . . . . . . . . . . . . . . . . 1.0 kelvin (K) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . degree Celsius (⬚C) . . . . . . . . . . . . . . . . . . . . . . . . . . . .t / ⬚C = T / K ⫺ 273.15 E+03 kilocalorieIT (kcalIT) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . joule (J). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1868 E+03 kilocalorieth (kcalth) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . joule (J). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.184 kilocalorie (mean) (kcal) . . . . . . . . . . . . . . . . . . . . . . . . . . . . joule (J). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.190 02 E+03 E+01 kilocalorieth per minute (kcalth /min) . . . . . . . . . . . . . . . . watt (W) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.973 333 kilocalorieth per second (kcalth / s) . . . . . . . . . . . . . . . . . . . watt (W) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.184 E+03 kilogram-force (kgf) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . newton (N) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.806 65 kilogram-force meter (kgf ⭈ m) . . . . . . . . . . . . . . . . . . . . . . newton meter (N ⭈ m). . . . . . . . . . . . . . . . . . . . . . . . . . 9.806 65


E+00 E+00

To convert from

to

Multiply by

kilogram-force per square centimeter (kgf/cm 2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . pascal (Pa) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.806 65

E+04

kilogram-force per square centimeter (kgf/cm 2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . kilopascal (kPa). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.806 65

E+01

kilogram-force per square meter (kgf/m2). . . . . . . . . . . pascal (Pa) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.806 65

E+00

kilogram-force per square millimeter (kgf/mm 2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . pascal (Pa) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.806 65

E+06

kilogram-force per square millimeter (kgf/mm 2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . megapascal (MPa) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.806 65

E+00

kilogram-force second squared per meter (kgf ⭈ s 2 /m). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . kilogram (kg) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.806 65

E+00

kilometer per hour (km/h) . . . . . . . . . . . . . . . . . . . . . . . . . . meter per second (m/s) . . . . . . . . . . . . . . . . . . . . . . . . 2.777 778 kilopond (kilogram-force) (kp) . . . . . . . . . . . . . . . . . . . . . . newton (N) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.806 65

E⫺01 E+00

kilowatt hour (kW ⭈ h) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . joule (J). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6

E+06

kilowatt hour (kW ⭈ h) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . megajoule (MJ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 kip (1 kip=1000 lbf) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . newton (N) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.448 222

E+00 E+03

kip (1 kip=1000 lbf) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . kilonewton (kN) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.448 222

E+00

kip per square inch (ksi) (kip/in2). . . . . . . . . . . . . . . . . . . pascal (Pa) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.894 757 kip per square inch (ksi) (kip/in2). . . . . . . . . . . . . . . . . . . kilopascal (kPa). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.894 757 knot (nautical mile per hour) . . . . . . . . . . . . . . . . . . . . . . . . meter per second (m/s) . . . . . . . . . . . . . . . . . . . . . . . . 5.144 444

E+06 E+03 E⫺01

lambert 18 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . candela per square meter (cd/m2). . . . . . . . . . . . . . 3.183 099 langley (cal th /cm 2 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . joule per square meter (J/m2) . . . . . . . . . . . . . . . . . . 4.184 light year (l.y.)19 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . meter (m) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.460 73 liter (L) 20 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . cubic meter (m3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.0

E+03 E+04 E+15 Eⴚ03

lumen per square foot (lm/ft 2) . . . . . . . . . . . . . . . . . . . . . . lux (lx) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.076 391

E+01

maxwell (Mx) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . weber (Wb) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.0 mho . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . siemens (S) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.0

Eⴚ08 E+00

microinch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . meter (m) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.54 microinch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . micrometer (␮m) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.54 micron (␮) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . meter (m) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.0

Eⴚ08 Eⴚ02 Eⴚ06

micron (␮) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . micrometer (␮m) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.0 mil (0.001 in). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . meter (m) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.54 mil (0.001 in). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . millimeter (mm) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.54

E+00 Eⴚ05 Eⴚ02

mil (angle) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . radian (rad) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.817 477 mil (angle) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . degree (⬚) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.625 mile (mi) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . meter (m) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.609 344

E⫺04 Eⴚ02 E+03

mile (mi) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . kilometer (km). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.609 344 mile (based on U.S. survey foot) (mi) 9 . . . . . . . . . . . . . . meter (m) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.609 347 mile (based on U.S. survey foot) (mi) 9 . . . . . . . . . . . . . . kilometer (km). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.609 347

E+00 E+03

E+00 mile, nautical 21 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . meter (m) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.852 E+03 mile per gallon (U.S.) (mpg) (mi/gal) . . . . . . . . . . . . . . . meter per cubic meter (m/m 3) . . . . . . . . . . . . . . . . . 4.251 437 E+05 mile per gallon (U.S.) (mpg) (mi/gal) . . . . . . . . . . . . . . . kilometer per liter (km /L) . . . . . . . . . . . . . . . . . . . . . 4.251 437 E⫺01 mile per gallon (U.S.) (mpg) (mi/gal) 22 . . . . . . . . . . . . . liter per 100 kilometer (L/100 km) . . . . . . . .divide 235.215 by number of miles per gallon mile per hour (mi/h) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . meter per second (m/s) . . . . . . . . . . . . . . . . . . . . . . . . 4.4704 Eⴚ01 mile per hour (mi/h) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . kilometer per hour (km/h) . . . . . . . . . . . . . . . . . . . . . 1.609 344 E+00

18

The exact conversion factor is 10 4 /␲. This conversion factor is based on 1 d = 86 400 s; and 1 Julian century = 36 525 d. (See The Astronomical Almanac for the Year 1995 , page K6, U.S. Government Printing Office, Washington, DC, 1994). 20 In 1964 the General Conference on Weights and Measures reestablished the name ‘‘liter’’ as a special name for the cubic decimeter. Between 1901 and 1964 the liter was slightly larger (1.000 028 dm3); when one uses high-accuracy volume data of that time, this fact must be kept in mind. 21 The value of this unit, 1 nautical mile = 1852 m, was adopted by the First International Extraordinary Hydrographic Conference, Monaco, 1929, under the name ‘‘International nautical mile.’’ 22 For converting fuel economy, as used in the U.S., to fuel consumption. 19


To convert from

to

Multiply by

mile per minute (mi/min) . . . . . . . . . . . . . . . . . . . . . . . . . . . meter per second (m/s) . . . . . . . . . . . . . . . . . . . . . . . . 2.682 24

E+01

mile per second (mi/s) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . meter per second (m/s) . . . . . . . . . . . . . . . . . . . . . . . . 1.609 344 millibar (mbar) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . pascal (Pa) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.0

E+03 E+02

millibar (mbar) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . kilopascal (kPa). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.0 1.333 224 9.806 65

Eⴚ01 E+02 E+00

2.908 882 6.0

E⫺04 E+01

minute (sidereal) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . second (s). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.983 617

E+01

oersted (Oe) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ampere per meter (A/m). . . . . . . . . . . . . . . . . . . . . . . 7.957 747 ohm centimeter (⍀ ⭈ cm). . . . . . . . . . . . . . . . . . . . . . . . . . . . ohm meter (⍀ ⭈ m). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.0

E+01 Eⴚ02

ohm circular-mil per foot . . . . . . . . . . . . . . . . . . . . . . . . . . . ohm meter (⍀ ⭈ m). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.662 426

E⫺09

ohm circular-mil per foot . . . . . . . . . . . . . . . . . . . . . . . . . . . ohm square millimeter per meter (⍀ ⭈ mm 2 / m). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.662 426

E⫺03

ounce (avoirdupois) (oz). . . . . . . . . . . . . . . . . . . . . . . . . . . . . kilogram (kg) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ounce (avoirdupois) (oz). . . . . . . . . . . . . . . . . . . . . . . . . . . . . gram (g) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ounce (troy or apothecary) (oz) . . . . . . . . . . . . . . . . . . . . . kilogram (kg) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ounce (troy or apothecary) (oz) . . . . . . . . . . . . . . . . . . . . . gram (g) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

E⫺02 E+01 E⫺02 E+01

millimeter of mercury, conventional (mmHg)13 . . . . . . . pascal (Pa) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . millimeter of water, conventional (mmH 2 O)13 . . . . . . . . pascal (Pa) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . minute (angle) (') . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . radian (rad) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . minute (min) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . second (s). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.834 952 2.834 952 3.110 348 3.110 348

ounce [Canadian and U.K. fluid (Imperial)] (fl oz). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . cubic meter (m3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.841 306 ounce [Canadian and U.K. fluid (Imperial)] (fl oz) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . milliliter (mL) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.841 306

E+01

ounce (U.S. fluid) (fl oz). . . . . . . . . . . . . . . . . . . . . . . . . . . . cubic meter (m3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.957 353 ounce (U.S. fluid) (fl oz). . . . . . . . . . . . . . . . . . . . . . . . . . . . millimeter (mL) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.957 353

E⫺05 E+01

ounce (avoirdupois)-force (ozf). . . . . . . . . . . . . . . . . . . . . . newton (N) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ounce (avoirdupois)-force inch (ozf ⭈ in) . . . . . . . . . . . . newton meter (N ⭈ m). . . . . . . . . . . . . . . . . . . . . . . . . . ounce (avoirdupois)-force inch (ozf ⭈ in) . . . . . . . . . . . . millinewton meter (mN ⭈ m) . . . . . . . . . . . . . . . . . . . ounce (avoirdupois) per cubic inch (oz / in3) . . . . . . . . . kilogram per cubic meter (kg/m3) . . . . . . . . . . . . . ounce (avoirdupois) per gallon [Canadian and U.K. (Imperial)] (oz / gal). . . . . . . . . . . . . . . . . . . . . . . . . kilogram per cubic meter (kg/m3) . . . . . . . . . . . . . ounce (avoirdupois) per gallon [Canadian and U.K. (Imperial)] (oz / gal). . . . . . . . . . . . . . . . . . . . . . . . . gram per liter (g / L) . . . . . . . . . . . . . . . . . . . . . . . . . . . ounce (avoirdupois) per gallon (U.S.)(oz / gal) . . . . . . . kilogram per cubic meter (kg/m3) . . . . . . . . . . . . . ounce (avoirdupois) per gallon(U.S.)(oz / gal). . . . . . . . gram per liter (g / L) . . . . . . . . . . . . . . . . . . . . . . . . . . . ounce (avoirdupois) per square foot (oz / ft2). . . . . . . . . kilogram per square meter (kg/m2) . . . . . . . . . . . . ounce (avoirdupois) per square inch (oz / in2) . . . . . . . . kilogram per square meter (kg/m2) . . . . . . . . . . . . ounce (avoirdupois) per square yard (oz / yd2) . . . . . . . . kilogram per square meter (kg/m2) . . . . . . . . . . . .

2.780 139 7.061 552 7.061 552 1.729 994

E⫺01 E⫺03 E+00 E+03

6.236 023

E+00

6.236 023 7.489 152 7.489 152 3.051 517

E+00 E+00 E+00 E⫺01

4.394 185 3.390 575

E+01 E⫺02

parsec (pc) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . meter (m) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . peck (U.S.) (pk) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . cubic meter (m3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . peck (U.S.) (pk) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . liter (L) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . pennyweight (dwt). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . kilogram (kg) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . pennyweight (dwt). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . gram (g) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . perm (0 ⬚C) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . kilogram per pascal second square meter [kg/(Pa ⭈ s ⭈ m2)] . . . . . . . . . . . . . . . . . . . . . . . . . . . . perm (23 ⬚C) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . kilogram per pascal second square meter [kg/(Pa ⭈ s ⭈ m2)] . . . . . . . . . . . . . . . . . . . . . . . . . . . . perm inch (0 ⬚C) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . kilogram per pascal second meter [kg/(Pa ⭈ s ⭈ m)] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . perm inch (23 ⬚C) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . kilogram per pascal second meter [kg/(Pa ⭈ s ⭈ m)] . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.085 678 8.809 768 8.809 768 1.555 174 1.555 174

E+16 E⫺03 E+00 E⫺03 E+00

5.721 35

E⫺11

5.745 25

E⫺11

1.453 22

E⫺12

1.459 29

E⫺12


E⫺05

To convert from

to

phot (ph) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . lux (lx) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . pica (computer) (1/6 in) . . . . . . . . . . . . . . . . . . . . . . . . . . . . meter (m) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . pica (computer) (1/6 in) . . . . . . . . . . . . . . . . . . . . . . . . . . . . millimeter (mm) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . pica (printer’s) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . meter (m) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Multiply by 1.0 4.233 333 4.233 333 4.217 518

E+04 E⫺03 E+00 E⫺03

pica (printer’s) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . millimeter (mm) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.217 518 pint (U.S. dry) (dry pt). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . cubic meter (m3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.506 105 pint (U.S. dry) (dry pt). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . liter (L) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.506 105 pint (U.S. liquid) (liq pt) . . . . . . . . . . . . . . . . . . . . . . . . . . . . cubic meter (m3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.731 765

E+00

pint (U.S. liquid) (liq pt) . . . . . . . . . . . . . . . . . . . . . . . . . . . . liter (L) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.731 765 point (computer) (1/72 in). . . . . . . . . . . . . . . . . . . . . . . . . . meter (m) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.527 778 point (computer) (1/72 in). . . . . . . . . . . . . . . . . . . . . . . . . . millimeter (mm) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.527 778

E⫺04 E⫺01 E⫺04 E⫺01 E⫺04 E⫺01

point (printer’s) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . meter (m) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . point (printer’s) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . millimeter (mm) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . poise (P) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . pascal second (Pa ⭈ s) . . . . . . . . . . . . . . . . . . . . . . . . . . pound (avoirdupois) (lb)23 . . . . . . . . . . . . . . . . . . . . . . . . . . . kilogram (kg) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.514 598 3.514 598 1.0

E⫺04 E⫺01 Eⴚ01

4.535 924 pound (troy or apothecary) (lb) . . . . . . . . . . . . . . . . . . . . . kilogram (kg) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.732 417 poundal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . newton (N) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.382 550 poundal per square foot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . pascal (Pa) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.488 164

E⫺01 E⫺01 E⫺01 E+00

poundal second per square foot . . . . . . . . . . . . . . . . . . . . . pascal second (Pa ⭈ s) . . . . . . . . . . . . . . . . . . . . . . . . . . pound foot squared (lb ⭈ ft2) . . . . . . . . . . . . . . . . . . . . . . . . . kilogram meter squared (kg ⭈ m2) . . . . . . . . . . . . . . pound-force (lbf)24 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . newton (N) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . pound-force foot (lbf ⭈ ft) . . . . . . . . . . . . . . . . . . . . . . . . . . . newton meter (N ⭈ m). . . . . . . . . . . . . . . . . . . . . . . . . . pound-force foot per inch (lbf ⭈ ft/in) . . . . . . . . . . . . . . . newton meter per meter (N ⭈ m/m) . . . . . . . . . . . .

1.488 164

E+00

4.214 011 4.448 222 1.355 818 5.337 866

E⫺02 E+00 E+00 E+01

pound-force inch (lbf ⭈ in). . . . . . . . . . . . . . . . . . . . . . . . . . . newton meter (N ⭈ m). . . . . . . . . . . . . . . . . . . . . . . . . . 1.129 848 pound-force inch per inch (lbf ⭈ in/in) . . . . . . . . . . . . . . newton meter per meter (N ⭈ m/m) . . . . . . . . . . . . 4.448 222 pound-force per foot (lbf/ft) . . . . . . . . . . . . . . . . . . . . . . . . newton per meter (N/m). . . . . . . . . . . . . . . . . . . . . . . 1.459 390

E⫺01 E+00 E+01

pound-force per inch (lbf/in). . . . . . . . . . . . . . . . . . . . . . . . newton per meter (N/m). . . . . . . . . . . . . . . . . . . . . . . 1.751 268 pound-force per pound (lbf/lb) (thrust to mass ratio) . . . . . . . . . . . . . . . . . . . . . newton per kilogram (N/kg) . . . . . . . . . . . . . . . . . . . 9.806 65 pound-force per square foot (lbf/ft2) . . . . . . . . . . . . . . . . pascal (Pa) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.788 026 pound-force per square inch (psi) (lbf/in2 ) . . . . . . . . . . pascal (Pa) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.894 757

E+02

pound-force per square inch (psi) (lbf/in2 ) . . . . . . . . . . kilopascal (kPa). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . pound-force second per square foot (lbf ⭈ s/ft2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . pascal second (Pa ⭈ s) . . . . . . . . . . . . . . . . . . . . . . . . . . pound-force second per square inch (lbf ⭈ s/in2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . pascal second (Pa ⭈ s) . . . . . . . . . . . . . . . . . . . . . . . . . . pound inch squared (lb ⭈ in2) . . . . . . . . . . . . . . . . . . . . . . . . kilogram meter squared (kg ⭈ m2) . . . . . . . . . . . . . . pound per cubic foot (lb/ft3) . . . . . . . . . . . . . . . . . . . . . . . . kilogram per cubic meter (kg/m3) . . . . . . . . . . . . . pound per cubic inch (lb/in3) . . . . . . . . . . . . . . . . . . . . . . . kilogram per cubic meter (kg/m3) . . . . . . . . . . . . . pound per cubic yard (lb/yd3) . . . . . . . . . . . . . . . . . . . . . . . kilogram per cubic meter (kg/m3) . . . . . . . . . . . . . pound per foot (lb/ft) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . kilogram per meter (kg/m) . . . . . . . . . . . . . . . . . . . . pound per foot hour [lb/(ft ⭈ h)] . . . . . . . . . . . . . . . . . . . . . pascal second (Pa ⭈ s) . . . . . . . . . . . . . . . . . . . . . . . . . .

6.894 757

E+00

4.788 026

E+01

6.894 757

E+03

2.926 397 1.601 846

E⫺04 E+01

2.767 990 5.932 764 1.488 164 4.133 789

E+04 E⫺01 E+00 E⫺04

pound per foot second [lb/(ft ⭈ s)]. . . . . . . . . . . . . . . . . . . pascal second (Pa ⭈ s) . . . . . . . . . . . . . . . . . . . . . . . . . . 1.488 164 pound per gallon [Canadian and U.K. (Imperial)] (lb/gal) . . . . . . . . . . . . . . . . . . . . . . . . . kilogram per cubic meter (kg/m3) . . . . . . . . . . . . . 9.977 637 pound per gallon [Canadian and U.K. (Imperial)] (lb/gal) . . . . . . . . . . . . . . . . . . . . . . . . . kilogram per liter (kg/L) . . . . . . . . . . . . . . . . . . . . . . 9.977 637 pound per gallon (U.S.) (lb/gal). . . . . . . . . . . . . . . . . . . . . kilogram per cubic meter (kg/m3) . . . . . . . . . . . . . 1.198 264

E+00

E+00 E+01 E+03

E+01 E⫺02

pound per gallon (U.S.) (lb/gal). . . . . . . . . . . . . . . . . . . . . kilogram per liter (kg/L) . . . . . . . . . . . . . . . . . . . . . . 1.198 264

E+02 E⫺01

pound per horsepower hour [lb/(hp ⭈ h)] . . . . . . . . . . . . kilogram per joule (kg/J) . . . . . . . . . . . . . . . . . . . . . . 1.689 659 pound per hour (lb/h) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . kilogram per second (kg/s) . . . . . . . . . . . . . . . . . . . . 1.259 979

E⫺07 E⫺04

23

The exact conversion factor is 4.535 923 7 E⫺01. All units that contain the pound refer to the avoirdupois pound.

24

If the local value of the acceleration of free fall is taken as g n= 9.806 65 m / s 2 (the standard value), the exact conversion factor is 4.448 221 615 260 5 E+00.


To convert from

to

Multiply by

pound per inch (lb/in). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . kilogram per meter (kg/m) . . . . . . . . . . . . . . . . . . . . 1.785 797 pound per minute (lb/min) . . . . . . . . . . . . . . . . . . . . . . . . . . kilogram per second (kg/s) . . . . . . . . . . . . . . . . . . . . 7.559 873

E+01 E⫺03

pound per second (lb/s) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . kilogram per second (kg/s) . . . . . . . . . . . . . . . . . . . . pound per square foot (lb/ft2) . . . . . . . . . . . . . . . . . . . . . . . kilogram per square meter (kg/m2) . . . . . . . . . . . . pound per square inch (not pound-force) (lb/in2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . kilogram per square meter (kg/m2) . . . . . . . . . . . . pound per yard (lb/yd) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . kilogram per meter (kg/m) . . . . . . . . . . . . . . . . . . . . psi (pound-force per square inch) (lbf/in 2) . . . . . . . . . . pascal (Pa) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . psi (pound-force per square inch) (lbf/in 2) . . . . . . . . . . kilopascal (kPa). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.535 924 4.882 428

E⫺01 E+00

7.030 696 4.960 546 6.894 757 6.894 757

E+02 E⫺01 E+03 E+00

quad (10 15 Btu IT ) 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . joule (J). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.055 056 quart (U.S. dry) (dry qt) . . . . . . . . . . . . . . . . . . . . . . . . . . . . cubic meter (m3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.101 221 quart (U.S. dry) (dry qt) . . . . . . . . . . . . . . . . . . . . . . . . . . . . liter (L) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.101 221 quart (U.S. liquid) (liq qt). . . . . . . . . . . . . . . . . . . . . . . . . . . cubic meter (m3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.463 529 quart (U.S. liquid) (liq qt). . . . . . . . . . . . . . . . . . . . . . . . . . . liter (L) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.463 529

E+18

rad (absorbed dose) (rad) . . . . . . . . . . . . . . . . . . . . . . . . . . . gray (Gy) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . rem (rem) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . sievert (Sv) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . revolution (r) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . radian (rad) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . revolution per minute (rpm) (r/min). . . . . . . . . . . . . . . . . radian per second (rad/s) . . . . . . . . . . . . . . . . . . . . . . rhe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . reciprocal pascal second [(Pa ⭈ s) ⫺1 ]. . . . . . . . . . . rod (based on U.S. survey foot) (rd)9. . . . . . . . . . . . . . . . meter (m) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . roentgen (R) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . coulomb per kilogram (C/kg). . . . . . . . . . . . . . . . . . rpm (revolution per minute) (r/min). . . . . . . . . . . . . . . . . radian per second (rad/s) . . . . . . . . . . . . . . . . . . . . . .

1.0 1.0 6.283 185 1.047 198 1.0 5.029 210 2.58 1.047 198

Eⴚ02 Eⴚ02 E+00 E⫺01 E+01 E+00 Eⴚ04 E⫺01

second (angle) (") . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . radian (rad) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . second (sidereal) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . second (s). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . shake . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . second (s). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . shake . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . nanosecond (ns) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . slug (slug). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . kilogram (kg) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . slug per cubic foot (slug/ft 3 ). . . . . . . . . . . . . . . . . . . . . . . . kilogram per cubic meter (kg/m3) . . . . . . . . . . . . . slug per foot second [slug/(ft ⭈ s)] . . . . . . . . . . . . . . . . . . pascal second (Pa ⭈ s) . . . . . . . . . . . . . . . . . . . . . . . . . . square foot (ft2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . square meter (m2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . square foot per hour (ft2 /h) . . . . . . . . . . . . . . . . . . . . . . . . . square meter per second (m2 /s) . . . . . . . . . . . . . . . . square foot per second (ft2 /s) . . . . . . . . . . . . . . . . . . . . . . . square meter per second (m2 /s) . . . . . . . . . . . . . . . . square inch (in2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . square meter (m2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . square inch (in2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . square centimeter (cm2). . . . . . . . . . . . . . . . . . . . . . . . square mile (mi2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . square meter (m2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . square mile (mi2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . square kilometer (km2). . . . . . . . . . . . . . . . . . . . . . . . . square mile (based on U.S. survey foot) (mi 2)9 . . . . . . . . . . . . . . . . square meter (m2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . square mile (based on U.S. survey foot) (mi 2)9 . . . . . . . . . . . . . . . . square kilometer (km2). . . . . . . . . . . . . . . . . . . . . . . . . square yard (yd2 ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . square meter (m2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . statampere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ampere (A) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.848 137 9.972 696 1.0 1.0 1.459 390 5.153 788 4.788 026 9.290 304 2.580 64

E⫺06 E⫺01 Eⴚ08 E+01 E+01 E+02 E+01 Eⴚ02 Eⴚ05

9.290 304 6.4516 6.4516 2.589 988 2.589 988

Eⴚ02 Eⴚ04 E+00 E+06 E+00

2.589 998

E+06

2.589 998 8.361 274 3.335 641

E+00 E⫺01 E⫺10

statcoulomb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . coulomb (C) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.335 641 statfarad . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . farad (F) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.112 650

E⫺10 E⫺12

stathenry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . henry (H) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . statmho . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . siemens (S) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . statohm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ohm (⍀) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . statvolt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . volt (V) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . stere (st) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . cubic meter (m3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . stilb (sb). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . candela per square meter (cd/m2). . . . . . . . . . . . . .

E+11 E⫺12 E+11 E+02 E+00

8.987 552 1.112 650 8.987 552 2.997 925 1.0

1.0 stokes (St) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . meter squared per second (m2 /s). . . . . . . . . . . . . . . 1.0


E⫺03 E+00 E⫺04 E⫺01

E+04 Eⴚ04

To convert from

to

Multiply by

tablespoon. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . cubic meter (m 3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.478 676

E⫺05

tablespoon. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . milliliter (mL) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.478 676

E+01

teaspoon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . cubic meter (m 3). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.928 922

E⫺06

teaspoon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . milliliter (mL) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . tex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . kilogram per meter (kg/m) . . . . . . . . . . . . . . . . . . . . therm (EC)25 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . joule (J). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . therm (U.S.)25 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . joule (J). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.928 922 1.0 1.055 06

E+00 Eⴚ06 E+08

1.054 804 2.916 667 2.916 667 8.896 443 8.896 443 1.016 047

E+08 E⫺02 E+01 E+03 E+00 E+03

1.328 939 1.0 1.0

E+03 E+03 E+03

3.516 853 4.184

E+03 E+09

2.831 685 9.071 847 1.186 553 2.519 958 1.333 224

E+00 E+02 E+03 E⫺01 E+02

unit pole. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . weber (Wb) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.256 637

E⫺07

watt hour (W ⭈ h) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . joule (J). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . watt per square centimeter (W/cm2 ). . . . . . . . . . . . . . . . watt per square meter (W/m2) . . . . . . . . . . . . . . . . . watt per square inch (W/in2 ). . . . . . . . . . . . . . . . . . . . . . . . watt per square meter (W/m2) . . . . . . . . . . . . . . . . . watt second (W ⭈ s) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . joule (J). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.6

E+03

1.0 1.550 003 1.0

E+04 E+03 E+00

yard (yd) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . meter (m) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . year (365 days). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . second (s). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . year (sidereal). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . second (s). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . year (tropical) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . second (s). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9.144 3.1536 3.155 815 3.155 693

Eⴚ01 E+07 E+07 E+07

ton, assay (AT) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . kilogram (kg) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ton, assay (AT) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . gram (g) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ton-force (2000 lbf) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . newton (N) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ton-force (2000 lbf) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . kilonewton (kN) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ton, long (2240 lb) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . kilogram (kg) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ton, long, per cubic yard . . . . . . . . . . . . . . . . . . . . . . . . . . . . kilogram per cubic meter (kg/m3) . . . . . . . . . . . . . ton , metric (t). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . kilogram (kg) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . tonne (called ‘‘metric ton’’ in U.S.) (t) . . . . . . . . . . . . . kilogram (kg) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ton of refrigeration (12 000 Btu IT /h) . . . . . . . . . . . . . . . . watt (W) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ton of TNT (energy equivalent) 26 . . . . . . . . . . . . . . . . . . . joule (J). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ton, register . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . cubic meter (m3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ton, short (2000 lb) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . kilogram (kg) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ton, short, per cubic yard . . . . . . . . . . . . . . . . . . . . . . . . . . . kilogram per cubic meter (kg/m3) . . . . . . . . . . . . . ton, short, per hour. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . kilogram per second (kg/s) . . . . . . . . . . . . . . . . . . . . torr (Torr) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . pascal (Pa) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

25 The therm (EC) is legally defined in the Council Directive of 20 December 1979, Council of the European Communities (now the European Union, EU). The therm (U.S.) is legally defined in the Federal Register of July 27, 1968. Although the therm (EC), which is based on the International Table Btu, is frequently used by engineers in the United States, the therm (U.S.) is the legal unit used by the U.S. natural gas industry. 26 Defined (not measured) value.


CONVERSION OF TEMPERATURES From

To

Celsius

Fahrenheit Kelvin Rankine

tF/˚F = (9/5) t/˚C + 32 T/K = t/˚C + 273.15 T/˚R = (9/5) (t/˚C + 273.15)

Fahrenheit

Celsius Kelvin Rankine

t/˚C = (5/9) [(tF/˚F) - 32] T/K = (5/9) [(tF/˚F) - 32] + 273.15 T/˚R = tF/˚F + 459.67

Kelvin

Celsius Rankine

t/˚C = T/K - 273.15 T/˚R = (9/5) T/K

Rankine

Fahrenheit Kelvin

tF/˚F = T/˚R - 459.67 T/K = (5/9) T/˚R

Definition of symbols: T = thermodynamic (absolute) temperature t = Celsius temperature (the symbol q is also used for Celsius temperature) tF = Fahrenheit temperature

DESIGNATION OF LARGE NUMBERS 106 109 1012 1015 1018 10100 10googol

U.S.A. million billion trillion quadrillion quintillion googol googolplex

1-46

Other Countries million milliard billion billiard trillion

CONVERSION FACTORS FOR ENERGY UNITS If greater accuracy is required, use the Energy Equivalents section of the Fundamental Physical Constants table.

v: 1 cm-1 v: 1 MHz

Wavenumber v cm-1 ⬟1 ⬟ 3.33564 × 10-5

Frequency v MHz 2.997925 × 104 1

Energy E aJ 1.986447 × 10-5 6.626076 × 10-10

Energy E eV 1.239842 × 10-4 4.135669 × 10-9

Energy E Eh 4.556335 × 10-6 1.519830 × 10-10

Molar energy Em kJ/mol 11.96266 × 10-3 3.990313 × 10-7

Molar energy Em kcal/mol 2.85914 × 10-3 9.53708 × 10-8

Temperature T K 1.438769 4.79922 × 10-5

1 aJ E: 1 eV Eh

⬟ 50341.1 ⬟ 8065.54 ⬟ 219474.63

1.509189 × 109 2.417988 × 108 6.579684 × 109

1 0.1602177 4.359748

6.241506 1 27.2114

0.2293710 3.674931 × 10-2 1

602.2137 96.4853 2625.500

143.9325 23.0605 627.510

7.24292 × 104 1.16045 × 104 3.15773 × 105

Em: 1 kJ/mol 1 kcal/mol

⬟ 83.5935 ⬟ 349.755

2.506069 × 106 1.048539 × 107

1.660540 × 10-3 6.947700 × 10-3

1.036427 × 10-2 4.336411 × 10-2

3.808798 × 10-4 1.593601 × 10-3

1 4.184

0.239006 1

120.272 503.217

T: 1 K

⬟ 0.695039

2.08367 × 104

1.380658 × 10-5

8.61738 × 10-5

3.16683 × 10-6

8.31451 × 10-3

1.98722 × 10-3

1

Examples of the use of this table:

1 aJ ⬟ 50341 cm-1 1 eV ⬟96.4853 kJ mol-1

The symbol ⬟ should be read as meaning “corresponds to” or “is equivalent to”. E = hv = hcv = kT; Em = NAE ; Eh is the Hartree energy


CONVERSION FACTORS FOR PRESSURE UNITS Pa Pa kPa MPa bar atmos Torr µmHg psi

1 1000 1000000 100000 101325 133.322 0.133322 6894.757

kPa 0.001 1 1000 100 101.325 0.133322 0.000133322 6.894757

MPa 0.000001 0.001 1 0.1 0.101325 0.000133322 1.33322 × 10–7 0.006894757

bar 0.00001 0.01 10 1 1.01325 0.00133322 1.33322 × 10–6 0.06894757

atmos

Torr

µmHg

9.8692 × 10–6 0.0098692 9.8692 0.98692 1 0.00131579 1.31579 × 10–6 0.068046

0.0075006 7.5006 7500.6 750.06 760 1 0.001 51.7151

7.5006 7500.6 7500600 750060 760000 1000 1 51715.1

psi 0.0001450377 0.1450377 145.0377 14.50377 14.69594 0.01933672 1.933672 × 10–5 1

To convert a pressure value from a unit in the left hand column to a new unit, multiply the value by the factor appearing in the column for the new unit. For example: 1 kPa = 9.8692 × 10–3 atmos 1 Torr = 1.33322 × 10–4 MPa

1-34

Notes: µmHg is often referred to as “micron” Torr is essentially identical to mmHg psi is an abbreviation for the unit pound–force per square inch psia (as a term for a physical quantity) implies the true (absolute) pressure psig implies the true pressure minus the local atmospheric pressure

CONVERSION FACTORS FOR THERMAL CONDUCTIVITY UNITS MULTIPLY ↓ by appropriate factor to OBTAIN→

BtuIT h-1 ft-1 °F-1

BtuIT in. h-1 ft-2 °F-1

Btuth h-1 ft-1 °F-1

Btuth in. h-1 ft-2 °F-1

calIT s-1 cm-1 °C-1

calth s-1 cm-1 °C-1

kcalth h-1 m-1 °C-1

J s-1 cm-1 K-1

W cm-1 K-1

W m-1 K-1

mW cm-1 K-1

BtuIT h-1 ft-1 °F-1 BtuIT in. h-1 ft-2 °F-1 Btuth h-1 ft-1 °F-1 Btuth in. h-1 ft-2 °F-1 calIT s-1 cm-1 °C-1 calth s-1 cm-1 °C-1 kcalth h-1 m-1 °C-1 J s-1 cm-1 K-1 W cm-1 K-1 W m-1 K-1 mW cm-1 K-1

1 8.33333 × 10-2 0.999331 8.32776 × 10-2 2.41909 × 102 2.41747 × 102 0.671520 57.7789 57.7789 0.577789 5.77789 × 10-2

12 1 11.9920 0.999331 2.90291 × 103 2.90096 × 103 8.05824 6.93347 × 102 6.93347 × 102 6.93347 0.693347

1.00067 8.33891 × 10-2 1 8.33333 × 10-2 2.42071 × 102 2.41909 × 102 0.671969 57.8176 57.8176 0.578176 5.78176 × 10-2

12.0080 1.00067 12 1 2.90485 × 103 2.90291 × 103 8.06363 6.93811 × 102 6.93811 × 102 6.93811 0.693811

4.13379 × 10-3 3.44482 × 10-4 4.13102 × 10-3 3.44252 × 10-4 1 0.999331 2.77592 × 10-3 0.238846 0.238846 2.38846 × 10-3 2.38846 × 10-4

4.13656 × 10-3 3.44713 × 10-4 4.13379 × 10-3 3.44482 × 10-4 1.00067 1 2.77778 × 10-3 0.239006 0.239006 2.39006 × 10-3 2.39006 × 10-4

1.48916 0.124097 1.48816 0.124014 3.60241 × 102 3.6 × 102 1 86.0421 86.0421 0.860421 8.60421 × 10-2

1.73073 × 10-2 1.44228 × 10-3 1.72958 × 10-2 1.44131 × 10-3 4.1868 4.184 1.16222 × 10-2 1 1 1 × 10-2 1 × 10-3

1.73073 × 10-2 1.44228 × 10-3 1.72958 × 10-2 1.44131 × 10-3 4.1868 4.184 1.16222 × 10-2 1 1 1 × 10-2 1 × 10-3

1.73073 0.144228 1.72958 0.144131 4.1868 × 102 4.184 × 102 1.16222 1 × 102 1 × 102 1 0.1

17.3073 1.44228 17.2958 1.44131 4.1868 × 103 4.184 × 103 11.6222 1 × 103 1 × 103 10 1

© 2000 CRC Press LLC

CONVERSION FACTORS FOR ELECTRICAL RESISTIVITY UNITS To convert from multiply by appropriate factor to Obtain

↓

→

abohm centimeter microohm centimeter ohm centimeter statohm centimeter (esu) ohm meter ohm circular mil per foot ohm inch ohm foot

abΩ cm 1 103 108 8.987 × 1020 1011 1.662 × 102 2.54 × 109 3.048 × 1010

µΩ cm

Ω cm

1 × 10–3 1 106 8.987 × 1017 108 1.662 × 10–1 2.54 × 106 3.048 × 107

10–9 10–6 1 8.987 × 1011 102 1.662 × 10–7 2.54 3.048 × 10–1

StatΩ cm 1.113 × 10–21 1.113 × 10–18 1.113 × 10–12 1 1.113 × 10–10 1.850 × 10–19 2.827 × 10–12 3.3924 × 10–11

Ωm 10–11 10–8 1 × 10–2 8.987 × 109 1 1.662 × 10–9 2.54 × 10–2 3.048 × 10–1

Ω cir. mil ft–1 6.015 × 10–3 6.015 6.015 × 106 5.406 × 1018 6.015 × 108 1 1.528 × 107 1.833 × 108

Ω in. 3.937 × 10–10 3.937 × 10–7 3.937 × 10–1 3.538 × 1011 3.937 × 101 6.54 × 10–6 1 12

Ω ft 3.281 × 10–11 3.281 × 10–6 3.281 × 10–2 2.949 × 1010 3.281 5.45 × 10–9 8.3 × 10–2 1

1-50

CONVERSION FACTORS FOR IONIZING RADIATION CONVERSION BETWEEN SI AND OTHER UNITS

Expression in SI units

Special name for SI units

s–1 J kg–1 J kg–1 s–1

becquerel gray

1-39

Activity Absorbed dose Absorbed dose rate

A D D·

Average energy per ion pair Dose equivalent Dose equivalent rate

W

Electric current Electric potential difference Exposure Exposure rate

I U, V

Fluence Fluence rate

φ Φ

Kerma Kerma rate

K K·

Lineal energy

y

coulomb per kilogram coulomb per kilogram second 1 per meter squared 1 per meter squared second joule per kilogram joule per kilogram second joule per meter

Linear energy transfer

L

joule per meter

J m–1

Mass attenuation coefficient Mass energy transfer coefficient Mass energy absorption coefficient Mass stopping power

µ/ρ

m2 kg–1

S/ρ

Power Pressure

P p

Radiation chemical yield Specific energy

G

meter squared per kilogram meter squared per kilogram meter squared per kilogram joule meter squared per kilogram joule per second newton per meter squared mole per joule

mol J–1

z

joule per kilogram

J kg–1

H H·

X X·

µtr/ρ µen/ρ

1 per second joule per kilogram joule per kilogram second joule

Expression in symbols for SI units

joule per kilogram joule per kilogram second ampere watt per ampere

Symbols using special names Bq Gy Gy s–1

J J kg–1 J kg–1 s–1

Conventional units

Symbol for conventional unit

Value of conventional unit in SI units

curie rad rad

Ci rad rad s–1

3.7 × 1010 Bq 0.01 Gy 0.01 Gy s–1

electronvolt

eV

1.602 × 10–19 J

sievert

Sv Sv s–1

rem rem per second

rem rem s–1

0.01 Sv 0.01 Sv s–1

volt

V

ampere volt

A V

1.0 A 1.0 A

C kg–1 C kg–1 s–1

roentgen roentgen

R R s–1

2.58 × 10–4 C kg–1 2.58 × 10–4 C kg–1 s–1

m–2 m–2 s–1

cm–2 cm–2 s–1

1.0 × 104 m–2 1.0 × 104 m–2 s–1

Gy Gy s–1

1 per centimeter squared 1 per centimeter squared second rad rad per second

rad rad s–1

0.01 Gy 0.01 Gy s–1

keV µm–1

1.602 × 10–10 J m–1

keV µm–1

1.602 × 10–10 J m–1

cm2 g–1

0.1 m2 kg–1

cm2 g–1

0.1 m2 kg–1

cm2 g–1

0.1 m2 kg–1

MeV cm2 g–1

1.602 × 10–14 J m2 kg–1

W Pa

kiloelectron volt per micrometer kiloelectron volt per micrometer centimeter squared per gram centimeter squared per gram centimeter squared per gram MeV centimeter squared per gram watt torr

W torr

1.0 W (101325/760)Pa

molecules (100 eV)–1 rad

1.04 × 10–7 mol J–1

Gy

molecules per 100 electron volts rad

A W A–1

J kg–1 J kg–1 s–1

gray

J m–1

m2 kg–1 m2 kg–1 J m2 kg–1 J s–1 N m–2

watt pascal

gray

0.01 Gy

CONVERSION FACTORS FOR IONIZING RADIATION (continued)

Quantity

Symbol for quantity

CONVERSION FACTORS FOR IONIZING RADIATION (continued) CONVERSION OF RADIOACTIVITY UNITS FROM MBq TO mCi AND µCi MBq

mCi

MBq

7000 6000 5000 4000 3000 2000 1000 900 800 700 600

189. 162. 135. 108. 81. 54. 27. 24. 21.6 18.9 16.2

500 400 300 200 100 90 80 70 60 50 40

mCi 13.5 10.8 8.1 5.4 2.7 2.4 2.16 1.89 1.62 1.35 1.08

MBq

µCi

MBq

µCi

30 20 10 9 8 7 6 5 4 3 2

810 540 270 240 220 189 162 135 108 81 54

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

27 24 21.6 18.9 16.2 13.5 10.8 8.1 5.4 2.7

CONVERSION OF RADIOACTIVITY UNITS FROM mCi AND µCi TO MBq mCi

MBq

mCi

MBq

200 150 100 90 80 70 60 50 40 30 20

7400 5550 3700 3330 2960 2590 2220 1850 1480 1110 740

10 9 8 7 6 5 4 3 2 1

370 333 296 259 222 185 148 111 74.0 37.0

µCi

MBq

µCi

MBq

1000 900 800 700 600 500 400 300 200 100 90

37.0 33.3 29.6 25.9 22.2 18.5 14.8 11.1 7.4 3.7 3.33

80 70 60 50 40 30 20 10 5 2 1

2.96 2.59 2.22 1.85 1.48 1.11 0.74 0.37 0.185 0.074 0.037

CONVERSION OF RADIOACTIVITY UNITS 100 TBq (1014 Bq) 10 TBq (1013 Bq) 1 TBq (1012 Bq) 100 GBq (1011 Bq) 10 GBq (1010 Bq) 1 GBq (109 Bq) 100 MBq (108 Bq) 10 MBq (107 Bq) 1 MBq (106 Bq)

= = = = = = = = =

2.7 kCi (2.7 × 103 Ci) 270 Ci (2.7 × 102 Ci) 27 Ci (2.7 × 101 Ci) 2.7 Ci (2.7 × 100 Ci) 270 mCi (2.7 × 10–1 Ci) 27 mCi (2.7 × 10–2 Ci) 2.7 mCi (2.7 × 10–3 Ci) 270 µCi (2.7 × 10–4 Ci) 27 µCi (2.7 × 10–5 Ci)

100 kBq (105 Bq) 10 kBq (104 Bq) 1 kBq (103 Bq) 100 Bq (102 Bq) 10 Bq (101 Bq) 1 Bq (100 Bq) 100 mBq (10–1 Bq) 10 mBq (10–2 Bq) 1 mBq (10–3 Bq)

CONVERSION OF ABSORBED DOSE UNITS SI Units 100 Gy (102 Gy) 10 Gy (101 Gy) 1 Gy (100 Gy) 100 mGy (10–1 Gy) 10 mGy (10–2 Gy) 1 mGy (10–3 Gy) 100 µGy (10–4 Gy) 10 µGy (10–5 Gy) 1 µGy (10–6 Gy) 100 nGy (10–7 Gy) 10 nGy (10–8 Gy) 1 nGy (10–9 Gy)

CONVERSION OF DOSE EQUIVALENT UNITS 100 Sv (102 Sv) 10 Sv (101 Sv) 1 Sv (100 Sv) 100 mSv (10–1 Sv) 10 mSv (10–2 Sv) 1 mSv (10–3 Sv) 100 µSv (10–4 Sv) 10 µSv (10–5 Sv) 1 µSv (10–6 Sv) 100 nSv (10–7 Sv) 10 nSv (10–8 Sv) 1 nSv (10–9 Sv)

Conventional = = = = = = = = = = = =

10,000 rad (104 rad) 1,000 rad (103 rad) 100 rad (102 rad) 10 rad (101 rad) 1 rad (100 rad) 100 mrad (10–1 rad) 10 mrad (10–2 rad) 1 mrad (10–3 rad) 100 µrad (10–4 rad) 10 µrad (10–5 rad) 1 µrad (10–6 rad) 100 nrad (10–7 rad)

2.7 µCi (2.7 × 10–6Ci) 270 nCi (2.7 × 10–7 Ci) 27 nCi (2.7 × 10–8 Ci) 2.7 nCi (2.7 × 10–9 Ci) 270 pCi (2.7 × 10–10 Ci) 27 pCi (2.7 × 10–11 Ci) 2.7 pCi (2.7 × 10–12 Ci) 270 fCi (2.7 × 10–13 Ci) 27 fCi (2.7 × 10–14 Ci)

= = = = = = = = =

1-40

= = = = = = = = = = = =

10,000 rem (104 rem) 1,000 rem (103 rem) 100 rem (102 rem) 10 rem (101 rem) 1 rem (100 rem) 100 mrem (10–1 rem) 10 mrem (10–2 rem) 1 mrem (10–3 rem) 100 µrem (10–4 rem) 10 µrem (10–5 rem) 1 µrem (10–6 rem) 100 nrem (10–7 rem)

VALUES OF THE GAS CONSTANT IN DIFFERENT UNIT SYSTEMS In SI units the value of the gas constant, R, is: R =8.314510 Pa m3 K-1 mol-1 = 8314.510 Pa L K-1 mol-1 = 0.08314510 bar L K-1 mol-1 This table gives the appropriate value of R for use in the ideal gas equation, PV = nRT, when the variables are expressed in other units. The following conversion factors for pressure units were used in generating the table: 1 atm = 101325 Pa 1 psi = 6894.757 Pa 1 torr (mmHg) = 133.322 Pa [at 0°C] 1 in Hg = 3386.38 Pa [at 0°C] 1 in H2O = 249.082 Pa [at 4°C] 1 ft H2O = 2988.98 Pa [at 4°C] The advice of Prabir K. Chandra is appreciated.

V

Units of V, T, n T n

ft3

K °R

cm3

K °R

L

K °R

m3

K °R

mol lb⋅mol mol lb⋅mol mol lb⋅mol mol lb⋅mol mol lb⋅mol mol lb⋅mol mol lb⋅mol mol lb⋅mol

kPa 0.2936241 133.1857 0.1631245 73.99204 8314.510 3771398 4619.172 2095221 8.314510 3771.398 4.619172 2095.221 0.008314510 3.771398 0.004619172 2.095221

atm

psi

0.00289785 1.31444 0.00160991 0.730245 82.0578 37220.8 45.5877 20678.2 0.0820578 37.2208 0.0455877 20.6782 0.0000820578 0.0372208 0.0000455877 0.0206782

0.0425866 19.3169 0.0236592 10.7316 1205.92 546995 669.954 303886 1.20592 546.995 0.669954 303.886 0.00120592 0.546995 0.000669954 0.303886

1-40

Units of P mmHg 2.20237 998.978 1.22354 554.987 62364.1 28287900 34646.7 15715500 62.3641 28287.9 34.6467 15715.5 0.0623641 28.2879 0.0346467 15.7155

in Hg

in H2O

ft H2O

0.0867074 39.3298 0.0481708 21.8499 2455.28 1113700 1364.04 618720 2.45528 1113.70 1.36404 618.720 0.00245528 1.11370 0.00136404 0.618720

1.17882 534.706 0.654903 297.059 33380.6 15141200 18544.8 8411770 33.3806 15141.2 18.5448 8411.77 0.0333806 15.1412 0.0185448 8.41177

0.0982355 44.5589 0.0545753 24.7549 2781.72 1261770 1545.40 700982 2.78172 1261.77 1.54540 700.982 0.00278172 1.26177 0.00154540 0.700982

PERIODIC TABLE OF THE ELEMENTS

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dx 1 xc , ¼ log x2 c2 2c xþc

ðx2 > c2 Þ

x dx 1 ¼ log ðc2 x2 Þ c2 x2 2 x dx 1 ¼ 2nðc2 x2 Þn ðc2 x2 Þnþ1

Z

Z dx 1 x dx ¼ þ ð2n 3Þ n ðc2 x2 Þ 2c2 ðn 1Þ ðc2 x2 Þn1 ðc2 x2 Þn1

Z

Z dx 1 x dx ¼ ð2n 3Þ ðx2 c2 Þn 2c2 ðn 1Þ ðx2 c2 Þn1 ðx2 c2 Þn1

48.

49. Z 50.

dx 1 x ¼ tan1 c2 þ x2 c c

x dx 1 ¼ log ðx2 c2 Þ x2 c2 2

A-22

n 6¼ 1, 2, 3

INTEGRALS (Continued)

Z 51.

x dx 1 ¼ 2n ðx2 c2 Þn ðx2 c2 Þnþ1 FORMS CONTAINING a þ bx and c þ dx u ¼ a þ bx, v ¼ c þ dx, k ¼ ad bc

52. 53. 54. 55. 56. 57. 58.

59.

c If k ¼ 0, then v ¼ u a Z v dx 1 ¼ log u v k u Z h i x dx 1 a c ¼ log ðuÞ log ðvÞ u v k b d Z dx 1 1 d v þ log ¼ u2 v k u k u Z x dx a c v log ¼ u2 v bku k2 u

Z 2 x dx a2 1 c2 aðk bcÞ ¼ þ log ðvÞ þ log ðuÞ u2 v b2 ku k2 d b2

Z Z dx 1 1 dx ¼ ðm þ n 2Þb un vm kðm 1Þ un1 vm1 un vm1 Z u bx k dx ¼ þ 2 log ðvÞ v d d 8 mþ1

Z m 1 u u > > > þ bðn m 2Þ dx > n1 n1 > kðn 1Þ v v > > > > or > >

> Z Z m 1 um um1 u dx < þ mk dx ¼ dðn m 1Þ vn1 vn > vn > > > or > > > m > Z m1

> > 1 u u > > mb dx : n1 dðn 1Þ v vn1

FORMS CONTAINING (a þ bxn) pffiffiffiffiffi dx 1 1 x ab p ffiffiffiffiffi , ðab > 0Þ ¼ tan a þ bx2 a ab 8 pffiffiffiffiffiffiffiffiffi > 1 a þ x ab > > pffiffiffiffiffiffiffiffiffi log p ffiffiffiffiffiffiffiffiffi , ðab < 0Þ > > > Z a x ab < 2 ab dx or 61. ¼ a þ bx2 > pffiffiffiffiffiffiffiffiffi > > > 1 x ab > > , ðab < 0Þ : pffiffiffiffiffiffiffiffiffi tanh1 a ab Z dx 1 bx 62. ¼ tan1 a2 þ b2 x2 ab a Z x dx 1 63. ¼ logða þ bx2 Þ a þ bx2 2b Z Z x2 dx x a dx 64. ¼ 2 a þ bx b b a þ bx2 Z Z dx x 1 dx þ 65. ¼ ða þ bx2 Þ2 2aða þ bx2 Þ 2a a þ bx2 Z

60.

A-23


Z 66.

Z 67.

Z 68. Z 69. Z 70. Z 71.

Z 72.

Z 73.

dx ða þ bx2 Þmþ1

Z 8 1 x 2m 1 dx > > þ m > 2 > 2ma ða þ bx Þ 2ma ða þ bx2 Þm > > < or ¼ " # > Z > m > > ð2mÞ! x X r!ðr 1Þ! 1 dx > > þ : mr r ð4aÞm a þ bx2 ðm!Þ2 2a r¼1 ð4aÞ ð2rÞ!ða þ bx2 Þ

x dx 1 ¼ 2bmða þ bx2 Þm ða þ bx2 Þmþ1 x2 dx x 1 ¼ mþ 2mb ða þ bx2 Þmþ1 2mbða þ bx2 Þ

Z

dx ða þ bx2 Þm

dx 1 x2 ¼ log 2 xða þ bx Þ 2a a þ bx2 dx 1 b ¼ x2 ða þ bx2 Þ ax a

dx xða þ bx2 Þmþ1

Z

dx a þ bx2

Z 8 1 1 dx > > þ m > 2 > 2amða þ bx a xða þ bx2 Þm Þ > > < or ¼ " # > > m > X > 1 ar x2 > > þ log : mþ1 r 2a rða þ bx2 Þ a þ bx2 r¼1

dx 1 ¼ x2 ða þ bx2 Þmþ1 a

Z

dx b x2 ða þ bx2 Þm a

Z

dx ða þ bx2 Þmþ1

Z

dx k 1 ðk þ xÞ3 pffiffiffi 2x k log ¼ þ 3 tan1 pffiffiffi , 3 3 a þ bx 3a 2 a þ bx k 3

Z

x dx 1 1 a þ bx3 pffiffiffi 2x k log ¼ þ 3 tan1 pffiffiffi , 3 3 a þ bx 3bk 2 k 3 ðk þ xÞ

74.

75. Z 76. Z 77. Z 78. Z 79. Z 80. Z 81.

dx 1 a þ bx log ¼ a2 b2 x2 2ab a bx

rffiffiffi 3 a k¼ b k¼

rffiffiffi 3 a b

x2 dx 1 ¼ logða þ bx3 Þ a þ bx3 3b

dx k 1 x2 þ 2kx þ 2k2 2kx 1 log ¼ þ tan , a þ bx4 2a 2 2k2 x2 x2 2kx þ 2k2

dx k 1 xþk 1 x log þ tan , ¼ a þ bx4 2a 2 xk k x dx 1 x2 tan1 , ¼ 4 a þ bx 2bk k x dx 1 x2 k log 2 , ¼ a þ bx4 4bk x þk

ab < 0, k ¼

ab > 0, k ¼

rffiffiffiffiffi 4 a 4b

rffiffiffiffiffiffiffi a 4 b

rffiffiffi a ab > 0, k ¼ b ab < 0, k ¼

rffiffiffiffiffiffiffi a b

x2 dx 1 1 x2 2kx þ 2k2 2kx 1 log ¼ þ tan , a þ bx4 4bk 2 2k2 x2 x2 þ 2kx þ 2k2

A-24

rffiffiffiffiffi 4 a ab > 0, k ¼ 4b


Z 82. Z 83. Z 84. Z 85. Z 86.

x3 dx 1 ¼ logða þ bx4 Þ a þ bx4 4b dx 1 xn ¼ log xða þ bxn Þ an a þ bxn dx 1 ¼ ða þ bxn Þmþ1 a xm dx 1 ¼ ða þ bxn Þpþ1 b

Z 87.

rffiffiffiffiffiffiffi a 4 ab < 0, k ¼ b

x2 dx 1 xk 1 x log þ 2 tan ¼ , a þ bx4 4bk xþk k

xm ða

Z

Z

dx 1 ¼ þ bxn Þpþ1 a

dx b ða þ bxn Þm a xmn dx a ða þ bxn Þp b Z xm ða

Z

Z

xn dx ða þ bxn Þmþ1 xmn dx ða þ bxn Þpþ1

dx b þ bxn Þp a

Z

dx xmn ða þ bxn Þpþ1

Z 8 1 > mn n p mnþ1 n pþ1 > x ða þ bx Þ aðm n þ 1Þ x ða þ bx Þ dx > > bðnp þ m þ 1Þ > > > > > > > > > or > > > > > >

> Z > > 1 > > xmþ1 ða þ bxn Þp þ anp xm ða þ bxn Þp1 dx > > > > np þ m þ 1 > Z < m n p 88. x ða þ bx Þ dx ¼ or > > > > >

Z > > 1 > > xmþ1 ða þ bxn Þpþ1 ðm þ 1 þ np þ nÞb xmþn ða þ bxn Þp dx > > > aðm þ 1Þ > > > > > > > or > > > > > >

Z > > 1 > > : xmþ1 ða þ bxn Þpþ1 þ ðm þ 1 þ np þ nÞ xm ða þ bxn Þpþ1 dx anðp þ 1Þ

FORMS CONTAINING c3 x3 Z 89.

dx 1 ðc xÞ3 1 2x c ¼ 2 log 3 þ pffiffiffi tan1 pffiffiffi c3 x3 6c c x3 c2 3 c 3

Z 90.

ðc3 Z

91. Z 92. Z 93.

dx x 2 þ ¼ x3 Þ2 3c3 ðc3 x3 Þ 3c3

Z

dx c3 x3

Z dx 1 x dx ¼ þ ð3n 1Þ n ðc3 x3 Þn ðc3 x3 Þnþ1 3nc3 ðc3 x3 Þ x dx 1 c3 x3 1 2x c ¼ log pffiffiffi tan1 pffiffiffi c3 x3 6c c 3 ðc xÞ3 c 3 x dx x2 1 þ ¼ ðc3 x3 Þ2 3c3 ðc3 x3 Þ 3c3

Z

x dx c3 x3

A-25


Z 94. Z 95.

Z x dx 1 x2 x dx ¼ þ ð3n 2Þ n ðc3 x3 Þn ðc3 x3 Þnþ1 3nc3 ðc3 x3 Þ x2 dx 1 ¼ logðc3 x3 Þ c3 x3 3

Z 96.

ðc3 Z

97.

x2 dx 1 ¼ 3nðc3 x3 Þn x3 Þnþ1 dx 1 x3 ¼ 3 log 3 3 x Þ 3c c x3

xðc3 Z

dx 1 1 x3 þ ¼ log 3 c x3 xðc3 x3 Þ2 3c3 ðc3 x3 Þ 3c6 Z Z dx 1 1 dx 99. ¼ n þ 3 nþ1 3 3 3 3 3 3nc ðc x Þ c xðc3 x3 Þn xðc x Þ Z Z dx 1 1 x dx ¼ 100. x2 ðc3 x3 Þ c3 x c3 c3 x3 Z Z Z dx 1 dx 1 x dx 101. ¼ n c3 ðc3 x3 Þnþ1 x2 ðc3 x3 Þnþ1 c3 x2 ðc3 x3 Þ 98.

FORMS CONTAINING c4 x4 " pffiffiffi pffiffiffi # dx 1 1 x2 þ cx 2 þ c2 1 cx 2 pffiffiffi log pffiffiffi 102. ¼ þ tan c4 þ x4 2c3 2 2 c2 x2 x2 cx 2 þ c2

Z dx 1 1 cþx x þ tan1 ¼ 3 log 103. 4 4 c x 2c 2 cx c Z 2 x dx 1 x ¼ tan1 2 104. c4 þ x4 2c2 c Z x dx 1 c2 þ x2 105. ¼ log 2 c4 x4 4c2 c x2 " pffiffiffi pffiffiffi # Z x2 dx 1 1 x2 cx 2 þ c2 1 cx 2 pffiffiffi 106. ¼ pffiffiffi log þ tan c4 þ x4 2c 2 2 c2 x2 x2 þ cx 2 þ c2 Z

Z 107. Z 108.

x2 dx 1 1 cþx 1 x log tan ¼ c4 x4 2c 2 cx c x3 dx 1 ¼ log ðc4 x4 Þ c4 x4 4 FORMS CONTAINING (a þ bx þ cx2) X ¼ a þ bx þ cx2 and q ¼ 4ac b2

b 2 If q ¼ 0, then X ¼ c x þ , and formulas starting with 23 should 2c be used in place of these. Z dx 2 2cx þ b ¼ pffiffiffi tan1 pffiffiffi , 109. ðq > 0Þ X q q 8 2 1 2cx þ b > > ffi tanh pffiffiffiffiffiffiffi > > pffiffiffiffiffiffi Z q q < dx or ¼ 110. pffiffiffiffiffiffiffi > X > 2cx þ b q 1 > > : pffiffiffiffiffiffiffi log ðq < 0Þ pffiffiffiffiffiffiffi , q 2cx þ b þ q Z Z dx 2cx þ b 2c dx 111. þ ¼ X2 qX q X

A-26


Z 112.

Z dx 2cx þ b 1 3c 6c2 dx ¼ þ þ 2 3 2 X q 2X qX X q

Z 8 2cx þ b 2ð2n 1Þc dx > > þ > n > nqX qn Xn > > Z < dx or ¼ 113. # X nþ1 > " Z > n > > ð2nÞ! c n 2cx þ b X q r ðr 1Þ!r! dx > > þ : q cX ð2rÞ! X ðn!Þ2 q r¼1 Z

Z x dx 1 b dx ¼ log X X 2c 2c X Z Z x dx bx þ 2a b dx 115. ¼ 2 X qX q X Z Z x dx 2a þ bx bð2n 1Þ dx 116. ¼ nþ1 n X nqX nq Xn 114.

Z 117. Z 118. Z 119.

Z 120. Z 121. Z 122.

x2 x b b2 2ac dx ¼ 2 log X þ c 2c 2c2 X x2 ðb2 2acÞx þ ab 2a þ dx ¼ cqX q X2

dx X

dx 1 x2 b ¼ log xX 2a X 2a

Z

Z

xm1 dx X nþ1

dx X

2 Z dx b X 1 b c dx ¼ 2 log 2 þ 2 2 x X 2a x ax a X 2a dx 1 b ¼ xX n 2aðn 1ÞX n1 2a dx xm X nþ1

Z

dx 1 þ Xn a

Z

dx xX n1

1 nþm1 b

ðm 1Þaxm1 X n m1 a Z 2n þ m 1 c dx

m1 a xm2 X nþ1

¼

FORMS CONTAINING 124.

dx X

xm dx xm1 nmþ1 b

¼ nþ1 X ð2n m þ 1ÞcX n 2n m þ 1 c Z m2 m1 a x dx

þ 2n m þ 1 c X nþ1

Z 123.

Z

Z

Z

dx xm1 X nþ1

pffiffiffiffiffiffiffiffiffiffiffiffiffiffi a þ bx

Z pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ða þ bxÞ3 a þ bx dx ¼ 3b Z

125.

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ð2a 3bxÞ ða þ bxÞ3 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x a þ bx dx ¼ 15b2

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ð8a2 12abx þ 15b2 x2 Þ ða þ bxÞ3 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 126. x a þ bx dx ¼ 105b3 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Z 8 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 3 > m m1 > x ða þ bxÞ ma x a þ bx dx > > bð2m þ 3Þ > > Z < pffiffiffiffiffiffiffiffiffiffiffiffiffiffi m or 127. x a þ bx dx ¼ > > m > > 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi X m!ðaÞmr > > ða þ bxÞrþ1 : mþ1 a þ bx b r!ðm rÞ!ð2r þ 3Þ r¼0 Z

2

A-27


Z Z pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi dx a þ bx pffiffiffiffiffiffiffiffiffiffiffiffiffiffi dx ¼ 2 a þ bx þ a 128. x x a þ bx 129.

pffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z Z pffiffiffiffiffiffiffiffiffiffiffiffiffiffi a þ bx a þ bx b dx pffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ dx ¼ 2 x x 2 x a þ bx

2qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 Z pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ða þ bxÞ3 ð2m 5Þb Z a þ bx 1 a þ bx 4 dx ¼ þ dx5 130. ðm 1Þa 2 xm xm1 xm1 Z

pffiffiffiffiffiffiffiffiffiffiffiffiffiffi dx 2 a þ bx pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ b a þ bx

Z

x dx 2ð2a bxÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ a þ bx 3b2 a þ bx

Z

x2 dx 2ð8a2 4abx 3b2 x2 Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ a þ bx 15b3 a þ bx

131.

132.

133.

8 Z m1

pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 x dx > m > pffiffiffiffiffiffiffiffiffiffiffiffiffiffi x a þ bx ma > > > ð2m þ 1Þb a þ bx > > Z < xm dx or pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 134. a þ bx > > pffiffiffiffiffiffiffiffiffiffiffiffiffiffi m > > > 2ðaÞm a þ bx X ð1Þr m!ða þ bxÞr > > : mþ1 b ð2r þ 1Þr!ðm rÞ!ar r¼0 Z 135. Z 136. Z 137.

pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi dx 1 a þ bx a pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ pffiffiffi log pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi , a x a þ bx a þ bx þ a dx 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ pffiffiffiffiffiffiffi tan1 a x a þ bx dx pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ x2 a þ bx

rffiffiffiffiffiffiffiffiffiffiffiffiffiffi a þ bx , a

ða > 0Þ

ða < 0Þ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z dx a þ bx b pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2a x a þ bx ax

pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8 Z ð2n 3Þb dx a þ bx > > pffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > n1 > ð2n 2Þa xn1 a þ bx ðn 1Þax > > > > > " pffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > n1 Z < ð2n 2Þ! r!ðr 1Þ! b nr1 a þ bx X dx p ffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ¼ 138. xr 2ðrÞ! 4a a ½ðn 1Þ! 2 xn a þ bx > r¼1 > > > > > # > Z > > > b n1 dx > > p ffiffiffiffiffiffiffiffiffiffiffiffiffi ffi þ : 4a x a þ bx Z 139. Z

n2

xða þ bxÞ

140. Z 141.

2n

n

ða þ bxÞ2 dx ¼

2ða þ bxÞ 2 bð2 nÞ

" # 4n 2n 2 ða þ bxÞ 2 aða þ bxÞ 2 dx ¼ 2 b 4n 2n

dx 1 m ¼ a xða þ bxÞ2

Z

dx xða þ bxÞ

m2 2

b a

A-28

Z

dx m

ða þ bxÞ 2


Z 142. Z 143.

ða þ bxÞn=2 dx ¼b x

Z

ða þ bxÞ ðn2Þ =2 dx þ a

Z

ða þ bxÞ ðn2Þ =2 dx x

Z 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 z a f ,z z dz, ðz ¼ a þ bxÞ f ðx, a þ bxÞ dx ¼ b b

FORMS CONTAINING u ¼ a þ bx

pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi a þ bx and c þ dx

v ¼ c þ dx

k ¼ ad bc

c If k ¼ 0, then, v ¼ u, and formulas starting with 124 should a be used in place of these. 8 pffiffiffiffiffiffiffiffiffiffi > bduv 2 > 1 > p ffiffiffiffiffi ffi , bd > 0, k < 0 tanh > > bv > bd > > > > > > or > > > > pffiffiffiffiffiffiffiffiffiffi Z < dx 2 bduv pffiffiffiffiffi ¼ 144. pffiffiffiffiffiffi tanh1 , bd > 0, k > 0: uv > du bd > > > > > > or > > > > pffiffiffiffiffiffiffiffiffiffi > > > 1 ðbv þ bduvÞ2 > > , ðbd > 0Þ : pffiffiffiffiffiffi log v bd 8 pffiffiffiffiffiffiffiffiffiffiffiffiffi > 2 bduv > 1 > p ffiffiffiffiffiffiffiffiffi tan > > Z < bd bv dx pffiffiffiffiffi ¼ 145. or uv > > > 1 2bdx þ ad þ bc > pffiffiffiffiffiffiffiffiffi > , sin1 : jkj bd Z 146.

pffiffiffiffiffi k þ 2bvpffiffiffiffiffi k2 uv dx ¼ uv 4bd 8bd

Z

dx pffiffiffiffiffi uv

8 pffiffiffi pffiffiffiffiffiffi 1 d u kd > > pffiffiffiffiffiffi log pffiffiffi pffiffiffiffiffiffi > > > d u þ kd kd > > Z < dx pffiffiffi ¼ 147. or v u > > > pffiffiffi pffiffiffiffiffiffi > > 1 > ðd u kd Þ2 > : pffiffiffiffiffiffi log , v kd Z 148. Z

pffiffiffiffiffi Z uv ad þ bc x dx dx pffiffiffiffiffi ¼ pffiffiffiffiffi bd uv 2bd uv

Z

pffiffiffiffiffi dx 2 uv pffiffiffiffiffi ¼ kv v uv

Z

pffiffiffiffiffi Z uv k v dx dx pffiffiffiffiffi ¼ pffiffiffiffiffi b 2b uv uv

149.

150.

151.

152.

ðkd < 0Þ

Z Z rffiffiffi v v v dx pffiffiffiffiffi dx ¼ u jvj uv Z

153.

pffiffiffi dx 2 d u pffiffiffi ¼ pffiffiffiffiffiffiffiffiffi tan1 pffiffiffiffiffiffiffiffiffi , v u kd kd

ðkd > 0Þ

pffiffiffi vm u dx ¼

Z m pffiffiffi 1 v dx pffiffiffi 2vmþ1 u þ k ð2m þ 3Þd u

A-29

ðbd < 0Þ


Z 154.

Z pffiffiffi dx 1 3 dx u pffiffiffi ¼ pffiffiffi þ m b m1 m1 ðm 1Þk v 2 v u u

vm

8 Z m1

> 2 v > m pffiffiffi > pffiffiffi dx v u mk > > bð2m þ 1Þ u > > > > Z m < v dx or pffiffiffi ¼ 155. > u > > pffiffiffi m > > > 2ðm!Þ2 u X 4k mr ð2rÞ! r > > v > : bð2m þ 1Þ! b ðr!Þ2 r¼0 FORMS CONTAINING 156.

Z pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 a2 dx ¼ 12½x x2 a2 a2 log ðx þ x2 a2 Þ Z

157. Z 158. Z 159.

160.

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 a2

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dx pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ log ðx þ x2 a2 Þ 2 2 x a dx 1 x pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ sec1 a x x2 a2 jaj dx 1 aþ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ log a x x2 þ a2

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! x2 þ a2 x

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! Z pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a þ x2 þ a2 x2 þ a2 dx ¼ x2 þ a2 a log x x

Z pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 a2 x dx ¼ x2 a2 jaj sec1 161. a x Z pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x dx pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ x2 a2 162. x2 a2 Z pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 163. x x2 a2 dx ¼ 13 ðx2 a2 Þ3 164.

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 3a2 x pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3a4 x2 a2 þ logðx þ x2 a2 Þ ðx2 a2 Þ3 dx ¼ x ðx2 a2 Þ3 4 2 2 Z

165. Z 166.

x dx 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 a2 3 x ðx2 a2 Þ

Z

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x ðx2 a2 Þ3 dx ¼ 15 ðx2 a2 Þ5

Z

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x a2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a4 ðx2 a2 Þ3 x x2 a2 log ðx þ x2 a2 Þ x2 x2 a2 dx ¼ 4 8 8

Z

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 a Þ ða2 þ x2 Þ3 x3 x2 þ a2 dx ¼ ð15x2 15

Z

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 a2 ðx2 a2 Þ5 þ ðx2 a2 Þ3 x3 x2 a2 dx ¼ 5 3

Z

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 dx x pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ x2 a2 log ðx þ x2 a2 Þ 2 2 2 2 x a

167.

168.

169.

170.

171.

dx x qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 ðx2 a2 Þ3 a x a

A-30


Z 172.

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x3 dx 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ ðx2 a2 Þ3 a2 x2 a2 2 2 3 x a

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 a2 dx pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 173. a2 x x2 x2 a2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z dx 1 a þ x2 þ a2 x2 þ a2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 174. þ log 2a3 2a2 x2 x x3 x2 þ a2 Z

Z 175.

dx pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ x3 x2 a2

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 a2 1 x þ 3 sec1 2a2 x2 2ja j a

Z

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x a2 x a4 x pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðx2 a2 Þ5 ðx2 a2 Þ3 x2 a2 x2 ðx2 a2 Þ3 dx ¼ 6 24 16 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a6 log ðx þ x2 a2 Þ 16

Z

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 a2 ðx2 a2 Þ7 ðx2 a2 Þ5 x3 ðx2 a2 Þ3 dx ¼ 7 5

176.

177.

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 a2 dx x2 a2 þ log ðx þ x2 a2 Þ ¼ 178. x2 x 179.

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 a þ x2 þ a2 x2 þ a2 x2 þ a2 log dx ¼ 2a x3 2x2 x

180.

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 a2 x2 a2 1 x sec1 dx ¼ þ 3 x 2x2 2jaj a

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðx2 a2 Þ3 x2 a2 dx ¼ 181. 4 x 3a2 x3 Z

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 dx x qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ log ðx þ x2 a2 Þ 2 2 3 x a ðx2 a2 Þ

Z

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x3 dx a2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ x2 a2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 x a2 ðx2 a2 Þ3

182.

183. Z 184. Z 185. Z 186. Z 187. Z 188. Z 189.

dx 1 1 aþ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 log 2 x2 þ a2 a 3 a 2 2 x ðx þ a Þ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 þ a2 x

dx 1 1 x qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 sec1 2 x2 a2 ja j a 3 a 2 2 x ðx a Þ dx 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 4 a 3 x2 ðx2 a2 Þ

"pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi # x2 a2 x þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x x2 a2

dx 1 3 3 aþ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ log 2a2 x2 x2 þ a2 2a4 x2 þ a2 2a5 x3 ðx2 þ a2 Þ3 dx 1 3 3 x pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5 sec1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 2 x2 x2 a2 4 x2 a2 2ja a j 3 2a 2a x3 ðx2 a2 Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m 1 xm 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dx ¼ xm1 x2 a2 a2 2 2 m m x a

A-31

Z

xm2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dx x2 a2

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 þ a2 x


Z 190.

Z 191. Z 192. Z 193. Z 194.

Z 195. Z 196.

" m X x2m ð2mÞ! pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r!ðr 1Þ! pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dx ¼ ða2 Þmr ð2xÞ2r1 x2 a2 2 ð2rÞ! 22m ðm!Þ x2 a2 r¼1 # pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ ð a2 Þm log ðx þ x2 a2 Þ m pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X x2mþ1 ð2rÞ!ðm!Þ2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dx ¼ x2 a2 ð4a2 Þmr x2r 2 x2 a2 r¼0 ð2m þ 1Þ!ðr!Þ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z x2 a2 dx ðm 2Þ dx pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðm 1Þa2 xm1 ðm 1Þa2 xm2 x2 a2 xm x2 a2 1 X pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m dx ðm 1Þ!m!ð2rÞ!22m2r1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ x2 a2 2 2 mr x2rþ1 x2m x2 a2 r¼0 ðr!Þ ð2mÞ!ða Þ

dx ð2mÞ! pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ x2mþ1 x2 þ a2 ðm!Þ2

dx ð2mÞ! pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ x2mþ1 x2 a2 ðm!Þ2

"pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m r!ðr 1Þ! x2 þ a2 X ð1Þmrþ1 2 Þmr x2r 2ð2rÞ!ð4a a2 r¼1 # pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1Þmþ1 x2 þ a2 þ a þ 2m 2mþ1 log 2 a x "pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi # m x2 a2 X r!ðr 1Þ! 1 1 x þ sec mr a2 2ð2rÞ!ð4a2 Þ x2r 22m jaj2mþ1 a r¼1

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 a2 dx pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 2 2 aðx aÞ ðx aÞ x a

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 a2 dx pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 2 2 aðx þ aÞ ðx þ aÞ x a Z Z pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 198. f ðx, x2 þ a2 Þ dx ¼ a f ða tan u, a sec uÞ sec2 u du, Z

197.

Z f ðx,

199.

Z pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 a2 Þ dx ¼ a f ða sec u, a tan uÞ sec u tan u du,

FORMS CONTAINING

200.

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 x2

Z pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x a2 x2 dx ¼ x a2 x2 þ a2 sin1 2 jaj Z

201.

Z 202.

8 1 x > > > sin jaj < dx pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ or a2 x2 > > > cos1 x : jaj dx 1 aþ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ log a x a2 x2

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! a2 x2 x

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! Z pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 x2 a þ a2 x2 2 2 dx ¼ a x a log 203. x x Z

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x dx pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ a2 x2 a2 x2 Z pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 205. x a2 x2 dx ¼ 13 ða2 x2 Þ3

204.

A-32

x u ¼ tan1 , a > 0 a x u ¼ sec1 , a > 0 a


qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Z qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 3a2 x pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3a4 1 x a2 x2 þ sin 206. ða2 x2 Þ3 dx ¼ x ða2 x2 Þ3 þ 4 2 jaj 2 Z dx x qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 207. 2 2 2 ða2 x2 Þ3 a a x Z 208. Z 209.

x dx 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 x2 3 a 2 2 ða x Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x ða2 x2 Þ3 dx ¼ 15 ða2 x2 Þ5

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x a2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x ða2 x2 Þ3 þ x a2 x2 þ a2 sin1 x2 a2 x2 dx ¼ 4 jaj 8 Z pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 211. x3 a2 x2 dx ¼ ð15x2 15 a Þ ða2 x2 Þ3 Z

210.

Z

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 a2 x a4 x pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a6 x ða2 x2 Þ3 þ a2 x2 þ sin1 x2 ða2 x2 Þ3 dx ¼ x ða2 x2 Þ5 þ 6 24 16 jaj 16

Z

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 a2 ða2 x2 Þ7 ða2 x2 Þ5 x3 ða2 x2 Þ3 dx ¼ 7 5

Z

x2 dx x pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 x pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ a2 x2 þ sin1 2 jaj 2 a2 x2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 x2 dx pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ a2 x x2 a2 x2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 x2 a2 x2 x sin1 dx ¼ x2 x jaj pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 x2 a2 x2 1 a þ a2 x2 log dx ¼ þ x3 2x2 x 2a qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ða2 x2 Þ3 a2 x2 dx ¼ 4 3a2 x3 x

212.

213.

214. Z 215. Z 216. Z 217. Z 218. Z 219.

x2 dx x x qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sin1 2 x2 jaj 3 a 2 2 ða x Þ

Z

x3 dx 2 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ ða2 x2 Þ3=2 x2 ða2 x2 Þ1=2 ¼ a2 x2 ðx2 þ 2a2 Þ 2 2 3 3 a x

Z

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x3 dx x2 a2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 2ða2 x2 Þ1=2 þ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ a2 x2 1=2 2 2 2 2 ða x Þ a x ða2 x2 Þ3

220.

221.

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 x2 dx 1 a þ a2 x2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ log 2a2 x2 x 2a3 x3 a2 x2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z dx 1 1 a þ a2 x2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 log 223. 2 a2 x2 x a 3 a 2 2 x ða x Þ Z

222.

Z 224. Z 225.

" pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi # a2 x2 dx 1 x qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 4 þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x a a2 x2 x2 ða2 x2 Þ3 dx 1 3 3 aþ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5 log qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 2 x2 a2 x2 4 a2 x2 2a 3 2a 2a 3 2 2 x ða x Þ

A-33

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 x2 x


pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z Z xm xm1 a2 x2 ðm 1Þa2 xm2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dx ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dx þ 226. m m a2 x2 a2 x2 Z 227. Z 228. Z 229.

" # m pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X x2m ð2mÞ! r!ðr 1Þ! 2m2r 2r1 a2m 1 x 2 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dx ¼ a a x x þ 2m sin 22m2rþ1 ð2rÞ! jaj 2 ðm!Þ2 a2 x2 r¼1 m pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X x2mþ1 ð2rÞ!ðm!Þ2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dx ¼ a2 x2 ð4a2 Þmr x2r 2 2 2 a x r¼0 ð2m þ 1Þ!ðr!Þ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z a2 x2 dx m2 dx pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ 2 m1 2 m 2 2 ðm 1Þa x ðm 1Þa x a x xm2 a2 x2

Z 230.

x2m Z

" pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi# m a2 x2 X dx ð2mÞ! r!ðr 1Þ! 1 a a2 x2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ þ log 2ð2rÞ!ð4a2 Þmr x2r 22m a2mþ1 a2 x x2mþ1 a2 x2 ðm!Þ2 r¼1

Z

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dx 1 ðb a2 x2 þ x a2 b2 Þ2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi log , b2 x2 ðb2 x2 Þ a2 x2 2b a2 b2

Z

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dx 1 x b2 a2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi tan1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , b a2 x2 ðb2 x2 Þ a2 x2 b b2 a2

Z

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dx 1 x a2 þ b2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi tan1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðb2 þ x2 Þ a2 x2 b a2 þ b2 b a2 x2

231.

232.

233.

234.

235.

1 X pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m ax ðm 1Þ!m!ð2rÞ!22m2r1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ a2 x2 2 2m2r x2rþ1 2 2 a x r¼0 ðr!Þ ð2mÞ!a

ðb2 > a2 Þ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 x2 x a2 þ b2 1 x a2 þ b2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sin1 sin dx ¼ 2 2 b þx jaj jbj jaj x2 þ b2 Z f ðx,

236.

Z pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 x2 Þ dx ¼ a f ða sin u, a cos uÞ cos u du,

FORMS CONTAINING

x u ¼ sin1 , a > 0 a

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a þ bx þ cx2

X ¼ a þ bx þ cx2 , q ¼ 4ac b2 , and k ¼ pffiffiffiffi pffiffiffi b X ¼ cx þ 2c 8 1 pffiffiffiffiffiffi > pffiffiffi logð2 cX þ 2cx þ bÞ > > > > < c

If q ¼ 0, then

Z 237.

Z 238. Z 239.

ða2 > b2 Þ

dx or pffiffiffiffi ¼ X > > > 1 2cx þ b > > : pffiffiffi sinh1 pffiffiffi , q c

dx 1 2cx þ b pffiffiffiffi ¼ pffiffiffiffiffiffi sin1 pffiffiffiffiffiffiffi , q c X

ðc > 0Þ

ðc < 0Þ

dx 2ð2cx þ bÞ pffiffiffiffi ¼ pffiffiffiffi X X q X

A-34

4c q


Z 240.

dx 2ð2cx þ bÞ 1 pffiffiffiffi pffiffiffiffi ¼ þ 2k X X2 X 3q X

8 pffiffiffiffi Z 2ð2cx þ bÞ X 2kðn 1Þ dx > > > pffiffiffiffi þ > n > 2n 1 ð2n 1ÞqX > X n1 X Z < dx or pffiffiffiffi ¼ 241. Xn X > > n1 > ð2rÞ! > ð2cx þ bÞðn!Þðn 1Þ!4n kn1 X > > pffiffiffiffi : r 2 q½ð2nÞ! X r¼0 ð4kXÞ ðr!Þ 242.

pffiffiffiffi Z pffiffiffiffi Z ð2cx þ bÞ X 1 dx pffiffiffiffi X dx ¼ þ 4c 2k X Z

pffiffiffiffi Z pffiffiffiffi ð2cx þ bÞ X 3 3 dx þ 2 pffiffiffiffi Xþ X X dx ¼ 2k 8k 8c X

Z

pffiffiffiffi Z pffiffiffiffi ð2cx þ bÞ X 5X 15 5 dx pffiffiffiffi þ 2 þ X X dx ¼ X2 þ 3 4k 8k 16k 12c X

243.

244.

2

pffiffiffiffi 8 Z pffiffiffiffi ð2cx þ bÞX n X 2n þ 1 > > X n1 X dx þ > > > 2ðn þ 1Þk 4ðn þ 1Þc > > Z < pffiffiffiffi or 245. X n X dx ¼ > " # > pffiffiffiffi n Z > > > ð2n þ 2Þ! kð2cx þ bÞ X X r!ðr þ 1Þ!ð4kXÞr dx > > p ffiffiffiffi þ : ð2r þ 2Þ! c ½ðn þ 1Þ! 2 ð4kÞnþ1 X r¼0 Z 246. Z 247.

x dx 2ðbx þ 2aÞ pffiffiffiffi ¼ pffiffiffiffi X X q X

Z

pffiffiffiffi Z X x dx b dx pffiffiffiffi ¼ pffiffiffiffi n n ð2n 1ÞcX 2c X n X X X

Z

Z x2 dx x 3b pffiffiffiffi 3b2 4ac dx pffiffiffiffi ¼ pffiffiffiffi 2 Xþ 2 2c 4c 8c X X

248.

249. Z 250. Z 251.

x2 dx ð2b2 4acÞx þ 2ab 1 pffiffiffiffi ¼ pffiffiffiffi þ c X X cq X

Z

dx pffiffiffiffi X

x2 dx ð2b2 4acÞx þ 2ab 4ac þ ð2n 3Þb2 pffiffiffiffi ¼ pffiffiffiffi þ n ð2n 1Þcq X X ð2n 1Þcq X n1 X

Z

dx pffiffiffiffi X n1 X

Z

2 Z x3 dx x 5bx 5b2 2a pffiffiffiffi 3ab 5b3 dx pffiffiffiffi ¼ pffiffiffiffi X þ þ 4c2 16c3 3c 12c2 8c3 3c2 X X

Z

pffiffiffiffi ð2n 1Þb xn dx 1 pffiffiffiffi ¼ xn1 X nc 2nc X

Z

pffiffiffiffi Z pffiffiffiffi X X bð2cx þ bÞ pffiffiffiffi b dx pffiffiffiffi X x X dx ¼ 3c 8c2 4ck X

Z

pffiffiffiffi Z pffiffiffiffi pffiffiffiffi X2 X b X X dx xX X dx ¼ 2c 5c

252.

253.

254.

255.

pffiffiffiffi Z X x dx b dx pffiffiffiffi ¼ pffiffiffiffi 2c c X X

Z

xn1 dx ðn 1Þa pffiffiffiffi nc X

A-35

Z

xn2 dx pffiffiffiffi X


pffiffiffiffi Z Z pffiffiffiffi pffiffiffiffi X nþ1 X b X n X dx 256. xX n X dx ¼ ð2n þ 3Þc 2c Z

pffiffiffiffi Z pffiffiffiffi 5b X X 5b2 4ac pffiffiffiffi X dx x2 X dx ¼ x þ 2 6c 16c 4c

Z

pffiffiffiffiffiffiffi dx 1 2 aX þ bx þ 2a pffiffiffiffi ¼ pffiffiffi log , a x x X

257.

258. Z 259. Z 260. Z 261.

dx 1 bx þ 2a pffiffiffiffi ¼ pffiffiffiffiffiffiffi sin1 pffiffiffiffiffiffiffi , jxj q a x X pffiffiffiffi dx 2 X pffiffiffiffi ¼ , bx x X

ða > 0Þ

ða < 0Þ

ða ¼ 0Þ

pffiffiffiffi Z X dx b dx pffiffiffiffi pffiffiffiffi ¼ 2 2a x X ax x X

262.

Z Z Z pffiffiffiffi X dx pffiffiffiffi b dx dx pffiffiffiffi þ a pffiffiffiffi ¼ Xþ x 2 X x X

263.

pffiffiffiffi Z Z Z pffiffiffiffi X dx X b dx dx p ffiffiffiffi pffiffiffiffi þ ¼ þ c x2 x 2 x X X FORMS INVOLVING

264.

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ax x2

Z pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 xa 2ax x2 dx ¼ ðx aÞ 2ax x2 þ a2 sin1 2 jaj

8 ax > cos1 > > > jaj > Z < dx p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi or 265. ¼ 2ax x2 > > > > 1 x a > : sin jaj 8 Z pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi xn1 ð2ax x2 Þ3=2 ð2n þ 1Þa > > > xn1 2ax x2 dx þ > > n þ 2 n þ 2 > > > > > > or > Z pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi < " # n 2 n 266. x 2ax x dx ¼ X pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi xnþ1 ð2n þ 1Þ!ðr!Þ2 anrþ1 r > 2 > x > > 2ax x n þ 2 2nr ð2r þ 1Þ!ðn þ 2Þ!n! > > r¼0 > > > > ð2n þ 1Þ!anþ2 1 x a > > : þ n sin jaj 2 n!ðn þ 2Þ! 267.

Z pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 Z pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 2ax x 2ax x ð2ax x2 Þ3=2 n3 dx ¼ þ dx xn xn1 ð2n 3Þa ð3 2nÞaxn

8 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z xn1 2ax x2 að2n 1Þ xn1 > > > pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dx þ > > n n > 2ax x2 > Z < xn dx or pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 268. 2ax x2 > > > pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X n > ð2nÞ!r!ðr 1Þ!anr r1 ð2nÞ!an 1 x a > > 2 > x þ sin : 2ax x jaj 2nr ð2rÞ!ðn!Þ2 2n ðn!Þ2 r¼1

A-36


8 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z > n1 dx 2ax x2 > > pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ > n > ð2n 1Þa xn1 2ax x2 að1 2nÞx > > > Z < dx or pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 269. n 2 > x 2ax x > > n1 nr > pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X > 2 ðn 1Þ!n!ð2rÞ! > 2 > > : 2ax x 2 nr rþ1 x r¼0 ð2nÞ!ðr!Þ a Z 270. Z 271.

dx xa ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð2ax x2 Þ3=2 a2 2ax x2 x dx x ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð2ax x2 Þ3=2 a 2ax x2 MISCELLANEOUS ALGEBRAIC FORMS

Z 272.

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dx pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ logðx þ a þ 2ax þ x2 Þ 2ax þ x2

273.

Z pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c ax2 þ c þ pffiffiffi log ðx a þ ax2 þ cÞ, ax2 þ c dx ¼ 2 2 a

274.

rffiffiffiffiffiffiffi Z pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c a ax2 þ c þ pffiffiffiffiffiffiffi sin1 x , ax2 þ c dx ¼ 2 2 a c

275.

Z rffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi 1þx dx ¼ sin1 x 1 x2 1x

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi 8 1 axn þ c c > > p ffiffi ffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p log pffiffiffi > > n c > axn þ c þ c > Z < dx pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 276. or x axn þ c > > pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi > > > axn þ c c 2 > : pffiffiffi log pffiffiffiffiffi , n c xn Z 277. Z 278. Z 279.

dx 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ pffiffiffiffiffiffi sec1 x axn þ c n c

rffiffiffiffiffiffiffiffiffiffiffiffi axn , c

ða < 0Þ

ðc > 0Þ

ðc < 0Þ

pffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dx 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ pffiffiffi logðx a þ ax2 þ cÞ, a ax2 þ c rffiffiffiffiffiffiffi dx 1 a pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ pffiffiffiffiffiffiffi sin1 x , c a ax2 þ c

ða > 0Þ

ða > 0Þ

ða < 0Þ

8 Z 1 xðax2 þ cÞmþ1=2 ð2m þ 1Þc > > ðax2 þ cÞm 2 dx þ > > > 2ðm þ 1Þ 2ðm þ 1Þ > > > > > > or > > Z < m pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X 280. ðax2 þ cÞmþ1=2 dx ¼ ð2m þ 1Þ!ðr!Þ2 cmr 2 > ðax2 þ cÞr > > x ax þ c 2m2rþ1 m!ðm þ 1Þ!ð2r þ 1Þ! > 2 > r¼0 > > > > > mþ1 Z > > dx > þ ð2m þ 1Þ!c : pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2mþ1 2 m!ðm þ 1Þ! ax2 þ c Z 281.

1

xðax2 þ cÞmþ2 dx ¼

3

ðax2 þ cÞm þ 2 ð2m þ 3Þa

A-37


8 Z > ðax2 þ cÞmþ1=2 ðax2 þ cÞm1=2 > > þc dx > > 2m þ 1 x > > Z < ðax2 þ cÞmþ1=2 or dx ¼ 282. > x > Z > m mr > pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X c ðax2 þ cÞr dx > > pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > þ cmþ1 : ax2 þ c 2r þ 1 x ax2 þ c r¼0

Z 283.

dx ðax2 þ cÞmþ1=2

Z 8 x 2m 2 dx > > þ > > 2 þ cÞm1=2 2 þ cÞm1=2 ð2m 1Þc > ð2m 1Þcðax ðax > < ¼ or > > m1 X 22m2r1 ðm 1Þ!m!ð2rÞ! > x > > > : pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ax þ c r¼0 ð2mÞ!ðr!Þ2 cmr ðax2 þ cÞr

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z dx ðm 2Þa dx ax2 þ c pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðm 1Þcxm1 ðm 1Þc xm2 ax2 þ c xm ax2 þ c pffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi Z 1 þ x2 1 x 2 þ 1 þ x4 pffiffiffiffiffiffiffiffiffiffiffiffiffi dx ¼ pffiffiffi log 285. 1 x2 2 ð1 x2 Þ 1 þ x4 Z

284.

pffiffiffi 1 x2 1 x 2 pffiffiffiffiffiffiffiffiffiffiffiffiffi dx ¼ pffiffiffi tan1 pffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ð1 þ x2 Þ 1 þ x4 1 þ x4 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z dx 2 a þ xn þ a2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ log pffiffiffiffiffi 287. na xn x xn þ a2 Z dx 2 a pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ sin1 pffiffiffiffiffin 288. na x x xn a2 Z rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x3=2 x 2 289. dx ¼ sin1 a3 x3 3 a Z

286.

FORMS INVOLVING TRIGONOMETRIC FUNCTIONS Z 290. Z 291. Z 292. Z 293. Z 294. Z 295. Z 296. Z 297. Z 298. Z 299.

1 ðsin axÞ dx ¼ cos ax a 1 ðcos axÞ dx ¼ sin ax a 1 1 ðtan axÞ dx ¼ log cos ax ¼ log sec ax a a 1 1 ðcot axÞ dx ¼ log sin ax ¼ log csc ax a a p ax 1 1 ðsec axÞ dx ¼ logðsec ax þ tan axÞ ¼ log tan þ a a 4 2 1 1 ax ðcsc axÞ dx ¼ logðcsc ax cot axÞ ¼ log tan a a 2 ðsin2 axÞ dx ¼

1 1 1 1 cos ax sin ax þ x ¼ x sin 2ax 2a 2 2 4a

ðsin3 axÞ dx ¼

1 ðcos axÞðsin2 ax þ 2Þ 3a

ðsin4 axÞ dx ¼

3x sin 2ax sin 4ax þ 8 4a 32a

ðsinn axÞ dx ¼

sinn1 ax cos ax n 1 þ na n

A-38

Z

ðsinn2 axÞ dx


Z 300. Z 301. Z 302. Z 303. Z 304. Z 305. Z 306. Z 307. Z 308. Z 309. Z 310.

ðsin2m axÞ dx ¼

ðsin2mþ1 axÞ dx ¼

m cos ax X 22m2r ðm!Þ2 ð2rÞ! 2r sin ax a r¼0 ð2m þ 1Þ!ðr!Þ2

ðcos2 axÞ dx ¼

1 1 1 1 sin ax cos ax þ x ¼ x þ sin 2ax 2a 2 2 4a

ðcos3 axÞ dx ¼

1 ðsin axÞðcos2 ax þ 2Þ 3a

ðcos4 axÞ dx ¼

3x sin 2ax sin 4ax þ þ 8 4a 32a

ðcosn axÞ dx ¼

1 n1 cosn1 ax sin ax þ na n

ðcos2m axÞ dx ¼

dx ¼ sin2 ax

Z

dx ¼ sinm ax

sin

Z

1 cos ax m2

þ ðm 1Þa sinm1 ax m 1

Z

dx sinm2 ax

m 1 2m2r1 X 1 2 ðm 1Þ!m!ð2rÞ! ðcsc2m axÞ dx ¼ cos ax a ð2mÞ!ðr!Þ2 sin2rþ1 ax r¼0

¼

ax

m sin ax X 22m2r ðm!Þ2 ð2rÞ! cos2r ax a r¼0 ð2m þ 1Þ!ðr!Þ2

ðcscm axÞ dx ¼

Z

dx 2mþ1

ðcosn2 axÞ dx

1 ðcsc2 axÞ dx ¼ cot ax a

Z

dx ¼ sin2m ax

Z

1 X sin ax m ð2mÞ!ðr!Þ2 ð2mÞ! cos2rþ1 ax þ x a r¼0 22m2r ð2r þ 1Þ!ðm!Þ2 22m ðm!Þ2

ðcos2mþ1 axÞ dx ¼

Z 311.

1 X cos ax m ð2mÞ!ðr!Þ2 ð2mÞ! sin2rþ1 ax þ x a r¼0 22m2r ð2r þ 1Þ!ðm!Þ2 22m ðm!Þ2

ðcsc2mþ1 axÞ dx

m 1 X 1 ð2mÞ!ðr!Þ2 1 ð2mÞ! ax þ ¼ cos ax log tan 2m2r ðm!Þ2 ð2r þ 1Þ! sin2rþ2 ax 2m ðm!Þ2 a a 2 2 2 r¼0

Z 312. Z 313. Z 314. Z 315.

dx ¼ cos2 ax dx ¼ cosn ax

Z Z

dx ¼ cos2m ax

ðsecn axÞ dx ¼ Z

ax

¼

1 sin ax n2

þ ðn 1Þa cosn1 ax n 1

Z

dx cosn2 ax

m 1 2m2r1 X 1 2 ðm 1Þ!m!ð2rÞ! ðsec2m axÞ dx ¼ sin ax a ð2mÞ!ðr!Þ2 cos2rþ1 ax r¼0

Z

dx cos2mþ1

1 ðsec2 axÞ dx ¼ tan ax a

ðsec2mþ1 axÞ dx

m 1 X 1 ð2mÞ!ðr!Þ2 1 ð2mÞ! þ logðsec ax þ tan axÞ ¼ sin ax 2 2m2r ðm!Þ ð2r þ 1Þ! cos2rþ2 ax a a 22m ðm!Þ2 2 r¼0

Z ðsin mxÞ ðsin nxÞ dx ¼

316.

sinðm nÞx sinðm þ nÞx , 2ðm nÞ 2ðm þ nÞ

Z ðcos mxÞ ðcos nxÞ dx ¼

317. Z 318.

ðsin axÞ ðcos axÞ dx ¼

sinðm nÞx sinðm þ nÞx þ , 2ðm nÞ 2ðm þ nÞ

1 sin2 ax 2a

A-39

ðm2 6¼ n2 Þ ðm2 6¼ n2 Þ


Z ðsin mxÞ ðcos nxÞ dx ¼

319. Z 320. Z 321. Z 322.

cosðm nÞx cosðm þ nÞx , 2ðm nÞ 2ðm þ nÞ

ðsin2 axÞ ðcos2 axÞ dx ¼

ðsin axÞ ðcosm axÞ dx ¼

ðsinm axÞ ðcos axÞ dx ¼

ðm2 6¼ n2 Þ

1 x sin 4ax þ 32a 8

cosmþ1 ax ðm þ 1Þa

sinmþ1 ax ðm þ 1Þa

8 Z cosm1 ax sinnþ1 ax m 1 > > > þ ðcosm2 axÞ ðsinn axÞ dx > > ðm þ nÞa mþn > > Z < 323. ðcosm axÞ ðsinn axÞ dx ¼ or > > > Z > n1 > sin ax cosmþ1 ax n 1 > > : þ ðcosm axÞ ðsinn2 axÞ dx ðm þ nÞa mþn Z 8 cosmþ1 ax m n þ 2 cosm ax > > dx > n1 > n1 > ðn 1Þa sin ax sinn2 ax > > Z < cosm ax 324. dx ¼ or > sinn ax > > Z > > > cosm1 ax m 1 cosm2 ax > : dx þ n1 sinn ax aðm nÞ sin ax m n 8 Z sinmþ1 ax mnþ2 sinm ax > > dx > > n1 > aðn 1Þ cos ax n1 cosn2 ax > > Z < sinm ax 325. dx ¼ or > cosn ax > > Z > > > sinm1 ax m 1 sinm2 ax > : þ dx n1 aðm nÞ cos ax m n cosn ax Z 326. Z 327. Z 328. Z 329. Z 330. Z 331. Z 332.

p ax sin2 ax 1 1 dx ¼ sin ax þ log tan þ cos ax a a 4 2 cos ax 1 csc ax ¼ dx ¼ a sin ax a sin2 ax dx 1 ¼ log tan ax ðsin axÞ ðcos axÞ a dx 1 ax ¼ sec ax þ log tan 2 ðsin axÞ ðcos axÞ a 2 dx 1 ¼ þ ðsin axÞ ðcosn axÞ aðn 1Þ cosn1 ax

Z

dx ðsin axÞ ðcosn2 axÞ

p ax dx 1 1 ¼ csc ax þ log tan þ a a 4 2 ðsin axÞ ðcos axÞ 2

Z 333.

sin ax 1 sec ax ¼ dx ¼ cos2 ax a cos ax a

dx 2 ¼ cot 2ax a ðsin2 axÞ ðcos2 axÞ

A-40


8 1 > > > > > aðm 1Þ ðsinm1 axÞ ðcosn1 axÞ > > Z > > mþn2 dx > Z < þ dx m2 m 1 ¼ 334. ðsin axÞ ðcosn axÞ sinm ax cosn ax > > > or > > Z > > > 1 mþn2 dx > > : n1 sinm ax cosn2 ax aðn 1Þ sinm1 ax cosn1 ax Z 335. Z 336. Z 337. Z 338. Z 339.

1 sinða þ bxÞ dx ¼ cosða þ bxÞ b 1 cosða þ bxÞ dx ¼ sinða þ bxÞ b p ax dx 1 ¼ tan 1 sin ax a 4 2 dx 1 ax ¼ tan 1 þ cos ax a 2 dx 1 ax ¼ cot 1 cos ax a 2

8 x > a tan þ b > > 2 1 > 2 > ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p p tan > > > a2 b2 a2 b2 > > > Z < dx or ¼ *340. a þ b sin x > > pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > x > > a tan þ b b2 a2 > 1 > 2 > > pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > x > b2 a2 log : a tan þ b þ b2 a2 2 8 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x > a2 b2 tan > 2 > 1 2 > ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p > tan > > aþb a2 b2 > > > > Z < dx or *341. ¼ a þ b cos x > 0pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 > > x > 2 a2 tan þ a þ b > b > 1 > C 2 > > pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi logB @pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A > x > : b2 a2 b2 a2 tan a b 2 Z *342.

dx a þ b sin x þ c cos x 8 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x > b b2 þ c2 a2 þ ða cÞ tan > > 1 > 2 > ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p if a2 < b2 þ c2 , a 6¼ c log > pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x, > 2 þ c2 a2 > 2 þ c2 a2 þ ða cÞ tan b > b þ b > > 2 > > > > or > > > > < x b þ ða cÞ tan ¼ 2 1 > > pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi tan pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 , if a2 > b2 þ c2 > > > a2 b2 c2 a2 b2 c2 > > > > > > or > > >

> > > 1 a ðb þ cÞ cos x ðb cÞ sin x > > , if a2 ¼ b2 þ c2 , a 6¼ c: : a a ðb cÞ cos x þ ðb þ cÞ sin x

*See note 6 on page A-19.

A-41


rffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffi Z sin2 x dx 1 aþb a x tan1 tan x , ¼ *343. 2 a þ b cos x b a aþb b Z *344.

a2 Z

*345. Z 346. Z 347.

Z 348.

cos2

ðab > 0, or jaj > jbjÞ

dx 1 b tan x ¼ tan1 2 2 a x þ b sin x ab

cos2 cx dx ¼ 2 a þ b2 sin2 cx

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 þ b2 a2 þ b2 tan cx x 1 tan 2 2 b ab c a

sin cx cos cx 1 logða cos2 cx þ b sin2 cxÞ dx ¼ 2cðb aÞ a cos2 cx þ b sin2 cx cos cx dx ¼ a cos cx þ b sin cx

Z

dx a þ b tan cx 1 ½acx þ b logða cos cx þ b sin cxÞ ¼ cða2 þ b2 Þ

sin cx dx ¼ a sin cx þ b cos cx

Z

dx 1 ¼ ½acx b log ða sin cx þ b cos cxÞ a þ b cot cx cða2 þ b2 Þ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8 1 c tan x þ b b2 ac > > > ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , log ðb2 > acÞ > > > 2 b2 ac c tan x þ b þ b2 ac > > > > > or > > > Z < dx 1 xþb ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi tan1 cptan *349. ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , ðb2 < acÞ 2 a cos2 x þ 2b cos x sin x þ c sin2 x > > ac b ac b2 > > > > > or > > > > > > 1 > : , ðb2 ¼ acÞ c tan x þ b Z

p ax sin ax 1 dx ¼ x þ tan 1 sin ax a 4 2

Z

p ax 1 dx 1 ax ¼ tan þ log tan ðsin axÞ ð1 sin axÞ a 4 2 a 2

Z

p ax 1 p ax dx 1 tan3 ¼ tan 2 2a 4 2 6a 4 2 ð1 þ sin axÞ

Z

p ax 1 p ax dx 1 þ cot3 ¼ cot 2 2a 4 2 6a 4 2 ð1 sin axÞ

Z

p ax 1 p ax sin ax 1 þ tan3 dx ¼ tan 2 2a 4 2 6a 4 2 ð1 þ sin axÞ

Z

p ax 1 p ax sin ax 1 þ cot3 dx ¼ cot 2a 4 2 6a 4 2 ð1 sin axÞ2

350.

351.

352.

353.

354.

355. Z 356. Z 357. Z 358.

sin x dx x a ¼ a þ b sin x b b

Z

dx a þ b sin x

dx 1 x b ¼ log tan ðsin xÞ ða þ b sin xÞ a 2 a

Z

dx a þ b sin x

dx b cos x a þ ¼ ða þ b sin xÞ2 ða2 b2 Þ ða þ b sin xÞ a2 b2


A-42

Z

dx a þ b sin x


Z

Z sin x dx a cos x h dx þ ¼ ða þ b sin xÞ2 ðb2 a2 Þða þ b sin xÞ b2 a2 a þ b sin x pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z dx 1 a2 þ b2 tan cx ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi tan1 *360. 2 a a2 þ b2 sin cx ac a2 þ b2 8 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 b2 tan cx 1 > > > pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi tan1 , ða2 > b2 Þ > > 2 2 Z a < ac a b dx ¼ *361. or pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 b2 sin2 cx > > > b2 a2 tan cx þ a 1 > > pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi log pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , ða2 < b2 Þ : 2 2 b2 a2 tan cx a 2ac b a Z cos ax 1 ax 362. dx ¼ x tan 1 þ cos ax a 2 Z cos ax 1 ax dx ¼ x cot 363. 1 cos ax a 2 Z p ax 1 dx 1 ax ¼ log tan þ tan 364. ðcos axÞð1 þ cos axÞ a 4 2 a 2 Z p ax 1 dx 1 ax ¼ log tan þ cot 365. ðcos axÞð1 cos axÞ a 4 2 a 2 Z dx 1 ax 1 ax 366. ¼ tan þ tan3 2 6a 2 ð1 þ cos axÞ2 2a 359.

Z 367. Z 368.

dx 1 ax 1 ax ¼ cot cot3 2a 2 6a 2 ð1 cos axÞ2 cos ax 1 ax 1 ax dx ¼ tan tan3 2a 2 6a 2 ð1 þ cos axÞ2

Z

cos ax 1 ax 1 ax dx ¼ cot cot3 2a 2 6a 2 ð1 cos axÞ2 Z Z cos x dx x a dx ¼ 370. a þ b cos x b b a þ b cos x Z x p b Z dx 1 dx ¼ log tan þ 371. ðcos xÞða þ b cos xÞ a 2 4 a a þ b cos x Z Z dx b sin x a dx 372. ¼ ða þ b cos xÞ2 ðb2 a2 Þða þ b cos xÞ b2 a2 a þ b cos x 369.

Z cos x a sin x b dx dx ¼ ða2 b2 Þða þ b cos xÞ a2 b2 a þ b cos x ða þ b cos xÞ2 Z dx 2 cx 1 a þ b tan ¼ tan *374. a2 þ b2 2ab cos cx cða2 b2 Þ ab 2 Z dx 1 a tan cx ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi tan1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *375. a2 þ b2 cos2 cx ac a2 þ b2 a2 þ b2 8 1 a tan cx > > pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi tan1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , ða2 > b2 Þ > > > Z ac a2 b2 a2 b2 < dx ¼ *376. or pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > a2 b2 cos2 cx > 1 a tan cx b2 a2 > > pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi log ðb2 > a2 Þ > : a tan cx þ b2 a2 2ac b2 a2 Z

373.

Z 377.

sin ax 1 dx ¼ logð1 cos axÞ 1 cos ax a


A-43


Z 378. Z 379. Z 380. Z 381. Z 382. Z 383. Z 384.

cos ax 1 dx ¼ log ð1 sin axÞ 1 sin ax a dx 1 1 ax ¼ þ log tan ðsin axÞð1 cos axÞ 2að1 cos axÞ 2a 2 p dx 1 1 ax ¼ þ log tan þ ðcos axÞð1 sin axÞ 2að1 sin axÞ 2a 4 2 sin ax 1 dx ¼ logðsec ax 1Þ ðcos axÞð1 cos axÞ a cos ax 1 dx ¼ logðcsc ax 1Þ ðsin axÞð1 sin axÞ a p sin ax 1 1 ax dx ¼ log tan þ ðcos axÞð1 sin axÞ 2að1 sin axÞ 2a 4 2 cos ax 1 1 ax dx ¼ log tan ðsin axÞð1 cos axÞ 2að1 cos axÞ 2a 2

Z

ax p dx 1 ¼ pffiffiffi log tan sin ax cos ax 2 8 a 2

Z

dx 1 p tan ax ¼ 2 2a 4 ðsin ax cos axÞ

Z

dx 1 ax ¼ log 1 tan 1 þ cos ax sin ax a 2

385.

386.

387. Z 388.

dx 1 b tan cx þ a log ¼ 2abc b tan cx a a2 cos2 cx b2 sin2 cx

Z

1 x sin ax cos ax a2 a

xðsin axÞ dx ¼

389. Z 390. Z 391.

x2 ðsin axÞ dx ¼

2x a2 x2 2 sin ax cos ax 2 a a3

x3 ðsin axÞ dx ¼

3a2 x2 6 a2 x3 6x sin ax cos ax 4 a a3

Z 8 1 m > > xm 1 cos ax dx xm cos ax þ > > a a > > > > > or > > Z < ½X m=2 m! xm2r 392. xm sin ax dx ¼

2rþ1 cos ax ð1Þrþ1 > > > ðm 2rÞ! a > r¼0 > > > ½ðm1Þ=2 > X m! xm2r1 > > > þ sin ax

2rþ2 ð1Þr : ðm 2r 1Þ! a r¼0

% & %& Note: [s] means greatest integer s; 312 ¼ 3, 12 ¼ 0, etc.

Z

xðcos axÞ dx ¼

393. Z 394. Z 395.

1 x cos ax þ sin ax a2 a

x2 ðcos axÞ dx ¼

2x cos ax a2 x2 2 þ sin ax a2 a3

x3 ðcos axÞ dx ¼

3a2 x2 6 a2 x3 6x cos ax þ sin ax 4 a a3

A-44


Z 8 m x sin ax m > > xm1 sin ax dx > > a a > > > > > or > > Z < ½X m=2 m m! xm2r 396. x ðcos axÞ dx ¼

2rþ1 ð1Þr > > sin ax > ðm 2rÞ! a > r¼0 > > > ½ðm1Þ=2 > X > m! xm2r1 > >

2rþ2 þ cos ax ð1Þr : ðm 2r 1Þ! a r¼0 See note integral 392. Z r X sin ax ðaxÞ2nþ1 397. dx ¼ ð1Þn x ð2n þ 1Þð2n þ 1Þ! n¼0 Z 398. Z

r X cos ax ðaxÞ2n dx ¼ log x þ ð1Þn x 2nð2nÞ! n¼1

x2 x sin 2ax cos 2ax 4a 8a2 4 2 Z x3 x 1 x cos 2ax 3 sin 2ax 400. x2 ðsin2 axÞ dx ¼ 4a2 6 4a 8a Z x cos 3ax sin 3ax 3x cos ax 3 sin ax þ 401. xðsin3 axÞ dx ¼ 12a 36a2 4a 4a2 Z x2 x sin 2ax cos 2ax þ 402. xðcos2 axÞ dx ¼ þ 4a 8a2 4 2 Z x3 x 1 x cos 2ax 3 sin 2ax þ 403. x2 ðcos2 axÞ dx ¼ þ 4a2 6 4a 8a Z x sin 3ax cos 3ax 3x sin ax 3 cos ax þ þ 404. xðcos3 axÞ dx ¼ þ 12a 36a2 4a 4a2 Z Z sin ax sin ax a cos ax 405. dx ¼ þ dx xm ðm 1Þxm1 m 1 xm1 Z Z cos ax cos ax a sin ax 406. dx ¼ dx m m1 x ðm 1Þx m 1 xm1 Z x x cos ax 1 dx ¼ þ logð1 sin axÞ 407. 1 sin ax að1 sin axÞ a2 Z x x ax 2 ax dx ¼ tan þ 2 log cos 408. 1 þ cos ax a 2 a 2 Z x x ax 2 ax dx ¼ cot þ 2 log sin 409. 1 cos ax a 2 a 2 Z x þ sin x x dx ¼ x tan 410. 1 þ cos x 2 Z x sin x x dx ¼ x cot 411. 1 cos x 2 pffiffiffi Z pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ax 2 sin ax 2 2 412. 1 cos ax dx ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ cos 2 a a 1 cos ax pffiffiffi Z pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 sin ax 2 2 ax 413. 1 þ cos ax dx ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ sin 2 a a 1 þ cos ax Z pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x x 414. 1 þ sin x dx ¼ 2 sin cos , 2 2 p p ½use þ if ð8k 1Þ < x ð8k þ 3Þ , otherwise ; k an integer] 2 2 399.

xðsin2 axÞ dx ¼

A-45


Z pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x x 415. 1 sin x dx ¼ 2 sin þ cos , 2 2 h i p p use þ if ð8k 3Þ < x ð8k þ1Þ , otherwise ; k an integer 2 2 Z pffiffiffi dx x pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 2 log tan , 416. 4 1 cos x ½use þ if 4kp < x < ð4kþ2Þp, otherwise ; k an integer] Z

417.

x þ p pffiffiffi dx pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 2 log tan , 4 1 þ cos x ½use þ if ð4k 1Þp < x < ð4k þ 1Þp, otherwise ; k an integer] Z

x p pffiffiffi dx pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 2 log tan , 4 8 1 sin x p p ½use þ if ð8k þ 1Þ < x < ð8k þ 5Þ , otherwise ; k an integer] 2 2 Z p ffiffi ffi dx x p pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 2 log tan þ , 419. 4 8 1 þ sin x p p ½use þ if ð8k 1Þ < x < ð8k þ 3Þ , otherwise ; k an integer] 2 2 Z 1 420. ðtan2 axÞ dx ¼ tan ax x a 418.

Z 421. Z 422. Z 423. Z 424. Z 425. Z 426. Z 427. Z 428. Z 429.

Z 430. Z 431.

ðtan3 axÞ dx ¼

1 1 tan2 ax þ log cos ax 2a a

ðtan4 axÞ dx ¼

tan3 ax 1 tan x þ x 3a a

ðtann axÞ dx ¼

tann1 ax aðn 1Þ

Z

ðtann2 axÞ dx

1 ðcot2 axÞ dx ¼ cot ax x a ðcot3 axÞ dx ¼

1 1 cot2 ax log sin ax 2a a

ðcot4 axÞ dx ¼

1 1 cot3 ax þ cot ax þ x 3a a

ðcotn axÞ dx ¼

cotn1 ax aðn 1Þ

x dx ¼ sin2 ax

Z

x dx ¼ sinn ax

x dx ¼ cos2 ax x dx ¼ cosn ax

Z

Z Z

Z

ðcotn2 axÞ dx

xðcsc2 axÞ dx ¼

x cot ax 1 þ 2 log sin ax a a

x cos ax 1 aðn 1Þsinn1 ax a2 ðn 1Þðn 2Þsinn2 ax Z ðn 2Þ x dx þ ðn 1Þ sinn2 ax

xðcscn axÞ dx ¼

1 1 xðsec2 axÞ dx ¼ x tan ax þ 2 log cos ax a a xðsecn axÞ dx ¼

x sin ax 1 aðn 1Þ cosn1 ax a2 ðn 1Þðn 2Þ cosn2 ax Z n2 x þ dx n 1 cosn2 ax

A-46


Z 432.

sin ax 1 b cos ax pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dx ¼ sin1 pffiffiffiffiffiffiffiffiffiffiffiffiffi ab 1 þ b2 1 þ b2 sin2 ax

Z

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sin ax 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dx ¼ logðb cos ax þ 1 b2 sin2 axÞ 2 ab 1 b2 sin ax

Z

ffi 1 þ b2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cos ax pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b cos ax 1 þ b2 sin2 ax sin1 pffiffiffiffiffiffiffiffiffiffiffiffiffi ðsin axÞ 1 þ b2 sin2 ax dx ¼ 2a 2ab 1 þ b2

Z

ffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cos ax pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 b2 sin2 ax ðsin axÞ 1 b2 sin2 ax dx ¼ 2a pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 b2 logðb cos ax þ 1 b2 sin2 axÞ 2ab

Z

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cos ax 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dx ¼ logðb sin ax þ 1 þ b2 sin2 axÞ ab 1 þ b2 sin2 ax

433.

434.

435.

436. Z 437.

cos ax 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dx ¼ sin1 ðb sin axÞ 2 ab 2 1 b sin ax

Z

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi sin ax pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðcos axÞ 1 þ b2 sin2 ax dx ¼ 1 þ b2 sin2 ax 2a pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 logðb sin ax þ 1 þ b2 sin2 axÞ þ 2ab

Z

ffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sin ax pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 b2 sin2 ax þ sin1 ðb sin axÞ ðcos axÞ 1 b2 sin2 ax dx ¼ 2a 2ab

438.

439.

! rffiffiffiffiffiffiffiffiffiffiffi dx 1 ab pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ pffiffiffiffiffiffiffiffiffiffiffi sin1 sin cx , ða > jbjÞ a a þ b tan2 cx c a b p p ½use þ if ð2k 1Þ < x ð2k þ 1Þ , otherwise ; k an integer 2 2 Z

440.

FORMS INVOLVING INVERSE TRIGONOMETRIC FUNCTIONS Z 441. Z 442. Z 443. Z 444.

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 a2 x2 a

ðcos1 axÞ dx ¼ x cos1 ax

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 a2 x2 a

ðtan1 axÞ dx ¼ x tan1 ax

1 log ð1 þ a2 x2 Þ 2a

ðcot1 axÞ dx ¼ x cot1 ax þ

1 log ð1 þ a2 x2 Þ 2a

Z

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ðsec1 axÞ dx ¼ x sec1 ax log ðax þ a2 x2 1Þ a

Z

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ðcsc1 axÞ dx ¼ x csc1 ax þ log ðax þ a2 x2 1Þ a

445.

446.

ðsin1 axÞ dx ¼ x sin1 ax þ

447.

Z x x pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sin1 dx ¼ x sin1 þ a2 x2 , a a

448.

Z x x pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cos1 dx ¼ x cos1 a2 x2 , a a

A-47

ða > 0Þ

ða > 0Þ


Z x x a 449. tan1 dx ¼ x tan1 log ða2 þ x2 Þ a a 2 450.

Z x x a cot1 dx ¼ x cot1 þ logða2 þ x2 Þ a a 2 Z

451. Z 452. Z 453. Z 454. Z 455. Z 456. Z 457. Z 458. Z 459. Z 460. Z 461. Z 462. Z 463. Z 464.

x ½sin1 ðaxÞ dx ¼

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ½ð2a2 x2 1Þ sin1 ðaxÞ þ ax 1 a2 x2 4a2

x ½cos1 ðaxÞ dx ¼

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ½ð2a2 x2 1Þ cos1 ðaxÞ ax 1 a2 x2 4a2

xn ½sin1 ðaxÞ dx ¼

xnþ1 a sin1 ðaxÞ nþ1 nþ1

xn ½cos1 ðaxÞ dx ¼

xnþ1 a cos1 ðaxÞ þ nþ1 nþ1

xðtan1 axÞ dx ¼

Z

xnþ1 dx pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi, 1 a2 x2 xnþ1 dx pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi, 1 a2 x2

ðn 6¼ 1Þ

ðn 6¼ 1Þ

1 þ a2 x2 x tan1 ax 2a 2a2

xn ðtan1 axÞ dx ¼

xðcot1 axÞ dx ¼

Z

xnþ1 a tan1 ax nþ1 nþ1

Z

xnþ1 dx 1 þ a2 x2

1 þ a2 x2 x cot1 ax þ 2a 2a2

xn ðcot1 axÞ dx ¼

xnþ1 a cot1 ax þ nþ1 nþ1

sin1 ðaxÞ 1 dx ¼ a log x2

Z

xnþ1 dx 1 þ a2 x2

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! 1 a2 x2 sin1 ðaxÞ x x

cos1 ðaxÞ dx 1 1þ ¼ cos1 ðaxÞ þ a log x2 x

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 a2 x2 x

tan1 ðaxÞ dx 1 a 1 þ a2 x2 ¼ tan1 ðaxÞ log 2 x x 2 x2 cot1 ax 1 a x2 dx ¼ cot1 ax log 2 2 x2 x 2 a x þ1 ðsin1 axÞ2 dx ¼ xðsin1 axÞ2 2x þ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 a2 x2 sin1 ax a

ðcos1 axÞ2 dx ¼ xðcos1 axÞ2 2x

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 a2 x2 cos1 ax a

8 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z > n 1 a2 x2 > 1 n 1 n1 > ðsin xðsin axÞ þ axÞ 100 nðn 1Þ ðsin1 axÞn2 dx > > > a > > > > or > > Z < ½n=2 1 n X n! 465. ðsin axÞ dx ¼ > xðsin1 axÞn2r ð1Þr > > ðn 2rÞ! > r¼0 > > pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > ½ðn1Þ=2 > X > 1 a2 x2 > r n! > ðsin1 axÞn2r1 þ ð1Þ > : ðn 2r 1Þ!a r¼0

Note: [s] means greatest integer s. Thus [3.5] means 3; ½5 ¼ 5, ½12 ¼ 0.

A-48


8 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z > n 1 a2 x2 > n 1 > ðcos1 axÞn1 120 nðn 1Þ ðcos1 axÞn2 dx > xðcos axÞ > > a > > > > or > > Z < ½n=2 466. ðcos1 axÞn dx ¼ X n! > xðcos1 axÞn2r ð1Þr > > ðn 2rÞ! > > r¼0 > pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > ½ðn1Þ=2 > X > n! 1 a2 x2 > > ðcos1 axÞn2r1 ð1Þr > : ðn 2r 1Þ!a r¼0 Z 467. Z 468.

Z 469. Z 470.

Z 471. Z 472. Z 473. Z 474. Z 475. Z 476. Z 477. Z 478.

1 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðsin1 axÞ dx ¼ ðsin1 axÞ2 2a 1 a 2 x2 xn xn1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi xn pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðsin1 axÞ dx ¼ 2 1 a2 x2 sin1 ax þ 2 na n a 1 a2 x2 Z n1 xn2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sin1 ax dx þ 2 na 1 a2 x2 1 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðcos1 axÞ dx ¼ ðcos1 axÞ2 2a 1 a 2 x2 xn xn1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi xn pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðcos1 axÞ dx ¼ 2 1 a2 x2 cos1 ax 2 2 2 na n a 1a x Z n1 xn2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cos1 ax dx þ na2 1 a2 x2 tan1 ax 1 dx ¼ ðtan1 axÞ2 a2 x2 þ 1 2a cot1 ax 1 dx ¼ ðcot1 axÞ2 a2 x2 þ 1 2a x sec1 ax dx ¼

x2 1 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sec ax 2 a2 x2 1 2a 2

xn sec1 ax dx ¼

xnþ1 1 sec1 ax nþ1 nþ1

sec1 ax sec1 ax þ dx ¼ x2 x x csc1 ax dx ¼

Z

xn dx pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 x2 1

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 x2 1 x

x2 1 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi csc ax þ 2 a2 x2 1 2a 2

xn csc1 ax dx ¼

xnþ1 1 csc1 ax þ nþ1 nþ1

csc1 ax csc1 ax dx ¼ 2 x x

Z

xn dx pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 x2 1

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 x2 1 x

FORMS INVOLVING TRIGONOMETRIC SUBSTITUTIONS Z

Z 480.

f

2z dz , 2 1 þ z 1 þ z2

x z ¼ tan 2

f

1 z2 dz , 2 1 þ z 1 þ z2

x z ¼ tan 2

Z f ðsin xÞ dx ¼ 2

479.

Z f ðcos xÞ dx ¼ 2

A-49


Z

Z

du f ðuÞ pffiffiffiffiffiffiffiffiffiffiffiffiffi , ðu ¼ sin xÞ 1 u2 Z Z du *482. f ðcos xÞ dx ¼ f ðuÞ pffiffiffiffiffiffiffiffiffiffiffiffiffi , ðu ¼ cos xÞ 1 u2 Z Z pffiffiffiffiffiffiffiffiffiffiffiffiffi du *483. f ðsin x, cos xÞ dx ¼ f ðu, 1 u2 Þ pffiffiffiffiffiffiffiffiffiffiffiffiffi , ðu ¼ sin xÞ 1 u2 Z Z 2z 1 z2 dz x , , z ¼ tan 484. f ðsin x, cos xÞ dx ¼ 2 f 1 þ z2 1 þ z2 1 þ z2 2 *481.

f ðsin xÞ dx ¼

LOGARITHMIC FORMS

Z

ðlog xÞ dx ¼ x log x x

485: Z

xðlog xÞ dx ¼

486: Z 487: Z 488: Z 489:

x2 x2 log x 2 4

x2 ðlog xÞ dx ¼

x3 x3 log x 3 9

xn ðlog axÞ dx ¼

xnþ1 xnþ1 log ax nþ1 ðn þ 1Þ2

ðlog xÞ2 dx ¼ xðlog xÞ2 2x log x þ 2x

Z 8 > xðlog xÞn n ðlog xÞn1 dx, > > > > Z < or 490: ðlog xÞn dx ¼ > n > X > ð log xÞr > n > : ð1Þ n!x r! r¼0 Z 491: Z 492: Z 493: Z 494:

ðn 6¼ 1Þ

ðlog xÞn 1 ðlog xÞnþ1 dx ¼ nþ1 x dx ðlog xÞ2 ðlog xÞ3 ¼ logðlog xÞ þ log x þ þ þ

log x 2 2! 3 3! dx ¼ logðlog xÞ x log x dx 1 ¼ xðlog xÞn ðn 1Þðlog xÞn1

Z xm dx xmþ1 mþ1 xm dx þ n ¼ n1 ðlog xÞ n 1 ðlog xÞn1 ðn 1Þðlog xÞ 8 Z xmþ1 ðlog xÞn n > > > xm ðlog xÞn1 dx > > mþ1 > Z < mþ1 or 496: xm ðlog xÞn dx ¼ > n > X > n! ð log xÞr > n mþ1 > > : ð1Þ m þ 1 x r!ðm þ 1Þnr Z

495:

r¼0

Z 497: Z 498:

xp cosðb ln xÞ dx ¼

x

½b sinðb ln xÞ þ ð p þ 1Þ cosðb ln xÞ þ c ð p þ 1Þ2 þ b2

xp sinðb ln xÞ dx ¼

xpþ1

½ð p þ 1Þ sinðb ln xÞ b cosðb ln xÞ þ c ð p þ 1Þ2 þ b2

½logðax þ bÞ dx ¼

ax þ b logðax þ bÞ x a

Z 499:

pþ1

* The square roots appearing in these formulas may be plus or minus, depending on the quadrant of x. Care must be used to give them the proper sign.

A-50


Z 500. Z 501.

Z 502.

503.

logðax þ bÞ a ax þ b logðax þ bÞ dx ¼ log x x2 b bx xm ½logðax þ bÞ dx ¼

" mþ1 # 1 b xmþ1 logðax þ bÞ mþ1 a mþ1 m þ1 X 1 b 1 axr mþ1 a r b r¼1

logðax þ bÞ 1 logðax þ bÞ 1 am1 ax þ b dx ¼ þ log xm m1 xm1 m1 b x r m 2 X m1 1 a 1 b þ , ðm > 2Þ m1 b r ax r¼1

Z h x þ ai dx ¼ ðx þ aÞ logðx þ aÞ ðx aÞ logðx aÞ log xa Z

504.

h x þ ai xmþ1 ðaÞmþ1 xmþ1 amþ1 dx ¼ xm log logðx þ aÞ logðx aÞ xa mþ1 mþ1 ½ðmþ1Þ=2 xm2rþ2 2amþ1 X 1 þ m þ 1 r¼1 m 2r þ 2 a

See note integral 392. Z 1 h x þ ai 1 xa 1 x2 a2 dx ¼ log log log 505. x2 xa x xþa a x2 8 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > 4ac b2 b 2cx þ b > > tan1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; x þ log X 2x þ > > > c 2c 4ac b2 > > > > or > Z < pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 506. ðlog XÞ dx ¼ b2 4ac b 2cx þ b > tanh1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; xþ log X 2x þ > > > 2c c b2 4ac > > > where > > > > : X ¼ a þ bx þ cx2 Z 507.

Z 508. Z 509. Z 510.

xnþ1 2c log X nþ1 nþ1 where X ¼ a þ bx þ cx2 xn ðlog XÞ dx ¼

½logðx2 a2 Þ dx ¼ x logðx2 a2 Þ 2x þ a log

x ½logðx þ Z

xnþ1 dx X

x a

xþa xa

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 a2 Þ dx ¼ x logðx þ x2 a2 Þ x2 a2

Z

513.

Z

ðb2 4ac > 0Þ

x ½logðx2 a2 Þ dx ¼ 12ðx2 a2 Þ logðx2 a2 Þ 12x2

½logðx þ

512.

xnþ2 b dx nþ1 X

½logðx2 þ a2 Þ dx ¼ x logðx2 þ a2 Þ 2x þ 2a tan1

Z 511.

Z

ðb2 4ac < 0Þ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x a2 x x2 a2 logðx þ x2 a2 Þ x2 a2 Þ dx ¼ 4 2 4

xm ½logðx þ

Z pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi xmþ1 1 xmþ1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dx logðx þ x2 a2 Þ x2 a2 Þ dx ¼ mþ1 mþ1 x2 a2

A-51


pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z logðx þ x2 þ a2 Þ logðx þ x2 þ a2 Þ 1 a þ x2 þ a2 log 514. dx ¼ a x2 x x pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z 2 2 2 2 logðx þ x a Þ logðx þ x a Þ 1 x þ sec1 dx ¼ 515. jaj x2 x a 2 Z 1 4 nþ1 x logðx2 a2 Þ anþ1 logðx aÞ 516. xn logðx2 a2 Þ dx ¼ nþ1 nþ1

ðaÞ

3 a2r xn2rþ1 5 logðx þ aÞ 2 n 2r þ 1 r¼0 ½X n=2

See note integral 392. EXPONENTIAL FORMS Z 517. Z 518. Z 519. Z

ex dx ¼ ex ex dx ¼ ex eax dx ¼

eax a

eax ðax 1Þ a2 8 xm eax m Z > xm1 eax dx > > > a > Z < a or 521. xm eax dx ¼ > m > X > m!xmr > ax > ð1Þr :e ðm rÞ!arþ1 r¼0

520.

x eax dx ¼

Z

eax dx ax a2 x2 a3 þ x3 ¼ log x þ þ þ þ

x 1! 2 2! 3 3! Z ax Z e 1 eax a eax 523. dx ¼ þ dx m 1 xm1 m 1 xm1 xm Z Z eax log x 1 eax dx 524. eax log x dx ¼ a a x Z dx ex 525. ¼ x logð1 þ ex Þ ¼ log x 1þe 1 þ ex Z dx x 1 526. ¼ logða þ bepx Þ a þ bepx a ap rffiffiffi Z dx 1 a ; ða > 0; b > 0Þ ¼ pffiffiffiffiffi tan1 emx 527. mx mx ae þ be b m ab 8 pffiffiffi mx pffiffiffi > ae b 1 > > p ffiffiffiffiffi log p ffiffiffiffi mx pffiffiffi > > > Z a e þ b 2m ab < dx ¼ 528. or aemx bemx > > rffiffiffi > > a mx > 1 > e ; ða > 0; b > 0Þ : pffiffiffiffiffi tanh1 b m ab Z ax þ ax 529. ðax ax Þ dx ¼ log a Z ax e 1 530. dx ¼ logðb þ ceax Þ ac b þ ceax Z x eax eax 531. dx ¼ 2 a ð1 þ axÞ ð1 þ axÞ2 522.

A-52


Z 532. Z 533. Z 534.

2

2

x ex dx ¼ 12ex eax ½sinðbxÞ dx ¼

eax ½a sinðbxÞ b cosðbxÞ a2 þ b2

eax ½sinðbxÞ ½sinðcxÞ dx ¼

eax ½ðb cÞ sinðb cÞx þ a cosðb cÞx 2½a2 þ ðb cÞ2 eax ½ðb þ cÞ sinðb þ cÞx þ a cosðb þ cÞx 2½a2 þ ðb þ cÞ2

8 ax e ½a sinðb cÞx ðb cÞ cosðb cÞx > > > > > 2½a2 þ ðb cÞ2 > > > > > > > eax ½a sinðb þ cÞx ðb þ cÞ cosðb þ cÞx > > þ > > > 2½a2 þ ðb þ cÞ2 > > > > > > > or > > Z < ax ax 535. e ½sinðbxÞ ½cosðcxÞ dx ¼ e ½ða sin bx b cos bxÞ½cosðcx Þ > > > > > > > > cðsin bxÞ sinðcx Þ > > > > > > where > > > > qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > > > ¼ ða2 þ b2 c2 Þ2 þ 4a2 c2 , > > > > : cos ¼ a2 þ b2 c2 , sin ¼ 2ac Z 536. Z 537. Z 538. Z 539.

Z 540. Z 541. Z 542.

Z 543.

eax ½sinðbxÞ ½sinðbx þ cÞ dx ¼

eax cos c eax ½a cosð2bx þ cÞ þ 2b sinð2bx þ cÞ 2a 2ða2 þ 4b2 Þ

eax ½sinðbxÞ ½cosðbx þ cÞ dx ¼

eax sin c eax ½a sinð2bx þ cÞ 2b cosð2bx þ cÞ þ 2a 2ða2 þ 4b2 Þ

eax ½cosðbxÞ dx ¼

eax ½a cosðbxÞ þ b sinðbxÞ a2 þ b2

eax ½cosðbxÞ ½cosðcxÞ dx ¼

eax ½ðb cÞ sinðb cÞx þ a cosðb cÞx 2½a2 þ ðb cÞ2 ax e ½ðb þ cÞ sinðb þ cÞx þ a cosðb þ cÞx þ 2½a2 þ ðb þ cÞ2

eax ½cosðbxÞ ½cosðbx þ cÞ dx ¼

eax ½cosðbxÞ ½sinðbx þ cÞ dx ¼

eax ½sinn bx dx ¼

eax ½cosn bx dx ¼

eax cos c eax ½a cosð2bx þ cÞ þ 2b sinð2bx þ cÞ þ 2a 2ða2 þ 4b2 Þ

eax sin c eax ½a sinð2bx þ cÞ 2b cosð2bx þ cÞ þ 2a 2ða2 þ 4b2 Þ

1 ða sin bx nb cos bxÞeax sinn1 bx a2 þ n2 b2

Z þ nðn 1Þb2 eax ½sinn2 bx dx

a2

1 ða cos bx þ nb sin bxÞeax cosn1 bx 2 2 þn b

Z þ nðn 1Þb2 eax ½cosn2 bx dx

A-53


Z 544.

Z 545.

Z 546.

1 xm ex sin x dx ¼ xm ex ðsin x cos xÞ 2 Z Z m m xm1 ex sin x dx þ xm1 ex cos x dx 2 2

xm eax ½sin bx dx ¼

8 a sin bx b cos bx > > xm eax > > > a2 þ b2 > > > > > Z > > m > > > xm1 eax ða sin bx b cos bxÞ dx 2 > > a þ b2 > > > > > > < or > m > X > ð1Þr m!xmr > > sin½bx ðr þ 1Þ eax > > > rþ1 ðm rÞ! > r¼0 > > > > > > > > where > > > > > ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p > : ¼ a2 þ b2 ; cos ¼ a; sin ¼ b

1 xm ex cos x dx ¼ xm ex ðsin x þ cos xÞ 2 Z Z m m xm1 ex sin x dx xm1 ex cos x dx 2 2

8 a cos bx þ b sin bx > > xm eax > > > a2 þ b2 > > > > > Z > > m > > > xm1 eax ða cos bx þ b sin bxÞ dx > > a2 þ b2 > > Z < 547. xm eax cos bx dx ¼ or > > > > > m > X ð1Þr m!xmr > > > cos½bx ðr þ 1Þ > eax > > rþ1 ðm rÞ! > r¼0 > > > > > > pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : ¼ a2 þ b2 ; cos ¼ a; sin ¼ b

A-54


8 eax cosm1 x sinn x½a cos x þ ðm þ nÞ sin x > > > > > > ðm þ nÞ2 þ a2 > > > Z > > na > > > eax ðcosm1 xÞðsinn1 xÞ dx > 2 > > ðm þ nÞ þ a2 > > > Z > > > ðm 1Þðm þ nÞ > > þ eax ðcosm2 xÞðsinn xÞ dx > > > ðm þ nÞ2 þ a2 > > > > > or > > > > > ax m > e cos x ½sinn1 x½a sin x ðm þ nÞ cos x > > > > > ðm þ nÞ2 þ a2 > > > Z > > > ma > > þ eax ðcosm1 xÞðsinn1 xÞ dx > > > ðm þ nÞ2 þ a2 > > > > Z > > ðn 1Þðm þ nÞ > > eax ðcosm xÞðsinn2 xÞ dx þ > 2 > 2 > ðm þ nÞ þ a > Z < or 548. eax ðcosm xÞðsinn xÞ dx ¼ > > > eax ðcosm1 xÞðsinn1 xÞða sin x cos x þ m sin2 x n cos2 xÞ > > > > > > ðm þ nÞ2 þ a2 > > > > Z > > mðm 1Þ > ax m2 > xÞðsinn xÞ dx > > þ ðm þ nÞ2 þ a2 e ðcos > > > > > Z > > nðn 1Þ > > > þ eax ðcosm xÞðsinn2 xÞ dx > 2 > ðm þ nÞ þ a2 > > > > > or > > > > > ax > > e ðcosm1 xÞðsinn1 xÞða cos x sin x þ m sin2 x n cos2 xÞ > > > > > ðm þ nÞ2 þ a2 > > > Z > > > mðm 1Þ > > eax ðcosm2 xÞðsinn2 xÞ dx þ > > > ðm þ nÞ2 þ a2 > > > Z > > > ðn mÞðn þ m 1Þ > > eax ðcosm xÞðsinn2 xÞ dx : þ ðm þ nÞ2 þ a2 Z 549. Z 550. Z 551. Z 552. Z 553.

xeax ðsin bxÞ dx ¼

xeax ðcos bxÞ dx ¼

xeax eax ða sin bx b cos bxÞ ½ða2 b2 Þ sin bx 2ab cos bx a2 þ b2 ða2 þ b2 Þ2 xeax eax ða cos bx b sin bxÞ ½ða2 b2 Þ cos bx 2ab sin bx 2 2 þb ða þ b2 Þ2

a2

eax eax ½a sin x þ ðn 2Þ cos x a2 þ ðn 2Þ2 þ n dx ¼ sin x ðn 1Þðn 2Þ ðn 1Þðn 2Þ sinn1 x eax eax ½a cos x ðn 2Þ sin x a2 þ ðn 2Þ2 dx ¼ þ ðn 1Þðn 2Þ cosn1 x cosn x ðn 1Þðn 2Þ eax tann x dx ¼ eax

tann1 x a n1 n1

Z

eax tann1 x dx

HYPERBOLIC FORMS Z ðsinh xÞ dx ¼ cosh x

554. Z 555.

ðcosh xÞ dx ¼ sinh x

A-55

Z

Z

Z

eax dx sinn2 x eax dx cosn2 x

eax tann2 x dx


Z ðtanh xÞ dx ¼ log cosh x

556. Z

ðcoth xÞ dx ¼ log sinh x

557. Z 558. Z 559.

ðsech xÞ dx ¼ tan1 ðsinh xÞ x ðcsch xÞ dx ¼ log tanh 2

Z xðsinh xÞ dx ¼ x cosh x sinh x

560. Z 561.

xn ðsinh xÞ dx ¼ xn cosh x n

Z

xn1 ðcosh xÞ dx

Z xðcosh xÞ dx ¼ x sinh x cosh x

562. Z 563.

xn ðcosh xÞ dx ¼ xn sinh x n

Z

xn1 ðsinh xÞ dx

Z ðsech xÞðtanh xÞ dx ¼ sech x

564. Z

ðcsch xÞðcoth xÞ dx ¼ csch x

565. Z

sinh 2x x 4 2 8 1 > > ðsinhmþ1 xÞðcoshn1 xÞ > > m þn > > > Z > > n1 > > ðsinhm xÞðcoshn2 xÞ dx þ > > > mþn Z < or 567. ðsinhm xÞðcoshn xÞ dx ¼ > > > 1 > > sinhm1 x coshnþ1 x > > > mþn > > Z > > > > m 1 ðsinhm2 xÞðcoshn xÞ dx, ðm þ n 6¼ 0Þ : mþn 8 1 > > > m1 > > ðm nÞðsinh xÞðcoshn1 xÞ > > > Z > > > mþn2 dx > > , ðm 6¼ 1Þ > m2 > m 1 > ðsinh xÞðcoshn xÞ Z < dx or 568. ðsinhm xÞðcoshn xÞ > > > 1 > > > > m1 > ðn 1Þ sinh x coshn1 x > > > Z > > > mþn2 dx > > , ðn 6¼ 1Þ : þ n1 ðsinhm xÞðcoshn2 xÞ Z 569. ðtanh2 xÞ dx ¼ x tanh x

566.

Z 570. Z 571. Z 572.

ðsinh2 xÞ dx ¼

ðtanhn xÞ dx ¼

tanhn1 x þ n1

Z

ðtanhn2 xÞ dx,

ðsech2 xÞ dx ¼ tanh x ðcosh2 xÞ dx ¼

sinh 2x x þ 4 2

A-56

ðn 6¼ 1Þ


Z 573. Z 574. Z 575.

ðcoth2 xÞ dx ¼ x coth x ðcothn xÞ dx ¼

cothn1 x þ n1

ðsinh mxÞðsinh nxÞ dx ¼

579. 580.

581.

582.

583. 584. 585. 586. 587. 588. 589. 590. 591.

sinhðm þ nÞx sinhðm nÞx þ , 2ðm þ nÞ 2ðm nÞ

ðm2 6¼ n2 Þ

ðsinh mxÞðcosh nxÞ dx ¼

coshðm þ nÞx coshðm nÞx þ , 2ðm þ nÞ 2ðm nÞ

ðm2 6¼ n2 Þ

Z x x pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dx ¼ x sinh1 x2 þ a2 , ða > 0Þ sinh1 a a 2 Z x x a2 x x pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x sinh1 dx ¼ x2 þ a2 , ða > 0Þ þ sinh1 a a 4 2 4 nþ1 Z Z x 1 xnþ1 xn sinh1 x dx ¼ dx, ðn 6¼ 1Þ sinh1 x n þ 1 ð1 þ x2 Þ1=2 nþ1 8 x pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x > x cosh1 x2 a2 , cosh1 > 0 > > Z < a a x or dx ¼ cosh1 > a pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > x > : x cosh1 x þ x2 a2 , ða > 0Þ cosh1 < 0 , a a Z 1 x 2x2 a2 x x x cosh1 dx ¼ cosh1 ðx2 a2 Þ2 a a 4 4 Z Z xnþ1 1 xnþ1 cosh1 x xn ðcosh1 xÞ dx ¼ dx, ðn 6¼ 1Þ 2 n þ 1 ðx 1Þ1=2 nþ1 Z x x x a tanh1 dx ¼ x tanh1 þ logða2 x2 Þ, 1 a a 2 a Z 2 2 x x a x ax x dx ¼ tanh1 þ , x tanh1 1 a a 2 a 2 Z nþ1 Z xnþ1 1 x coth1 x þ dx, ðn 6¼ 1Þ xn coth1 x dx ¼ n þ 1 x2 1 nþ1 Z ðsech1 xÞ dx ¼ xsech1 x þ sin1 x Z

x2 1 pffiffiffiffiffiffiffiffiffiffiffiffiffi2 1x sech1 x 2 2 Z Z xnþ1 1 xn sech1 x þ 593. xn sech1 x dx ¼ dx, n þ 1 ð1 x2 Þ1=2 nþ1 Z x 594. csch1 x dx ¼ xcsch1 x þ sinh1 x jxj 592.

ðm2 6¼ n2 Þ

ðcosh mxÞðcosh nxÞ dx ¼ Z

578.

ðn 6¼ 1Þ

sinhðm þ nÞx sinhðm nÞx , 2ðm þ nÞ 2ðm nÞ

Z 577.

cothn2 x dx,

ðcsch2 xÞ dx ¼ ctnh x

Z 576.

Z

xsech1 x dx ¼

A-57

ðn 6¼ 1Þ


Z

x2 1 x pffiffiffiffiffiffiffiffiffiffiffiffiffi2 1þx csch1 x þ 2 jxj 2 Z Z xnþ1 1 x xn csch1 x þ dx, 596. xn csch1 x dx ¼ n þ 1 jxj ðx2 þ 1Þ12 nþ1 xcsch1 x dx ¼

595.

ðn 6¼ 1Þ

DEFINITE INTEGRALS Z

1

597. 0

Z

1 n 1þ 1 1 m xn1 ex dx ¼ log dx ¼ x n m¼1 1 þ n 0 m ¼ ðnÞ, n 6¼ 0, 1, 2, 3, . . . Z 1

1

tn pt dt ¼

598. 0

Z

n! , ðlog pÞnþ1

1

tn1 eðaþ1Þt dt ¼

599. 0

Z

1

600. 0

n1

1 Y

ðGamma FunctionÞ

ðn ¼ 0, 1, 2, 3, . . . and p > 0Þ

ðnÞ , ða þ 1Þn

ðn > 0, a > 1Þ

1 n ðn þ 1Þ xm log dx ¼ , x ðm þ 1Þnþ1

ðm > 1, n > 1Þ

601. ðnÞ is finite if n > 0, ðn þ 1Þ ¼ nðnÞ 602. ðnÞ ð1 nÞ ¼

p sin np

603. ðnÞ ¼ ðn 1Þ! if n ¼ integer > 0 Z1 pffiffiffi 2 604. ð12Þ ¼ 2 et dt ¼ p ¼ 1:7724538509 ¼ ð12Þ! 0

1 3 5 . . . ð2n 1Þ pffiffiffi p n ¼ 1, 2, 3, . . . 2n p ffiffiffi ð1Þn 2n p n ¼ 1, 2, 3, . . . 606. ðn þ 12Þ ¼ 1 3 5 . . . ð2n 1Þ Z1 Z1 xm1 ðmÞðnÞ ¼ Bðm, nÞ xm1 ð1 xÞn1 dx ¼ dx ¼ ðm þ nÞ ð1 þ xÞmþn 607. 0 0 ðBeta functionÞ 605. ðn þ 12Þ ¼

608. Bðm, nÞ ¼ Bðn, mÞ ¼ Z

ðmÞðnÞ , where m and n are any positive real numbers: ðm þ nÞ

b

ðx aÞm ðb xÞn dx ¼ ðb aÞmþnþ1

609. a

Z

1

610. 1

Z

1

611. 0

Z

1

612. 0

Z

1

613. 0

Z 614. 0

1

dx 1 , ¼ xm m 1

ðm þ 1Þ ðn þ 1Þ , ðm þ n þ 2Þ

½m > 1

dx ¼ p csc pp, ð1 þ xÞxp dx ¼ p cot pp, ð1 xÞxp

½ p < 1 ½ p < 1

xp1 dx p ¼ ð1 þ xÞ sin pp ¼ Bð p, 1 pÞ ¼ ð pÞð1 pÞ, m1

x dx p ¼ mp , 1 þ xn n sin n

½0 < m < n

A-58

½0 1, n > 1, b > aÞ

DEFINITE INTEGRALS (Continued)

3 2 aþ1 aþ1 c 1 7 xa dx mðaþ1bcÞ=b 6 b b 6 7 ¼ 4 5 b Þc ðm þ x b ðcÞ 0

Z 615.

Z

a > 1, b > 0, m > 0, c > aþ1 b

1

dx pffiffiffi ¼ p ð1 þ xÞ x

1

a dx p ¼ , a2 þ x2 2

616. 0

Z 617. 0

Z

p if a > 0; 0, if a ¼ 0; , if a < 0 2

a

ða2 x2 Þn=2 dx ¼

618. 0

1 2

Z

a

ða2 x2 Þn=2 dx ¼

a

1 3 5...n p

anþ1 2 4 6 . . . ðn þ 1Þ 2

ðn oddÞ

8 1 mþnþ1 m þ 1 n þ 2 > > > a , B > > 2 2 2 > > > > > Za or < xm ða2 x2 Þn=2 dx ¼ 619. mþ1 nþ2 > 0 > > 1 mþnþ1 2 2 > > > a > > mþnþ3 >2 > : 2

Z

p=2

ðsinn xÞ dx ¼

620. 0

Z

sin mx dx p ¼ , x 2

1

cos x dx ¼1 x

1

tan x dx p ¼ x 2

0

622. 0

Z 623. 0

Z

p if m > 0; 0, if m ¼ 0; , if m < 0 2

Z

p

p

sin ax sin bx dx ¼

624. 0

Z

cos ax cos bx dx ¼ 0,

ða 6¼ b; a, b integersÞ

0

Z

p=a

p

½sinðaxÞ ½cosðaxÞ dx ¼ 0

½sinðaxÞ ½cosðaxÞ dx ¼

625. 0

Z

ðn an even integer, n 6¼ 0Þ

2 4 6 8 . . . ðn 1Þ > > , ðn an odd integer, n 6¼ 1Þ > > 1 3 5 7 . . . ðnÞ > > > > > or > > > > > nþ1 > > > pffiffiffi > > p 2 > > , ðn > 1Þ > > : 2 nþ1 2

1

621. Z

8 Z p=2 > > ðcosn xÞ dx > > > > 0 > > > > or > > > > 1 3 5 7 . . . ðn 1Þ p > > > > 2 4 6 8 . . . ðnÞ 2 , > > > > > < or

0 p

½sinðaxÞ ½cosðbxÞ dx ¼

626. 0

Z 627. 0

1

sin x cos mx dx ¼ 0, x

2a , if a b is odd, or 0 if a b is even a2 b2 if m < 1 or m > 1;

A-59

p p , if m ¼ 1; , if m2 < 1 4 2

DEFINITE INTEGRALS (Continued)

Z

1

sin ax sin bx pa dx ¼ , ða bÞ x2 2 Zp Zp p 629: sin2 mx dx ¼ cos2 mx dx ¼ 2 0 0 628:

0

Z

1

sin2 ð pxÞ pp dx ¼ x2 2

1

sin x p , dx ¼ xp 2ð pÞ sinð pp=2Þ

0 0; , p ¼ q > 0 x 2 4

1

cosðmxÞ p jmaj e dx ¼ x2 þ a2 2jaj

633: 0

Z 634: 0

Z 635: 0

Z

1

cosðx2 Þ dx ¼

636:

Z

0

Z

sinðx2 Þ dx ¼

0 1

sin axn dx ¼

637: 0

Z

1

rffiffiffi 1 p 2 2

1 p ð1=nÞ sin , na1=n 2n

n>1

1

1 p ð1=nÞ cos , na1=n 2n rffiffiffi Z1 Z1 sin x cos x p pffiffiffi dx ¼ pffiffiffi dx ¼ 639: x x 2 0 0 cos axn dx ¼

638:

n>1

0

Z

1

640: ðaÞ 0

Z

1

sin3 x 3p dx ¼ x3 8

1

sin4 x p dx ¼ x4 3

641: 0

Z 642: 0

Z

p=2

643: 0

Z 644: 0

Z

p

2p

0

Z 0

Z

1

ða > b 0Þ ða2 < 1Þ

dx p ¼ a2 sin2 x þ b2 cos2 x 2ab

p=2

dx pða2 þ b2 Þ ¼ , 2 4a3 b3 ða2 sin x þ b2 cos2 xÞ 2

0

0

ða < 1Þ

p=2

648: Z

dx cos1 a ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi , 1 þ a cos x 1 a2

sin3 x 3 dx log 3 x2 4

cos ax cos bx b dx ¼ log x a

0

649:

1

0

dx 2p ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi , 1 þ a cos x 1 a2

647: Z

Z ðbÞ

dx p ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , a þ b cos x a2 b2

645:

646:

sin3 x p dx ¼ x 4

p=2

1 n m , sinn1 x cosm1 x dx ¼ B , 2 2 2

A-60

ða, b > 0Þ

m and n positive integers

DEFINITE INTEGRALS (Continued) Z

p=2

ðsin2nþ1 Þ d ¼

650: 0

Z

p=2

ðsin2n Þ d ¼

651: 0

Z

p=2

652: 0

Z

p=2

3 pffiffiffiffiffiffiffiffiffiffi ð2pÞ2 cos d ¼ % & 2 ð14Þ

654: 0

Z

p=2

ðtanh Þ d ¼

655: 0

Z

1

656: 0

ðn ¼ 1, 2, 3, . . .Þ

, x 1 1 1 1 dx ¼ 2 2 2 þ 2 2 þ sin x 1 3 5 7 dx p ¼ 1 þ tanm x 4

0

ðn ¼ 1, 2, 3, . . .Þ

1 3 5 . . . ð2n 1Þ p , 2 4 . . . ð2nÞ 2

p=2

653: Z

2 4 6 . . . ð2nÞ , 1 3 5 . . . ð2n þ 1Þ

p , hp 2 cos 2

ð0 < h < 1Þ

tan1 ðaxÞ tan1 ðbxÞ p a dx ¼ log , x 2 b

ða, b > 0Þ b

b

b

657. The area enclosed by a curve defined through the equation xc þ yc ¼ ac where a > 0, c a positive odd integer and b a positive even integer is given by h c i2 2 2ca b 2c b b ZZZ 658. I ¼ xh1 ym1 zn1 dv, where R denotes the region of space bounded by the R

xp yq zk þ þ ¼ 1, which a b c lies in the first octant and where h, m, n, p, q, k, a, b, c, denote positive real numbers is given by h m n Z h½1ðx=aÞp 1=e Z c½1ðx=aÞp ðy=bÞq 1=e Za ah bm cn p q k h1 m n1 x dx y dy z dz ¼ h m n pqk 0 0 0 þ þ þ1 p q k Z1 1 659: eax dx ¼ , ða > 0Þ a 0 co-ordinate planes and that portion of the surface

Z

1 ax

e

660: 0

ebx b dx ¼ log , a x

ða, b > 0Þ

8 ðn þ 1Þ > > , ðn > 1, a > 0Þ > > Z1 < anþ1 661: xn eax dx ¼ or > 0 > > n! > : , ða > 0, n positive integerÞ anþ1 Z1 ðkÞ nþ1 662: xn expðaxp Þ dx ¼ k , n > 1, p > 0, a > 0, k ¼ pa p 0 Z

1

2 2

ea

663:

x

0

Z 664: 0

1

dx ¼

1 pffiffiffi 1 1 p¼ , 2a 2a 2

ða > 0Þ

2

xex dx ¼ 12

A-61

DEFINITE INTEGRALS (Continued) pffiffiffi 1 p 2 x2 ex dx ¼ 4 0 rffiffiffi Z1 1 3 5 . . . ð2n 1Þ p 2 x2n eax dx ¼ 2nþ1 an a 0 Z1 n! 2 x2nþ1 eax dx ¼ nþ1 , ða > 0Þ 2a 0 " # Z1 m X m! ar xm eax dx ¼ mþ1 1 ea a r! 0 r¼0 Z

665:

666: 667:

668:

pffiffiffi e2a p , ða 0Þ 2 0 rffiffiffi Z1 pffiffiffi 1 p enx x dx ¼ 2n n 0 rffiffiffi Z 1 nx e p pffiffiffi dx ¼ n x 0 Z1 a eax ðcos mxÞ dx ¼ 2 , ða > 0Þ a þ m2 0 Z1 m eax ðsin mxÞ dx ¼ 2 , ða > 0Þ a þ m2 0 Z1 2ab xeax ½sinðbxÞ dx ¼ , ða > 0Þ ða2 þ b2 Þ2 0 Z1 a2 b2 xeax ½cosðbxÞ dx ¼ , ða > 0Þ ða2 þ b2 Þ2 0 Z

669:

670:

671: 672: 673: 674:

675:

Z

1

2

eðx

a2 =x2 Þ

dx ¼

xn eax ½sinðbxÞ dx ¼

n!½ða þ ibÞnþ1 ða ibÞnþ1 , 2iða2 þ b2 Þnþ1

ði2 ¼ 1, a > 0Þ

xn eax ½cosðbxÞ dx ¼

n!½ða ibÞnþ1 þ ða þ ibÞnþ1 , 2ða2 þ b2 Þnþ1

ði2 ¼ 1, a > 0Þ

1

676: 0

Z

1

677: 0

Z

1 ax

Z

1

sin x dx ¼ cot1 a, ða > 0Þ x 0 pffiffiffi Z1 b2 p 2 2 679: ea x cos bx dx ¼ exp 2 , ðab 6¼ 0Þ 4a 2a 0 Z1 p p et cos tb1 ½sinðt sin Þ dt ½ðbÞ sinðbÞ, b > 0, < < 680: 2 2 0 Z1 p p 681: et cos tb1 ½cosðt sin Þ dt ½ðbÞ cosðbÞ, b > 0, < < 2 2 0 Z1 bp 682: tb1 cos t dt ¼ ½ðbÞ cos , ð0 < b < 1Þ 2 0 Z1 bp tb1 ðsin tÞ dt ¼ ½ðbÞ sin , ð0 < b < 1Þ 683: 2 0 e

678:

684:

ðlog xÞn dx ¼ ð1Þn n!

0

log

pffiffiffi 1 p 1 2 dx ¼ x 2

log

1 pffiffiffi 1 2 dx ¼ p x

Z 1 685: 0

Z 1 686: 0

A-62

DEFINITE INTEGRALS (Continued) Z 1 1 n log dx ¼ n! 687: x 0 Z

1

x logð1 xÞ dx ¼ 34

688: 0

Z

1

x logð1 þ xÞ dx ¼ 14

689: 0

Z

1

xm ðlog xÞn dx ¼

690: 0

691:

692:

693:

694:

695:

696:

697:

698:

ð1Þn n! , ðm þ 1Þnþ1

m > 1, n ¼ 0, 1, 2, . . .

If n 6¼ 0, 1, 2, . . . replace n! by ðn þ 1Þ: Z1 log x p2 dx ¼ 1 þ x 12 0 Z1 log x p2 dx ¼ 1 x 6 0 Z1 logð1 þ xÞ p2 dx ¼ x 12 0 Z1 logð1 xÞ p2 dx ¼ x 6 0 Z1 p2 ðlog xÞ½logð1 þ xÞ dx ¼ 2 2 log 2 12 0 Z1 p2 ðlog xÞ½logð1 xÞ dx ¼ 2 6 0 Z1 log x p2 dx ¼ 2 1 x 8 0 Z1 1 þ x dx p2

¼ log 1 x x 4 0 Z

1

log x dx p pffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ log 2 2 1 x2 Z 1 n 1 ðn þ 1Þ 700: xm log dx ¼ , if m þ 1 > 0, n þ 1 > 0 x ðm þ 1Þnþ1 0 Z1 p ðx xq Þ dx pþ1 ¼ log 701: , ð p þ 1 > 0, q þ 1 > 0Þ log x qþ1 0 699:

0

Z

1

702: 0

Z

1

703: 0

Z

pffiffiffi dx sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ¼ p, ðsame as integral 686Þ 1 log x x e þ1 p2 log x dx ¼ e 1 4 Z

p=2

p=2

p log cos x dx ¼ log 2 2

p=2

p log csc x dx ¼ log 2 2

ðlog sin xÞ dx ¼

704: 0

Z

0

Z

p=2

ðlog sec xÞ dx ¼

705: 0

Z

0 p

xðlog sin xÞ dx ¼

706: 0

Z

p2 log 2 2

p=2

ðsin xÞðlog sin xÞ dx ¼ log 2 1

707: 0

A-63


p=2

ðlog tan xÞ dx ¼ 0

708: 0

Z

p

logða b cos xÞ dx ¼ p log

709:

aþ

0

Z 710:

p

logða2 2ab cos x þ b2 Þ dx ¼

0

Z

1

sin ax p ap dx ¼ tanh sinh bx 2b 2b

1

cos ax p p dx ¼ sech cosh bx 2b 2b

1

dx p ¼ cosh ax 2a

1

x dx p2 ¼ 2 sinh ax 4a

711: 0

Z 712: 0

Z 713: 0

Z 714: 0

Z

1

715: Z Z

ð0 jbj < aÞ

eax ðsinh bxÞ dx ¼

b , a2 b2

ð0 jbj < aÞ

1

1

sinh ax p ap 1 dx ¼ csc ebx þ 1 2b b 2a

1

sinh ax 1 p ap dx ¼ cot ebx 1 2a 2b b

0

Z 718: 0

Z

p=2

719: 0

Z

p=2

720: 0

Z

ab>0 ba>0

a , a2 b2

0

717:

2p log a, 2p log b,

ða bÞ

eax ðcosh bxÞ dx ¼

0

716:

,

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! a2 b2 , 2

" # 2 dx p 1 1 3 2 4 1 3 5 2 6 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 1þ k þ k2 þ k þ

, 2 2 4 2 4 6 1 k2 sin2 x 2 # " 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p 1 1 3 2 k4 1 3 5 2 k6 1

, 1 k2 sin2 x dx ¼ k2 2 2 2 4 3 2 4 6 5

1

ex log x dx ¼ ¼ 0:5772157 . . .

721: 0

Z

1

2

ex log x dx ¼

722: 0

Z

1

723: 0

Z 724: 0

1

pffiffiffi p ð þ 2 log 2Þ 4

1 1 ex dx ¼ ¼ 0:5772157 . . . x 1e x

[Euler’s Constant]

1 1 ex dx ¼ ¼ 0:5772157 . . . x 1þx

For n even: Z n=21 1 X n sinðn 2kÞx 1 n þ n x 725: cosn x dx ¼ n1 k ðn 2kÞ 2 n=2 2 k¼0

A-64

if k2 < 1

if k2 < 1


n

sin x dx ¼

726:

1 2n1

n=21 X k¼0

h p i 1 n sin ðn 2kÞ 2 x þ n k 2 2k n

For n odd: Z ðn1Þ=2 1 X n sinðn 2kÞx 727: cosn x dx ¼ n1 k n 2k 2 k¼0 Z 728:

sinn x dx ¼

1 2n1

ðn1Þ=2 X k¼0

h i sin ðn 2kÞ p x n 2 k 2k n

A-65

n n=2

! x

DIFFERENTIAL EQUATIONS SPECIAL FORMULAS Certain types of differential equations occur sufficiently often to justify the use of formulas for the corresponding particular solutions. The following set of tables I to XIV covers all first, second, and nth order ordinary linear differential equations with constant coefficients for which the right members are of the form PðxÞerx sin sx or PðxÞerx cos sx, where r and s are constants and P(x), is a polynomial of degree n. When the right member of a reducible linear partial differential equation with constant coefficients is not zero, particular solutions for certain types of right members are contained in tables XV to XXI. In these tables both F and P are used to denote polynomials, and it is assumed that no denominator is zero. In any formula the roles of x and y may be reversed throughout, changing a formula in which x dominates to one in which y dominates. Tables XIX, m m! stands for and is the XX, XXI are applicable whether the equations are reducible or not. The symbol n ðm nÞ!n! nþ1 st coefficient in the expansion of (a þ b)m. Also 0! ¼ 1 by definition. The tables as herewith given are those contained in the text Differential Equations by Ginn and Company (1955) and are published with their kind permission and that of the author, Professor Frederick H. Steen.

Solution of Linear Differential Equations with Constant Coefficients Any linear differential equation with constant coefficients may be written in the form pðDÞy ¼ RðxÞ where D is the differential operation Dy ¼

dy dx

p(D) is a polynomial in D, y is the dependent variable, x is the independent variable, R(x) is an arbitrary function of x. A power of D represents repeated differentiation, that is Dn y ¼

dny dxn

For such an equation, the general solution may be written in the form y ¼ yc þ yp where yp is any particular solution, and yc is called the complementary function. This complementary function is defined as the general solution of the homogeneous equation, which is the original differential equation with the right side replaced by zero, i.e. pðDÞy ¼ 0 The complementary function yc may be determined as follows: 1. 2.

Factor the polynomial p(D) into real and complex linear factors, just as if D were a variable instead of an operator. For each nonrepeated linear factor of the form (D a), where a is real, write down a term of the form

ceax where c is an arbitrary constant.

A-65

3.

For each repeated real linear factor of the form (D a)n, write down n terms of the form c1 eax þ c2 xeax þ c3 x2 eax þ þ cn xn1 eax

4.

where the ci’s are arbitrary constants. For each non-repeated conjugate complex pair of factors of the form (D a þ ib)(D a ib), write down 2 terms of the form c1 eax cos bx þ c2 eax sin bx

5.

For each repeated conjugate complex pair of factors of the form (D a þ ib)n(D a ib)n, write down 2n terms of the form c1 eax cos bx þ c2 eax sin bx þ c3 xeax cos bx þ c4 xeax sin bx þ þ c2n1 xn1 eax cos bx þ c2n xn1 eax sin bx

6.

The sum of all the terms thus written down is the complementary function yc.

To find the particular solution yp, use the following tables, as shown in the examples. For cases not shown in the tables, there are various methods of finding yp. The most general method is called variation of parameters. The following example illustrates the method: Find yp for (D2 4) y ¼ ex. This example can be solved most easily by use of equation 63 in the tables following. However it is given here as an example of the method of variation of parameters. The complementary function is yc ¼ c1 e2x þ c2 e2x To find yp, replace the constants in the complementary function with unknown functions, yp ¼ ue2x þ ve2x We now prepare to substitute this assumed solution into the original equation. We begin by taking all the necessary derivatives: yp ¼ ue2x þ ve2x y0p ¼ 2ue2x þ 2ve2x þ u0 e2x v0 e2x For each derivative of yp except the highest, we set the sum of all the terms containing u0 and v0 to 0. Thus the above equation becomes u0 e2x þ v0 e2x ¼ 0

and y0p ¼ 2ue2x 2ve2x

Continuing to differentiate, we have y00p ¼ 4ue2x þ 4ve2x þ 2u0 e2x 2v0 e2x When we substitute into the original equation, all the terms not containing u0 or v0 cancel out. This is a consequence of the method by which yp was set up. Thus all that is necessary is to write down the terms containing u0 or v0 in the highest order derivative of yp, multiply by the constant coefficient of the highest power of D in p(D), and set it equal to R(x). Together with the previous terms in u0 and v0 which were set equal to 0, this gives us as many linear equations in the first derivatives of the unknown functions as there are unknown functions. The first derivatives may then be solved for by algebra, and the unknown functions found by integration. In the present example, this becomes u0 e2x þ v0 e2x ¼ 0 0 2x

2u e 0

2v0 e2x ¼ ex

0

We eliminate v and u separately, getting 4u0 e2x ¼ ex 4v0 e2x ¼ ex Thus u0 ¼ 14ex v0 ¼ 14e3x Therefore, by integrating u ¼ 14ex 1 3x v ¼ 12 e

A constant of integration is not needed, since we need only one particular solution. Thus 1 3x 2x yp ¼ ue2x þ ve2x ¼ 14ex e2x 12 e e 1 x ¼ 14ex 12 e ¼ 13ex

A-66

and the general solution is y ¼ yc þ yp ¼ c1 e2x þ c2 e2x 13ex The following samples illustrate the use of the tables. Example 1. Solve (D2 4)y ¼ sin 3x. Substitution of q ¼ 4, s ¼ 3 in formula 24 gives yp ¼

sin 3x 9 4

wherefore the general solution is sin 3x 13 Example 2. Obtain a particular solution of ðD2 4D þ 5Þy ¼ x2 e3x sin x: Applying formula 40 with a ¼ 2, b ¼ 1, r ¼ 3, s ¼ 1, P(x) ¼ x2, s þ b ¼ 2, s b ¼ 0, a r ¼ 1, (a r)2 þ (s þ b)2 ¼ 5, (a r)2 þ (s b)2 ¼ 1, we have

e3x sin x 2 0 2 2ð1Þ2 2ð1Þ0 3 1 2 23 3 1 0 0 x þ 2x þ 2 yp ¼ 2 5 1 25 1 1 125

e3x cos x 1 1 2 14 10 1 3ð1Þ4 1 3ð1Þ0 x þ 2x þ 2 2 5 1 25 1 125 1 1 2 4 2 2 28 136 3x x x e3x sin x þ x2 þ x e cos x ¼ 5 25 125 5 25 125 y ¼ c1 e2x þ c2 e2x

The special formulas effect a very considerable saving of time in problems of this type. Example 3. Obtain a particular solution of (D2 4D þ 5)y ¼ x2e2x cos x. (Compare with Example 2.) Formula 40 is not applicable here since for this equation r ¼ a, s ¼ b, wherefore the denominator (a r)2 þ (s b)2 ¼ 0. We turn instead to formula 44. Substituting a ¼ 2, b ¼ 1, P(x) ¼ x2 and replacing sin by cos, cos by sin, we obtain Z e2x cos x 2 2 e2x sin x 1 x x2 yp ¼ þ dx 4 4 2 2 2 3 x 1 x x e2x cos x þ e2x sin x ¼ 4 8 6 4 which is the required solution. Example 4. Find zp for ðDx 3Dy Þ z ¼ lnðy þ 3xÞ. Referring to Table XV we note that formula 69 (not 68) is applicable. This gives zp ¼ x lnðy þ 3xÞ It is easily seen that y=3 lnðy þ 3xÞ would serve equally well. Example 5. Solve ðDx þ 2Dy 4Þ z ¼ y cosðy 2xÞ. Since R in formula 76 contains a polynomial in x, not y, we rewrite the given equation in the form ðDy þ12 Dx 2Þ z ¼12 y cos ðy 2xÞ Then

zc ¼ e2y F x 21 yÞ ¼ e2x f ð2x yÞ

and by the formula 1 y 12 þ zp ¼ cosðy 2xÞ 2 2 2 1 ¼ ð2y þ 1Þ cos ðy 2xÞ 8 3 Example 6. Find zp for ðDx þ 4Dy Þ z ¼ ð2x yÞ2 . Using formula 79, we obtain RRR 2 3 u du u5 ð2x yÞ5 zp ¼ ¼ ¼ 3 5 4 3 ð8Þ 480 ½2 þ 4ð1Þ Example 7. Find zp for ðD3x þ 5D2x Dy 7Dx þ 4Þz ¼ e2xþ3y . By formula 87 zp ¼

23

e2xþ3y e2xþ3y ¼ 2 þ5 2 37 2þ4 58

Example 8. Find zp for ðD4x þ 6D3x Dy þ Dx Dy þ D2y þ 9Þz ¼ sin ð3x þ 4yÞ

A-67

Since every term in the left member is of even degree in the two operators Dx and Dy, formula 90 is applicable. It gives sinð3x þ 4yÞ ð9Þ2 þ 6ð9Þð12Þ þ ð12Þ þ ð16Þ þ 9 sinð3x þ 4yÞ ¼ 710

zp ¼

TABLE I: (D a)y ¼ R R 1. erx 2. sin sx* 3. P(x) 4. erx sin sx* 5. P(x) erx 6. P(x) sin sx*

7. P(x)erx sin sx* 8. eax 9. eax sin sx* 10. P(x)eax 11. P(x)eax sin sx

yp erx ra a sin sx þ s cos sx 1 s ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sin sx þ tan1 2 2 2 2 a a þs " a þs # 1 P0 ðxÞ P00 ðxÞ PðnÞ ðxÞ PðxÞ þ þ 2 þ

þ a a an a Replace a by a r in formula 2 and multiply by erx. Replace a by a r in formula 3 and multiply by erx.

# ak k2 ak2 s2 þ k4 ak4 s4 a a2 s2 0 a3 3as2 00 ðk1Þ PðxÞ þ P ðxÞ þ P ðxÞ þ

þ P ðxÞ þ

a2 þ s2 ða2 þ s2 Þ2 ða2 þ s2 Þ3 ða2 þ s2 Þk # " k k1 2 3 a s k3 ak3 s3 þ 1 s 2as 3a s s 00 0 ðk1Þ PðxÞ þ P ðxÞ þ P ðxÞ þ

þ P ðxÞ þ

cos sx 2 a þ s2 ða2 þ s2 Þ2 ða2 þ s2 Þ3 ða2 þ s2 Þk "

sin sx

Replace a by a r in formula 6 and multiply by erx. xeax eax cos sx Z s eax PðxÞ dx " #

eax sin sx P0 ðxÞ P000 ðxÞ Pv ðxÞ eax cos sx P00 ðxÞ Piv ðxÞ þ

þ

PðxÞ s s s5 s3 s3 s2 s4 *For cos sx in R replace ‘‘sin’’ by ‘‘cos’’ and ‘‘cos’’ by ‘‘ sin’’ in yp.

Dn ¼

dn dxn

m n

¼

m! ðm nÞ!n!

0! ¼ 1

TABLE II: (D a)2y ¼ R R

yp

12. erx

erx ðr aÞ2

13. sin sx*

14. P(x) 15. erx sin sx* 16. P(x)erx 17. P(x) sin sx*

1 1 2 2 1 2as s Þ sin sx þ 2as cos sx ¼ sin sx þ tan ½ða ða2 þ s2 Þ a2 þ s 2 a2 s 2 " # 1 2P0 ðxÞ 3P00 ðxÞ ðn þ 1ÞPðnÞ ðxÞ PðxÞ þ þ

þ þ a an a2 a2 Replace a by a r in formula 13 and multiply by erx. rx Replace " a by a r in formula 14 and multiply by e . sin sx

a2 s2 a3 3as2 0 a4 6a2 s2 þ s4 00 P ðxÞ þ 3 PðxÞ þ 2 2 P ðxÞ þ ða2 þ s2 Þ2 ða þ s2 Þ3 ða2 þ s2 Þ4 ak

þ ðk 1Þ

k 2

ak2 s2 þ

k k4 4 s

4 a

ða2 þ s2 Þk

3 ðk2Þ

P

ðxÞ þ 5

" þ cos sx

2as 3a2 s s3 0 4a3 s 4as3 00 P ðxÞ þ 3 2 PðxÞ þ 2 2 P ðxÞ þ ða2 þ s2 Þ2 ða þ s2 Þ3 ða þ s2 Þ4 þ ðk 1Þ

18. P(x)erx sin sx* 19. eax 20. eax sin sx 21. P(x)eax ax

22. P(x)e

sin sx*

k 1

ak1 s 2

k 3

ak3 s3 þ 2 k

ða þ s Þ

3 Pðk2Þ ðxÞ þ 5

Replace a by a r in formula 17 and multiply by erx. 1 2 ax 2x e eax sin sx s2 Z Z eax PðxÞ dx dx " #

eax sin sx 3P00 ðxÞ 5Piv ðxÞ 7Pvi ðxÞ eax cos sx 2P0 ðxÞ 4P000 ðxÞ 6Pv ðxÞ PðxÞ þ þ

þ 5 2 2 4 6 2 3 s s s s s s s s *For cos sx in R replace ‘‘sin’’ by ‘‘cos’’ by ‘‘ sin’’ in yp.

A-68

DIFFERENTIAL EQUATIONS (Continued) TABLE III: (D2 þ q)y ¼ R R

yp

23. erx 24. sin sx* 25. P(x) 26. erx sin sx rx

27. P(x) e

28. P(x) sin sx*

erx r2 þ q sin sx s2 þ q " # ð2kÞ 1 P00 ðxÞ Piv ðxÞ ðxÞ kP PðxÞ þ

þ ð1Þ q q qk q2

ðr2 s2 þ qÞerx sin sx 2rserx cos sx erx 2rs 1 ¼ q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi sin sx tan 2 2 2 2 r s þq ðr2 s2 þ qÞ þ ð2rsÞ ðr2 s2 þ qÞ2 þ ð2rsÞ2 " rx 2 3 e 2r 3r q 00 4r 4qr 000 P ðxÞ 2 P ðxÞ þ PðxÞ 2 P0 ðxÞ þ 2 r þq r2 þ q ðr þ qÞ2 ðr þ qÞ3 # ðk Þrk1 ðk3 Þrk3 q þ ðk5 Þrk5 q2 ðk1Þ þ þ ð1Þk1 1 P ðxÞ þ

ðr2 þ qÞk1 " sin sx 3s2 þ q 5s4 þ 10s2 q þ q2 iv P00 ðxÞ þ P ðxÞ þ PðxÞ 2 2 2 ðs þ qÞ ðs2 þ qÞ4 ðs þ qÞ # 2kþ1 2k 2kþ1 2k2 2k4 2 q þ 2kþ1 q þ ð2kÞ s þ 3 s 5 s þ ð1Þk 1 P ðxÞ þ

ðs2 þ qÞ2k " # 2k 2k2 2k 2k4 0 2 þ 3 s q þ ð2k1Þ s cos sx 2P ðxÞ 4s þ 4q 000 kþ1 1 s P ðxÞ þ

þ ð1Þ P ðxÞ þ

ðs2 þ qÞ ðs2 þ qÞ ðs2 þ qÞ3 ðs2 þ qÞ2k1

TABLE IV: (D2 þ b2)y ¼ R 29. sin bx*

30. P(x) sin bx*

x cos bx 2b " #

Z sin bx P00 ðxÞ Piv ðxÞ cos bx P00 ðxÞ PðxÞ PðxÞ þ

þ dx 2b ð2bÞ2 ð2bÞ2 ð2bÞ4 ð2bÞ2 * For cos sx in R replace ‘‘sin’’ by ‘‘cos’’ and ‘‘cos’’ by ‘‘sin’’ in yp.

TABLE V: (D2 þ pD þ q)y ¼ R R

yp

erx r2 þ pr þ q

31. erx

32. sin sx*

33. P(x) 34. erx sin sx* 35. P(x) erx

36. P(x) sin sx*

rx 37. PðxÞe sin sx ax

38. PðxÞe

ðq s2 Þ sin sx ps cos sx 1 ps ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sin sx tan1 2 2 2 2 qs ðq s Þ þ ðpsÞ ðq s2 Þ2 þ ðpsÞ2 " # n n4 2 n1 n2 2 3 q þ ðn2 q ðnÞ 1 p p q 00 p 2pq 000 n p ð 1 Þp 2 Þp P ðxÞ P ðxÞ þ

þ ð1Þ P ðxÞ PðxÞ P0 ðxÞ þ qn q q q2 q3 Replace p by p þ 2r, q by q þ pr þ r2 in formula 32 and multiply by erx. Replace p by p þ 2r, q by q þ pr þ r2 in formula 33 and multiply by erx.

TABLE VI: (D b)(D a)y ¼ R

# ! ! a b a2 s2 b2 s2 a3 3as2 b3 3bs2 00 0 PðxÞ þ ðxÞ þ ðxÞ þ

P P a2 þ s2 b2 þ s2 ða2 þ s2 Þ2 ðb2 þ s2 Þ2 ða2 þ s2 Þ3 ðb2 þ s2 Þ3 " cos sx s s 2as 2bs PðxÞ þ :P0 ðxÞ þ ba a2 þ s2 b2 þ s2 ða2 þ s2 Þ2 ðb2 þ s2 Þ2 #y ! 3a2 s s2 3b2 s s3 00 þ ðxÞ þ

P ða2 þ s2 Þ3 ðb2 þ s2 Þ3

sin sx ba

"

rx Replace "Z a by a – r, b by b - r in formula 36 and multiply by e . # eax PðxÞ P0 ðxÞ P00 ðxÞ PðnÞ ðxÞ þ PðxÞ dx þ þ þ

þ ðb aÞ ðb aÞ2 ðb aÞ3 ab ðb aÞnþ1

*For cos sx in R replace ‘‘sin’’ by ‘‘cos’’ and ‘‘cos’’ by ‘‘sin’’ in yp. y For additional terms, compare with formula 6.

A-69

DIFFERENTIAL EQUATIONS (Continued) TABLE VII: ðD2 2aD þ a2 þ b2 Þy ¼ R R 39.

PðxÞ sin sx

yp " ! sin sx sþb sb 2aðs þ bÞ 2aðs bÞ P0 ðxÞ PðxÞ þ % & % & 2 2 2b a2 þ ðs þ bÞ2 a2 þ ðs bÞ2 a2 þ ðs þ bÞ2 a2 þ ðs bÞ2 þ

cos sx 2b

40. PðxÞerx sin sx 41. PðxÞeax ax 42. e sin sx ax 43. e sin bx

44. PðxÞeax sin bx

# ! 3a2 ðs þ bÞ ðs þ bÞ3 3a2 ðs bÞ ðs bÞ3 00 ðxÞ þ

P % 2 &3 % 2 &3 a þ ðs þ bÞ2 a þ ðs bÞ2

! a a a2 ðs þ bÞ2 a2 ðs bÞ2 0 PðxÞ þ % &2 % 2 &2 P ðxÞ a2 þ ðs þ bÞ2 a2 þ ðs bÞ2 a2 þ ðs þ bÞ2 a þ ðs bÞ2

#y ! a2 3aðs þ bÞ2 a3 3aðs bÞ2 00 þ % &3 % 2 &3 P ðxÞ þ a2 þ ðs þ bÞ2 a þ ðs bÞ2 rx Replace a by a r in formula 39 " # and multiply by e . eax P00 ðxÞ Piv ðxÞ PðxÞ 2 þ

b b4 b2 eax sin sx s2 þ b2 xeax cos bx 2b " # # Z " eax sin bx P00 ðxÞ Piv ðxÞ eax cos bx P00 ðxÞ Piv ðxÞ PðxÞ þ

þ

dx PðxÞ 2b ð2bÞ2 ð2bÞ2 ð2bÞ4 ð2bÞ2 ð2bÞ4 *For cos sx in R replace ‘‘sin’ by ‘‘cos’ and ‘‘cos’’ by ‘‘sin’’ in yp. y For additional terms, compare with formula 6.

TABLE VIII: f ðDÞy ¼ ½Dn þ an1 Dn1 þ þ a1 D þ a0 y ¼ R R

yp

rx 45. e

erx f ðrÞ

46. sin sx*

½a0 a2 s2 þ a4 s4 sin sx ½a1 s a3 s3 þ a5 s5 þ cos sx ½a0 a2 s2 þ a4 s4 2 þ ½a1 s a3 s3 þ a5 s5 2

TABLE IX: f ðD2 Þy ¼ R 47. sin sx*

sin sx sin sx ¼ f ðs2 Þ a0 a2 s2 þ s2n

TABLE X: ðD aÞn y ¼ R rx

48. erx 49. sin sx 50. PðxÞ 51. erx sin sx 52. erx PðxÞ 53. PðxÞ sin sx

e ðr aÞn ð1Þn f½an ðn2 Þan2 s2 þ ðn4 Þan4 s4 sin sx þ ½ðn1 Þan1 s ðn3 Þan3 s3 þ cos sxg ða2 þ s 2 Þ2

n P0 ðxÞ nþ1 P00 ðxÞ nþ2 P000 ðxÞ ð1Þn PðxÞ þ þ þ

þ 1 2 3 an a a2 a2 Replace a by a r in formula 49 and multiply by erx. Replace a by a r in formula 50 and multiply by erx. n n 0 000 nþ1 ð1Þ sin sx½An PðxÞ þ ð1 ÞAnþ1 P ðxÞ þ 2 Anþ2 P00 ðxÞ þ nþ2 3 Anþ3 P ðxÞ þ þ ð1Þn cos sx½Bn PðxÞ þ ðn1 ÞBnþ1 P0 ðxÞ þ A1 ¼

erx sin sx

2

Bnþ2 P00 ðxÞ þ

nþ2 000 3 Bnþ3 P ðxÞ þ

a a2 s2 ak ðk2 Þak2 s2 þ ðk4 Þak4 s4 , A2 ¼ 2 , . . . , Ak ¼ 2 2 2 a þs ða þ s Þ ða2 þ s2 Þk 2

a 2as ðk Þak1 s ðk3 Þak3 s3 þ , B2 ¼ 2 , . . . , Bk ¼ 1 a 2 þ s2 ða þ s2 Þ2 ða2 þ s2 Þk Replace a by a r in formula 53 and multiply by erx. B1 ¼

54.

nþ1

A-70

DIFFERENTIAL EQUATIONS (Continued) 55. eax PðxÞ

eax

Z Z

Z

PðxÞ dxn

n þ 4 Pv ðxÞ ð 1Þ e sin sx n P0 ðxÞ n þ 2 P000 ðxÞ þ

n 5 3 n1 s n1 s n1 s s " # 00 ðnþ1Þ=2 ax n þ 3 Piv ðxÞ ð 1Þ e cos sx n 1 n þ 1 P ðxÞ þ þ

PðxÞ n1 sn n1 n1 s2 s4

ðn1Þ=2 ax

ax 56. PðxÞe sin sx

ð 1Þn=2 eax sin sx sn þ

"

ðn oddÞ

# n þ 3 Piv ðxÞ n1 n þ 1 P00 ðxÞ þ

PðxÞ n1 n1 n1 s2 s4

n þ 4 Pv ðxÞ ð 1Þn=2 eax cos sx n P0 ðxÞ n þ 2 P000 ðxÞ þ

n1 sn n1 s n1 s5 s3

ðn evenÞ

*For cos sx in R replace ‘‘sin’’ by ‘‘cos’’ and ‘‘cos’’ by ‘‘sin’’ in yp.

TABLE XI: (D a)nf (D)y ¼ R

xn eax

n! f ðaÞ *For cos sx in R replace ‘‘sin’’ by ‘‘cos’’ and ‘‘cos’’ by ‘‘sin’’ in yp.

57. eax

TABLE XII: (D2 þ q)ny ¼ R R 58. e

yp rx

59. sin sx

erx =ðr2 þ qÞn

60. PðxÞ 61. erx sin sx

sin sx=ðq s2 Þn " # n P00 ðxÞ nþ1 Piv ðxÞ nþ2 Pvi ðxÞ 1 PðxÞ þ þ

1 3 2 qn q2 q2 q3 /% n n n2 2 n n4 4 & % & 0 erx A 2 A B þ 4 A B sin sx n1 An1 B n3 An3 B3 þ cos sx ðA2 þB2 Þn A ¼ r2 s2 þ q,

B ¼ 2rs

TABLE XIII: (D2 þ b2)ny ¼ R xn cos bx n!ð2bÞn

ð1Þn=2

xn sin bx n!ð2bÞn

62. sin bx

ð1Þðnþ1Þ=2

rx 63. e

erx =ðrn qÞ " # 1 PðnÞ ðxÞ Pð2nÞ ðxÞ PðxÞ þ

þ q q q2 ðn1Þ=2 n q sin sx þ ð 1Þ s cos sx sin sx ðn evenÞ ðn oddÞ, q2 þ s2n ð s2 Þn=2 q rx rx rx Ae sin sx Be cos sx e B ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sin sx tan1 A A 2 þ B2 A 2 þ B2 % & % & A ¼ rn n2 rn2 s2 þ n4 rn4 s4 q, B ¼ n1 rn1 s n3 rn3 s3 þ *For cos sx in R replace ‘‘sin’’ by ‘‘cos’’ and ‘‘cos’’ by ‘‘ sin’’ in yp.

ðn oddÞ,

ðn evenÞ

TABLE XIV: (Dn q)y ¼ R

64. PðxÞ 65. sin sx 66. erx sin sx

TABLE XV: (Dx þ mDy)z ¼ R R 67. eaxþby 68. f ðax þ byÞ

zp eaxþby a þ mb R f ðuÞ du , a þ mb

u ¼ ax þ by

69. f ðy mxÞ

xf ðy mxÞ R 70. ðx, yÞf ðy mxÞ f ðy mxÞ ðx, a þ mxÞ dx ða ¼ y mx after integration)

A-71

DIFFERENTIAL EQUATIONS (Continued) TABLE XVI: ðDx þ mDy kÞz ¼ R axþby

71. eaxþby

e a þ mb k

72. sinðax þ byÞ

73. exþ y sinðax þ byÞ 74. exk f ðax þ byÞ 75. f ðy mxÞ 76. pðxÞf ðy mxÞ 77. ekx f ðy mxÞ

ða þ bmÞ cosðax þ byÞ þ k sinðax þ byÞ ða þ bmÞ2 þ k2 Replace k in 72 by k m and multiply by exþ y R ekx f ðuÞ du , u ¼ ax þ by a þ mb f ðy mxÞ k " # 1 P0 ðxÞ P00 ðxÞ PðnÞ ðxÞ f ðy mxÞ pðxÞ þ þ 2 þ

þ k k kn k xekx f ðy mxÞ *For cosðax þ byÞ replace ‘‘sin’’ by ‘‘cos’’ and ‘‘cos’’ by ‘‘sin’’ in zp . @ @ @kþr Dx ¼ ; Dy ¼ ; Dxk Dyr ¼ @xk @y r @x @y

TABLE XVII: ðDz þ mDy Þn z ¼ R zp

R 78. e

axþby

79. f ðax þ byÞ 80. f ðy mxÞ 81.

ðx, yÞf ðy þ mxÞ

eaxþby ða þ mbÞn RR

R

f ðuÞ dun , ða þ mbÞn

xn f ðy mxÞ n! Z Z f ðy mxÞ

u ¼ ax þ by

Z

ðx, a þ mxÞ dxn ða ¼ y mx after integrationÞ

TABLE XVIII: ðDx þ mDy kÞn z ¼ R axþby

82. eaxþby

e ða þ mb kÞn

83. f ðy mxÞ

ð1Þn f ðy mxÞ kn

84. PðxÞf ðy mxÞ 85. ekz f ðax þ byÞ 86. ekx f ðy mxÞ

n P0 ðxÞ n þ 1 P00 ðxÞ n þ 2 P000 ðxÞ ð1Þn f ðy mxÞ PðxÞ þ þ þ

þ n 2 3 k 1 3 k 2 k k RR R kx n e

f ðuÞ du , u ¼ ax þ by ða þ mbÞn xn kx e f ðy mxÞ n!

h i TABLE XIX: Dnx þ a1 Dxn1 Dy þ a2 Dxn2 D2y þ þ an Dny z ¼ R 87. eaxþby

a þ a1 a

n1

eaxþby b þ a2 an2 b2 þ þ an bn

88. f ðax þ byÞ

RR

R

f ðuÞdun , ðu ¼ ax þ byÞ an þ a1 an1 b þ a2 an2 b2 þ þ an bn

89. eaxþby

eaxþby Fða, bÞ

TABLE XX: FðDx , Dy Þz ¼ R

TABLE XXI: F D2x , Dx Dy , D2y z ¼ R 90. sinðax þ byÞ

sinðax þ byÞ Fða2 , ab, b2 Þ *For cosðax þ byÞreplace ‘‘sin ’’ by ‘‘cos’’, and ‘‘cos’’ by ‘‘sin’’ in zp .

A-72

DIFFERENTIAL EQUATIONS Differential equation Separation of variables

Method of solution Z

f1 ðxÞg1 ðyÞ dx þ f2 ðxÞg2 ðyÞ dy ¼ 0 Exact equation

f1 ðxÞ dx þ f2 ðxÞ

Z M@xþ

Z

g2 ðyÞ dy ¼ c g1 ðyÞ

Z Z @ M@x dy ¼ c n @y

Mðx, yÞ dx þ Nðx, yÞ dy ¼ 0 where @M=@y ¼ @N=@x

Linear first order equation

where @x indicates that the integration is to be performed with respect to x keeping y constant. Z R R ye P dx ¼ Qe P dx dx þ c

dy þ PðxÞy ¼ QðxÞ dx R

Z

R

Bernoulli’s equation

veð1nÞ

dy þ PðxÞy ¼ QðxÞyn dx

where v ¼ y1n : If n ¼ 1, the solution is

Pdx

¼ ð1 nÞ

Qeð1nÞ

Pdx

dx þ c

Z ln y ¼

Homogeneous equation y dy ¼F dx x

Reducible to homogeneous ða1 x þ b1 y þ c1 Þ dx þ ða2 x þ b2 y þ c2 Þ

Z ln x ¼

ðQ PÞ dx þ c

dv þc FðvÞ v

where v ¼ y=x: If FðvÞ ¼ v, the solution is y ¼ cx

Set u ¼ a1 x þ b1 y þ c1 v ¼ a2 x þ b2 y þ c2

dy ¼ 0 a1 b1 6¼ a2 b2

Eliminate x and y and the equation becomes homogenous

Reducible to separable

Set u ¼ a1 x þ b1 y

ða1 x þ b1 y þ c1 Þ dx þ ða2 x þ b2 y þ c2 Þ

Eliminate x or y and equation becomes separable

dy ¼ 0 a1 b1 ¼ a2 b2

A-73

DIFFERENTIAL EQUATIONS (Continued)

yFðxyÞ dx þ x GðxyÞ dy ¼ 0

Z ln x ¼

GðvÞ dv þc vfGðvÞ FðvÞg

where v ¼ xy: If GðvÞ ¼ FðvÞ, the solution is xy ¼ c:

Linear, homogeneous second order equation

Let m1 , m2 be the roots of m2 þ bm þ c ¼ 0: Then there are 3 cases:

d2y dy þ b þ cy ¼ 0 dx dx2

Case 1. m1 , m2 real and distinct: y ¼ c1 em1 x þ c2 em2 x

b, c are real constants

Case 2. m1 , m2 real and equal: y ¼ c1 em1 x þ c2 xem1 x Case 3. m1 ¼ p þ qi, m2 ¼ p qi : y ¼ epx ðc1 cos qx þ c2 sin qxÞ where p ¼ b=2, q ¼

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4c b2 =2

Linear, nonhomogeneous second order equation

There are 3 cases corresponding to those immediately above:

d2y dy þ b þ cy ¼ RðxÞ dx dx2

Case 1. y ¼ c1 em1 x þ c2 em2 x em1 x m1 m2

Z

þ

em2 x m2 m1

Z

þ

b, c are real constants

em1 x RðxÞ dx

em2 x RðxÞ dx

Case 2. y ¼ c1 em1 x þ c2 xem1 x Z

þ xem1 x

em1 x

Z

em1 x RðxÞ dx

xem1 x RðxÞ dx

Case 3. y ¼ epx ðc1 cos qx þ c2 sin qxÞ

A-74

Z

þ

epx sin qx q

epx cos qx q

Z

epx RðxÞ cos qx dx

epx RðxÞ sin qx dx

DIFFERENTIAL EQUATIONS (Continued) Euler or Cauchy equation x2

d2y dy þ bx þ cy ¼ SðxÞ dx dx

Putting x ¼ et , the equation becomes d2y dy þ ðb 1Þ þ cy ¼ Sðet Þ dt dt2 and can then be solved as a linear second order equation.

Bessel’s equation x2

d2y dy þ x þ ð 2 x2 n2 Þy ¼ 0 dx dx2

Transformed Bessel’s equation x2

y ¼ c1 Jn ð xÞ þ c2 Yn ð xÞ

d2y dy þ ð2p þ 1Þx þ ð2 x2r þ 2 Þy ¼ 0 dx dx2

n o y ¼ xp c1 Jq=r xr þ c2 Yq=r xr r r qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where q ¼ p2 2 :

Legendre’s equation y ¼ c1 Pn ðxÞ þ c2 Qn ðxÞ d2y dy ð1 x Þ 2 2x þ nðn þ 1Þy ¼ 0 dx dx 2

FOURIER SERIES If f (x) is a bounded periodic function of period 2L ði:e: fx þ 2LÞ ¼ f ðxÞ, and satisfies the Dirichlet conditions: A. In any period f (x) is continuous, except possibly for a finite number of jump discontinuities. B. In any period f (x) has only a finite number of maxima and minima. Then f(x) may be represented by the Fourier series 1 a0 X npx npx þ bn sin þ an cos L L 2 n¼1

where an and bn are as determined below. This series will converge to f(x) at every point where f(x) is continuous, and to f ðxþ Þ þ f ðx Þ 2 (i.e., the average of the left-hand and right-hand limits) at every point where f(x) has a jump discontinuity. Z 1 L npx dx, n ¼ 0, 1, 2, 3, . . . , an ¼ f ðxÞ cos L L L ZL 1 npx bn ¼ dx, n ¼ 1, 2, 3, . . . f ðxÞ sin L L L we may also write an ¼

1 L

Z

þ2L

f ðxÞ cos

npx 1 dx and bn ¼ L L

where is any real number. Thus if ¼ 0, Z 1 2L npx dx, an ¼ f ðxÞ cos L 0 L Z 2L 1 npx dx, bn ¼ f ðxÞ sin L 0 L

A-75

Z

þ2L

f ðxÞ sin

n ¼ 0, 1, 2, 3, . . . , n ¼ 1, 2, 3, . . .

npx dx L

2. If in addition to the above restrictions, f (x), is even (i.e., f( x) ¼ f (x)) the Fourier series reduces to 1 a0 X npx þ an cos L 2 n¼1

That is, bn ¼ 0. In this case, a simpler formula for an is Z 2 L npx dx, an ¼ f ðxÞ cos L 0 L

n ¼ 0, 1, 2, 3, . . .

3. If in addition to the restrictions in (1), f(x) is an odd function (i.e., f( x) ¼ f(x)), then the Fourier series reduces to 1 X

npx L

bn sin

n¼1

That is, an ¼ 0. In this case, simpler formula for the bn is Z 2 L npx dx, bn ¼ f ðxÞsin L 0 L

n ¼ 1, 2, 3, . . .

4. If in addition to the restrictions in (2) above, f(x) ¼ f (L x), then an will be 0 for all even values of n, including n ¼ 0. Thus in this case, the expansion reduces to 1 X

a2m1 cos

m¼1

ð2m 1Þpx L

5. If in addition to the restrictions in (3) above, f(x), ¼ f(L x), then bn will be 0 for all even values of n. Thus in this case, the expansion reduces to 1 X

b2m1 sin

m¼1

ð2m 1Þpx L

(The series in (4) and (5) are known as odd-harmonic series, since only the odd harmonics appear. Similar rules may be stated for even-harmonic series, but when a series appears in the even-harmonic form, it means that 2L has not been taken as the smallest period of f(x). Since any integral multiple of a period is also a period, series obtained in this way will also work, but in general computation is simplified if 2L is taken to be the smallest period.) 6. If we write the Euler definitions for cos and sin , we obtain the complex form of the Fourier Series known either as the ‘‘Complex Fourier Series’’ or the ‘‘Exponential Fourier Series’’ of f(x). It is represented as f ðxÞ ¼

þ1 1 n ¼X cn ei!n x 2 n¼1

where cn ¼

1 L

Z

L

f ðxÞ ei!n x dx,

n ¼ 0, 1, 2, 3, . . .

L

np , n ¼ 0, 1, 2, . . . L The set of coefficients {cn} is often referred to as the Fourier spectrum. 7. If both sine and cosine terms are present and if f(x) is of period 2L and expandable by a Fourier series, it can be represented as with !n ¼

f ðxÞ ¼

1 npx a0 X þ n , þ cn sin L 2 n¼1

where an ¼ cn sin n , qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi bn ¼ cn cos n , cn ¼ a2n þ b2n ,

n ¼ arc tan

an bn

It can also be represented as f ðxÞ ¼

1 npx a0 X þ n , þ cn cos L 2 n¼1

where an ¼ cn cos n , qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi bn ¼ cn sin n , cn ¼ a2n þ b2n ,

where n is chosen so as to make an, bn, and cn hold.

A-76

bn n ¼ arc tan an

8. The following table of trigonometric identities should be helpful for developing Fourier Series.

n

n even

n odd

n/2 odd

n/2 even

sin np

0

0

0

0

0

cos np

ð1Þn

þ1

1

þ1

þ1

sin

np 2

0

ð1Þðn1Þ=2

0

0

cos

np 2

ð1Þn=2

0

1

þ1

pffiffiffi 2 2 ð1Þðn þ4nþ11Þ=8 2

ð1Þðn2Þ=4

0

sin

np 4

*A useful formula for sin

sin

np np and cos is given by 2 2

np ðiÞnþ1 np ðiÞn ¼ ½ð1Þn 1 and cos ¼ ½ð1Þn þ 1 , 2 2 2 2

where i2 ¼ 1:

AUXILIARY FORMULAS FOR FOURIER SERIES 1¼

4 px 1 3px 1 5px sin þ sin þ sin þ

p k 3 k 5 k

x¼

2k px 1 2px 1 3px sin sin þ sin

p k 2 k 3 k

x¼

k 4k px 1 3px 1 5px 2 cos þ 2 cos þ 2 cos þ

2 p k 3 k 5 k

½0 < x < k

½k < x < k

½0 < x < k

2 2 p 4 px p2 2px p 4 3px þ 3 sin sin sin k k k 1 1 2 3 3

p2 4px p2 4 5px þ þ

½0 < x < k sin sin k k 4 5 53

x2 ¼

2k2 p3

x2 ¼

k2 4k2 px 1 2px 1 3px 1 4px þ 2 cos 2 cos þ

2 cos 2 cos k 2 k 3 k 4 k 3 p

1 1 1 p 1 þ þ

¼ 3 5 7 4 1

1 1 1 p2 þ 2 þ 2 þ

¼ 2 2 3 4 6

1

1 1 1 p2 þ þ

¼ 22 32 42 12

1þ

1 1 1 p2 þ 2 2 þ

¼ 2 3 5 7 8

1 1 1 1 p2 þ þ þ þ

¼ 22 42 62 82 24

A-77

½k < x < k

FOURIER EXPANSIONS FOR BASIC PERIODIC FUNCTIONS

f(x) 1 0

L

−1

4 X 1 npx sin p n¼1, 3, 5, ... n L

f ðxÞ ¼

2L x

f (x) c

1 0

L

−1

f ðxÞ ¼

x

2L

1 2X ð1Þn npc npx 1 sin cos p n¼1 n L L

c

f(x) 2c

1 0

L

2L

f ðxÞ ¼

x

1 c 2X ð1Þn npc npx þ cos sin L p n¼1 n L L

f (x) 1/c

c 3L/ 2

0 1/c

L/ 2

L

f ðxÞ ¼

x

2L

1 2X np sin 12 npc=L npx sin sin 1 L n¼1 2 L 2npc=L

c

f (x) 1 2L

L

0

f ðxÞ ¼

x

−1

1 2X ð1Þnþ1 npx sin p n¼1 L n

f(x) 1 0

2L

L

f ðxÞ ¼

x

1 4 X 1 npx cos 2 p2 n¼1, 3, 5, ... n2 L

f (x) 1 3L/ 2

0 −1

L/ 2

L

2L

x

f ðxÞ ¼

8 X ð1Þðn1Þ=2 npx sin p2 n¼1, 3, 5, ... L n2

f(x) 1 0

2L

x

1 1 1X 1 npx sin f ðxÞ ¼ 2 p n¼1 n L

A-78

FOURIER EXPANSIONS FOR BASIC PERIODIC FUNCTIONS (Continued ) f(x)

1 X 1 2 1 npx ; f ðxÞ ¼ ð1 þ aÞ þ 2 ½ð1Þn cos npa 1 cos 2 p ð1 aÞ n¼1 n2 L c a¼ 2L

c

1 0

L

x

2L

f (x) c/ 2

1

f ðxÞ ¼

0

L

−1

2L

x

1 2X ð1Þn1 sin npa npx ; 1þ sin p n¼1 npð1 aÞ L n

c a¼ 2L

c/ 2

f (x) c

f ðxÞ ¼

1 0

c/2

L

2L − c/2 2L

X 1 1 4 npx ; cos npa cos 2 p2 ð1 2aÞ n¼1, 3, 5, ... n2 L

c a¼ 2L

x

f(x) c/2

1 0 −1

f ðxÞ ¼

2L − c/2 c/2

2L

L

x

1 2X ð1Þn 1 þ ð1Þn npx ; 1þ sin npa sin p n¼1 n L npð1 2aÞ

c a¼ 2L

c/2

f (x) 1 0 −1

c

f ðxÞ

7L /4 L /4

2L

L

x

1 4X 1 np npx sin sin npa sin ; p n¼1 n 4 L

c a¼ 2L

c

f (x) 1 5L/3 2L 0 −1

L /3

L

f ðxÞ ¼

1 9 X 1 np npx ; sin sin p2 n¼1 n2 3 L

f ðxÞ ¼

1 32 X 1 np npx ; sin sin 2 3p n¼1 n2 4 L

c a¼ 2L

x

f (x) 1 0 −1

7L /4 2 L L/4

L

c a¼ 2L

x

f (x) sin ωt T = 2π /ω

f ðxÞ ¼

1 0

π /ω

2π /ω

1 1 2 X 1 þ sin !t cos n!t p 2 p n¼2, 4, 6, ... n2 1

t

Extracted from graphs and formulas, pages 372, 373, Differential Equations in Engineering Problems, Salvadori and Schwarz, published by Prentice-Hall, Inc.,1954.

A-79

THE FOURIER TRANSFORMS* R. E. Gaskell For a piecewise continuous function F(x) over a finite interval 0 % x % p, the finite Fourier cosine transform of F(x) is Zp fc ðnÞ ¼ FðxÞ cos nx dx ðn ¼ 0, 1, 2, . . .Þ ð1Þ 0

If x ranges over the interval 0 % x % L, the substitution x0 ¼ px=L allows the use of this definition, also. The inverse transform is written. x 1 2X FðxÞ ¼ fc ð0Þ fc ðnÞ cos nx ð0 < x < pÞ p p n¼1 ½Fðx þ oÞ þ Fðx oÞ . We observe that FðxÞ ¼ FðxÞ ¼ at point of continuity. The formula where FðxÞ ¼ 2 Zp fcð2Þ ðnÞ ¼ F 00 ðxÞ cos nx dx

ð2Þ

0

¼ n2 f c ðnÞ F 0 ð0Þ þ ð1Þn F 0 ðpÞ makes the finite Fourier cosine transform useful in certain boundary value problems. Analogously, the finite Fourier sine transform of F(x) is Zp fs ðnÞ ¼ FðxÞ sin nx dx ðn ¼ 1, 2, 3, . . .Þ

ð3Þ

ð4Þ

0

and FðxÞ ¼

1 2X fs ðnÞ sin nx ð0 < x < pÞ p n¼1

ð5Þ

Corresponding to (3) we have fsð2Þ ðnÞ ¼

Z

p

F 00 ðxÞ sin nx dx

0

ð6Þ

¼ n2 fs ðnÞ n Fð0Þ nð1Þn FðpÞ Fourier Transforms If FðxÞ is defined for x ^ 0 and is piecewise continuous over any finite interval, and if Z

x

FðxÞ dx 0

is absolutely convergent, then fc ðÞ ¼

rffiffiffi Z x 2 FðxÞ cosðxÞ dx p 0

is the Fourier cosine transform of FðxÞ. Furthermore, rffiffiffi Z x 2 FðxÞ ¼ fc ðÞ cosðxÞ d p 0 n d F if limx!1 n ¼ 0, an important property of the Fourier cosine transform dx rffiffiffi Z x 2 d 2r F cosðxÞ dx fcð2rÞ ðÞ ¼ p 0 dx2r rffiffiffi r1 2X ð1Þn a2r2n1 2n þ ð1Þr 2r fc ðÞ ¼ p n¼0 d rF where limx!1 r ¼ ar, makes it useful in the solution of many problems. dx Under the same conditions. rffiffiffi Z x 2 FðxÞ sinðxÞ dx fs ðÞ ¼ p 0

ð7Þ

ð8Þ

ð9Þ

ð10Þ

*From Beyer, W. H., Ed., CRC Handbook of Mathematical Sciences, 5th ed., CRC Press, Boca Raton, 1978, 592–598. With permission.

A-80

defines the Fourier sine transform of FðxÞ, and FðxÞ ¼

rffiffiffi Z x 2 fs ðÞ sinðxÞ d p 0

ð11Þ

Corresponding to (9) we have

rffiffiffi Z 1 2 d 2r F sinðxÞ dx p 0 dx2r rffiffiffi r 2X ¼ ð1Þn 2n1 a2r2n þ ð1Þr1 2r fs ðÞ p n¼1 R1 Similarly, if FðxÞ is defined for 1 < x < 1, and if 1 FðxÞ dx is absolutely convergent, then Z1 1 FðxÞeiax dx f ðÞ ¼ pffiffiffiffiffiffi 2p 1 is the Fourier transform of FðxÞ, and Z1 1 FðxÞ ¼ pffiffiffiffiffiffi f ðÞeiax d 2p 1 Also, if n d F lim n ¼ 0 ðn ¼ 1, 2, . . . , r 1Þ jxj!1 dx fsð2rÞ ðÞ ¼

then 1 f ðrÞ ðÞ ¼ pffiffiffiffiffiffi 2p

Z

F ðrÞ ðxÞeix dx ¼ ðiÞr f ðÞ

1

FðxÞ

fs ðnÞ 1

p

FðxÞ sin nx dx ðn ¼ 1, 2, . . .Þ

fs ðnÞ ¼

FðxÞ

0

2

ð1Þnþ1 fs ðnÞ

Fðp xÞ

3

1 n

px p x p

4 5 6

ð1Þnþ1 n 1 ð1Þn n 2 np sin 2 n2

1 ,

x when 0 < x < p=2 p x when p=2 < x < p

7

ð1Þnþ1 n3

xðp2 x2 Þ 6p

8

1 ð1Þn n3

xðp xÞ 2

p2 ð1Þn1 2½1 ð1Þn n n3 ! 2 6 p 10 pð1Þn 3 n n 9

11 12

ð13Þ

ð14Þ

1

Finite Sine Transforms

Z

ð12Þ

x2 x3

n ½1 ð1Þn ecp n2 þ c2 n n2 þ c2

ecx sinh cðp xÞ sinh cp

n ðk 6¼ 0, 1, 2, . . .Þ n2 k2 8 ½1 < n2 m2 ð 1Þ 16 > when n 6¼ m ¼ 1, 2, . . . : 0 when n ¼ m n 17 ðk 6¼ 0, 1, 2, . . . Þ ðn2 k2 Þ2 bn 18 ðjbj % 1Þ n 1 ð 1Þn n b ðjbj % 1Þ 19 n

cos kx

cos mx

p sin kx x cos kðp xÞ 2k sin kp 2k sin2 kp 2 bsinx arc tan p 1 b cos x 2 2b sinx arc tan p 1 b2

Finite Cosine Transforms fc ðnÞ Z 1

fc ðnÞ ¼

2 3

n

4 5 6 7 8 9 10

11

12 13 14

FðxÞ

p

FðxÞ cos nx dx ðn ¼ 0, 1, 2, . . . Þ

FðxÞ

0

ð 1Þ fc ðnÞ 0 when n ¼ 1, 2, . . . ; fc ð0Þ ¼ p 2 np sin ; fc ð0Þ ¼ 0 n 2 1 ð 1Þn p2 ; fc ð0Þ ¼ 2 2 n ð 1Þn p2 ; f ð0Þ ¼ c 6 n2 1 ; f ð0Þ ¼ 0 c n2 ð 1Þn 1 ð 1Þn p4 3p2 6 ; fc ð0Þ ¼ 2 4 4 n n ð 1Þn ec p 1 2 2 n þc 1 2 n þ c2

Fðp xÞ 1 , 1 when 0 < x < p=2 1 when p=2 < x < p x

k ½ð 1Þn cos pk 1 n 2 k2 ðk 6¼ 0, 1, 2, Þ

sin kx

ð 1Þnþm 1 ; fc ðmÞ ¼ 0 ðm ¼ 1, 2, Þ n2 m 2 1 ðk 6¼ 0, 1, 2, . . . Þ n 2 k2

1 sin mx m

0 when n ¼ 1, 2, . . . ; p ðm ¼ 1, 2, Þ fc ðmÞ ¼ 2

cos mx

x2 2p ðp xÞ2 p 2p 6 x3 1 cx e c cosh cðp xÞ c sinh cp

cos kðp xÞ k sin kp

Fourier Sine Transforms* FðxÞ , 1

2 3

1 0

ð0 < x < aÞ ðx > aÞ

xp1 ð0 aÞ

4

ex

5

xex =2 x2 cos 2

6

2

fs ðÞ rffiffiffi

2 1 cos p rffiffiffi 2 ðpÞ pp sin p p 2

1 sin½að1 Þ sin½að1 þ Þ pffiffiffiffiffiffi 1 1þ 2p rffiffiffi

2 p 1 þ 2 2

e" =2 ! !# pffiffiffi 2 2 2 2 2 sin c cos S 2 2 2 2

A-82

FðxÞ

fs ðÞ " ! !# pffiffiffi 2 2 2 2 þ sin S 2 cos C 2 2 2 2

2

7

sin

x 2

*C(y) and S(y) are the Fresnel integrals

1 CðyÞ ¼ pffiffiffiffiffiffi 2p

Z

1 SðyÞ ¼ pffiffiffiffiffiffi 2p

y

1 pffiffi cos t dt, t

0

Z

y

0

1 pffiffi sin t dt t

*More extensive tables of the Fourier sine and cosine transforms can be found in Fritz Oberhettinger, Tabellen zur-Fourier Transformation, Springer, 1957. Fourier Cosine Transforms FðxÞ , 1

1 0

f c ðÞ rffiffiffi 2 sin a p rffiffiffi 2 ðpÞ pp cos p p 2

1 sin½að1 Þ sin½að1 þ Þ pffiffiffiffiffiffi þ 1 1 þ 2p rffiffiffi 2 1 p 1 þ 2

ð0 < x < aÞ ðx > aÞ

xp1 ð0 aÞ 2

4 ex 2

5 ex

2

=2

e

=2

6

x2 cos 2

cos

2 p 2 4

7

sin

x2 2

cos

2 p þ 2 4

! !

Fourier Transforms FðxÞ

1

sin ax x ,

2 , 3 4

fs ðÞ 8 rffiffiffi > : 0

eiwx 0

ðp, x < qÞ ðx < p, x > qÞ

ecxþiwx 0

epx

2

ðx > 0Þ ðc > 0Þ ðx < 0Þ

RðpÞ > 0 2

5

cos px

6

sin px2

7 jxjp

ð0 < pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðjxj < aÞ 11 a 2 x2 > : 0 ðjxj > aÞ

jj < a jj > a

i eipðwþÞ eiqðwþÞ pffiffiffiffiffiffi ðw þ Þ 2p i pffiffiffiffiffiffi 2pðw þ þ icÞ 1 2 =4p pffiffiffiffiffi e 2p " # 1 2 p pffiffiffiffiffi cos 4p 4 2p " # 1 2 p pffiffiffiffiffi cos þ 4p 4 2p rffiffiffi pp 2 ð1 pÞ sin 2 p jjð1pÞ qp ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ða2 þ 2 Þ þ a pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 þ 2 rffiffiffi 2 cos a2 cosh 2 p cosh þ cos a 1 sin a pffiffiffiffiffiffi 2p cosh þ cos a

10

rffiffiffi p J ðaÞ 2 0

A-83

FðxÞ

f ðÞ 8 0 < rffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p J ða b2 2 Þ : 2 0

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sin½b a2 þ x2 12 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 þ x2 ( pn ðxÞ ðjxj < 1Þ 13 0 ðjxj > 1Þ 8 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > a 2 x2 < cos½b pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 14 a x2 > : 0 8 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > a 2 x2 < cosh½b pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 x2 15 a > : 0

ðjj > bÞ ðjj < bÞ

n

i pffiffiffi J 1 ðÞ nþ2 ðjxj < aÞ ðjxj > aÞ ðjxj < aÞ ðjxj > aÞ

rffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p J ða a2 þ b2 Þ 2 0 rffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p J ða 2 b2 Þ 2 0

* More extensive tables of Fourier transforms can be found in W. Magnus and F. Oberhettinger, Formulas and Theorems of the Special Functions of Mathematical Physics. Chelsea, 1949, 116–120. The following functions appear among the entries of the tables on transforms. Function EiðxÞ

Definition Z

Name

v

x

e dv; or sometimes defined v Z x v e dv as EiðxÞ ¼ v Zx x sin v dv v 0 Zx cos v dv; or sometimes defined v x as negative of this integral

Sine, Cosine, and Exponential Integral tables pages 548–556

x

SiðxÞ CiðxÞ

erf ðxÞ erfcðxÞ

2 pffiffiffi p

x


2

ev dv

Error function

0

2 1 erf ðxÞ ¼ pffiffiffi p x

Ln ðxÞ

Z


Z

1

e

v2

Complementary function to error function

dv

x

n

e d ðxn ex Þ, n! dxn

n ¼ 0, 1, . . .

Laguerre polynomial of degree n

SERIES EXPANSION The expression in parentheses following certain of the series indicates the region of convergence. If not otherwise indicated it is to be understood that the series converges for all finite values of x. BINOMIAL nðn 1Þ n2 2 nðn 1Þðn 2Þ n3 3 x y þ x y þ

ðx þ yÞn ¼ xn þ nxn1 y þ 2! 3! ð1 xÞn ¼ 1 nx þ

nðn 1Þx2 nðn 1Þðn 2Þx3 þ

2! 3!

ðx2 < 1Þ

ð1 xÞn ¼ 1 nx þ

nðn þ 1Þx2 nðn þ 1Þðn þ 2Þx3 þ

2! 3!

ðx2 < 1Þ

ð1 xÞ1 ¼ 1 x þ x2 x3 þ x4 x5 þ

ðx2 < 1Þ

ð1 xÞ2 ¼ 1 2x þ 3x2 4x3 þ 5x4 6x5 þ

ðx2 < 1Þ

REVERSION OF SERIES Let a series be represented by y ¼ a1 x þ a2 x2 þ a3 x3 þ a4 x4 þ a5 x5 þ a6 x6 þ to find the coefficients of the series x ¼ A1 y þ A2 y2 þ A3 y3 þ A4 y4 þ 1 a2 1 A2 ¼ 3 A3 ¼ 5 2a22 a1 a3 a1 a1 a1 1 A4 ¼ 7 5a1 a2 a3 a21 a4 5a32 a1 A1 ¼

A-84

ðaj 6¼ 0Þ

ðy2 < x2 Þ

SERIES EXPANSION The expression in parentheses following certain of the series indicates the region of convergence. If not otherwise indicated it is to be understood that the series converges for all finite values of x. BINOMIAL nðn 1Þ n2 2 nðn 1Þðn 2Þ n3 3 x y þ x y þ

ðx þ yÞn ¼ xn þ nxn1 y þ 2! 3! ð1 xÞn ¼ 1 nx þ

nðn 1Þx2 nðn 1Þðn 2Þx3 þ

2! 3!

ðx2 < 1Þ

ð1 xÞn ¼ 1 nx þ

nðn þ 1Þx2 nðn þ 1Þðn þ 2Þx3 þ

2! 3!

ðx2 < 1Þ

ð1 xÞ1 ¼ 1 x þ x2 x3 þ x4 x5 þ

ðx2 < 1Þ

ð1 xÞ2 ¼ 1 2x þ 3x2 4x3 þ 5x4 6x5 þ

ðx2 < 1Þ

REVERSION OF SERIES Let a series be represented by y ¼ a1 x þ a2 x2 þ a3 x3 þ a4 x4 þ a5 x5 þ a6 x6 þ to find the coefficients of the series x ¼ A1 y þ A2 y2 þ A3 y3 þ A4 y4 þ 1 a2 1 A2 ¼ 3 A3 ¼ 5 2a22 a1 a3 a1 a1 a1 1 A4 ¼ 7 5a1 a2 a3 a21 a4 5a32 a1 A1 ¼

ðaj 6¼ 0Þ

ðy2 < x2 Þ

1 2 6a1 a2 a4 þ 3a21 a23 þ 14a42 a31 a5 21a1 a22 a3 a91 1 A6 ¼ 11 7a31 a2 a5 þ 7a31 a3 a4 þ 84a1 a32 a3 a41 a6 28a21 a22 a4 28a21 a2 a33 42a52 a1 1 A7 ¼ 13 8a41 a2 a6 þ 8a41 a3 a5 þ 4a41 a24 þ 120a21 a32 a4 þ 180a21 a22 a23 þ 132a62 a51 a7 a1 36a31 a22 a5 72a31 a2 a3 a4 12a31 a33 330a1 a42 a3 A5 ¼

TAYLOR 2

1: f ðxÞ ¼ f ðaÞ þ ðx aÞf 0 ðaÞ þ

ðx aÞ 00 ðx a)3 000 f ðaÞ f ðaÞ þ 3! 2!

ðx aÞn ðnÞ f ðaÞ þ ðTaylor0 s SeriesÞ n! (Increment form)

þ

þ

h2 00 h3 f ðxÞ þ f 000 ðxÞ þ 2! 3! x2 00 x3 000 0 ¼ f ðhÞ þ xf ðhÞ þ f ðhÞ þ f ðhÞ þ 2! 3! 3. If f (x) is a function possessing derivatives of all orders throughout the interval a % x % b, then there is a value X, with aXb, such that 2: f ðx þ hÞ ¼ f ðxÞ þ hf 0 ðxÞ þ

f ðbÞ ¼ f ðaÞ þ ðb aÞf 0 ðaÞ þ f ða þ hÞ ¼ f ðaÞ þ hf 0 ðaÞ þ

ðb aÞ2 00 ðb aÞn1 ðn1Þ ðb aÞn ðnÞ f ðaÞ þ þ f f ðXÞ ðaÞ þ 2! ðn 1Þ! n!

h2 00 hn1 ðn1Þ hn f ðaÞ þ þ f ðaÞ þ f ðnÞ ða þ hÞ, b ¼ a þ h, 0 < < 1: 2! ðn 1Þ! n!

or f ðxÞ ¼ f ðaÞ þ ðx aÞf 0 ðaÞ þ

ðx aÞ2 00 f ðn1Þ ðaÞ þ Rn , f ðaÞ þ þ ðx aÞn1 ðn 1Þ! 2!

where f ðnÞ ½a þ ðx aÞ ðx aÞn , 0 < < 1: n! The above forms are known as Taylor’s series with the remainder term. 4. Taylor’s series for a function of two variables @ @ @f ðx, yÞ @f ðx, yÞ þk ; If h þ k f ðx, yÞ ¼ h @x @y @x @y @ @ 2 @2 f ðx, yÞ @2 f ðx, yÞ @2 f ðx, yÞ þ k2 h þk f ðx, yÞ ¼ h2 þ 2hk 2 @x @y @x @x@y @y2 n @ @ etc., and if h þ k f ðx, yÞx¼a y¼b with the bar and subscripts means that after differentiation we are to replace x by @x @y a and y by b, @ @ 1 @ @ n f ðx, yÞx¼a þ þ h þk f ðx, yÞx¼a þ f ða þ h, b þ kÞ ¼ f ða, bÞ þ h þ k y¼b y¼b @x @y n! @x @y Rn ¼

MACLAURIN 2

f ðxÞ ¼ f ð0Þ þ xf 0 ð0Þ þ

x 00 x3 f ðn1Þ ð0Þ þ Rn , f ð0Þ þ f 000 ð0Þ þ þ xn1 ðn 1Þ! 2! 3!

where Rn ¼

xn f ðnÞ ðxÞ , n!

0 < < 1:

EXPONENTIAL 1 1 1 1 þ þ þ þ

1! 2! 3! 4! x2 x3 x4 ex ¼ 1 þ x þ þ þ þ 2! 3! 4! e¼1þ

A-85

ðall real values of xÞ

ðx loge aÞ2 ðx loge aÞ3 þ þ

2! 3!

ðx aÞ2 ðx aÞ3 þ þ

ex ¼ ea 1 þ ðx aÞ þ 2! 3! ax ¼ 1 þ x loge a þ

LOGARITHMIC

loge x ¼

x1 1 x1 þ x 2 x

2 þ

1 x1 3 þ

3 x

loge x ¼ ðx 1Þ 12ðx 1Þ2 þ 13ðx 1Þ3 " # x1 1 x1 3 1 x1 5 þ þ þ

loge x ¼ 2 xþ1 3 xþ1 5 xþ1 loge ð1 þ xÞ ¼ x 12x2 þ 13x3 14x4 þ

1 1 1 loge ðn þ 1Þ loge ðn 1Þ ¼ 2 þ 3 þ 5 þ n 3n 5n " # 3 5 x 1 x 1 x þ þ þ

loge ða þ xÞ ¼ loge a þ 2 2a þ x 3 2a þ x 5 2a þ x

1þx x3 x5 x2n1 loge ¼ 2 x þ þ þ

þ þ

1x 3 5 2n 1 2

loge x ¼ loge a þ

ðx > 12Þ ð2 x > 0Þ ðx > 0Þ ð1 < x 1Þ

ða > 0, a < x < þ1Þ 1 < x < 1

3

ðx aÞ ðx aÞ ðx aÞ þ þ

a 2a2 3a3

0 < x % 2a

TRIGONOMETRIC 3

5

7

x x x þ þ ðall real values of xÞ 3! 5! 7! x2 x4 x6 cos x ¼ 1 þ þ ðall real values of xÞ 2! 4! 6! x3 2x5 17x7 62x9 ð1Þn1 22n ð22n 1ÞB2n 2n1 þ þ þ

þ x tan x ¼ x þ þ þ

, 3 15 315 2835 ð2nÞ! h i 2 p2 x < 4 , and Bn represents the nth Bernoulli number. sin x ¼ x

cot x ¼

1 x x3 2x5 x7 ð1Þnþ1 22n

B2n x2n1 , x 3 45 945 4725 ð2nÞ! ½x2 < p2 , and Bn represents the nth Bernoulli number:

sec x ¼ 1 þ

x2 5 61 6 277 8 ð1Þn x þ x þ

þ þ x4 þ E2n x2n þ , 720 8064 2 24 ð2nÞ! h i 2 x2 < p4 , and En represents the nth Euler number.

1 x 7 3 31 127 þ þ x þ x5 þ x7 þ x 6 360 15; 120 604; 800 ð1Þnþ1 2ð22n1 1Þ B2n x2n1 þ , þ ð2nÞ! ½x2 < p2 , and Bn represents nth Bernoulli number.] x2 x2 x2 ðx2 < 1Þ sin x ¼ x 1 2 1 2 2 1 2 2 p 2 p 3 p 4x2 4x2 4x2 1 2 2 1 2 2

ðx2 < 1Þ l cos x ¼ 1 2 p 3 p 5 p x3 1 3 5 1 3 5 7 p p x þ x þ

x2 < 1, < sin1 x < þ sin1 x ¼ x þ 2 4 6 7 2 2 2 3 2 4 5 p x3 1 3 5 1 3 5x7 1 2 1 x þ cos x ¼ x þ þ þ

ðx < 1, 0 < cos x < pÞ 2 2 3 2 4 5 2 4 6 7 3 5 7 x x x ðx2 < 1Þ tan1 x ¼ x þ þ 3 5 7 p 1 1 1 1 ðx > 1Þ tan1 x ¼ þ 3 5 þ 7 2 x 3x 5x 7x p 1 1 1 1 tan1 x ¼ þ 3 5 þ 7 ðx < 1Þ 2 x 3x 5x 7x 3 5 7 p x x x ðx2 < 1Þ cot1 x ¼ x þ þ 2 3 5 7 csc x ¼

A-86

x2 x4 x6

6 180 2835 x2 x4 x6 17x8 loge cos x ¼

2 12 45 2520 2 4 x 7x 62x6 þ þ

loge tan x ¼ loge x þ þ 3 90 2835 x2 3x4 8x5 3x6 56x7 þ þ

esin x ¼ 1 þ x þ 2! 4! 5! 6! 7! 2 4 6 x 4x 31x ecos x ¼ e 1 þ þ

2! 4! 6! x2 3x3 9x4 37x5 þ þ þ

etan x ¼ 1 þ x þ þ 2! 3! 4! 5! loge sin x ¼ loge x

ðx aÞ2 sin a 2! 3 4 ðx aÞ ðx aÞ cos a þ sin a þ 3! 4!

sin x ¼ sin a þ ðx aÞ cos a

A-87

ðx2 < p2 Þ p2 x2 < 4 2 p x2 < 4

p2 x2 < 4

VECTOR ANALYSIS Definitions Any quantity which is completely determined by its magnitude is called a scalar. Examples of such are mass, density, temperature, etc. Any quantity which is completely determined by its magnitude and direction is called a vector. Examples of such are velocity, acceleration, force, etc. A vector quantity is represented by a directed line segment, the length of which represents the magnitude of the vector. A vector quantity is usually represented by a boldfaced letter such as V. Two vectors V1 and V2 are equal to one another if they have equal magnitudes and are acting in the same directions. A negative vectors written as V, is one which acts in the opposite direction to V, but is of equal magnitude to it. If we represent the magnitude of V by v, we write jVj ¼ v. A vector parallel to V, but equal 1 to the reciprocal of its magnitude is written as V1 or as . V V The unit vector ðv 6¼ 0Þ is that vector which has the same direction as V, but has a magnitude of unity (sometimes v represented as V0 or ^v). Vector Algebra The vector sum of V1 and V2 is represented by V1 þV2 . The vector sum of V1 and V2, or the difference of the vector V2 from V1 is represented by V1 V2 . If r is a scalar, then rV ¼ Vr, and represents a vector r times the magnitude of V, in the same direction as V if r is positive, and in the opposite direction if r is negative. If r and s are scalars, V1, V2, V3, vectors, then the following rules of scalars and vectors hold: V1 þ V2 ¼ V2 þ V1 ðr þ sÞV1 ¼ rV1 þ sV1 ;

rðV1 þ V2 Þ ¼ rV1 þ rV2

V1 þ ðV2 þ V3 Þ ¼ ðV1 þ V2 Þ þ V3 ¼ V1 þ V2 þ V3 Vectors in Space A plane is described by two distinct vectors V1 and V2. Should these vectors not intersect each other, then one is displaced parallel to itself until they do (fig. 1). Any other vector V lying in this plane is given by V ¼ rV1 þ sV2 A position vector specifies the position in space of a point relative to a fixed origin. If therefore V1 and V2 are the position vectors of the points A and B, relative to the origin O, then any point P on the line AB has a position vector V given by V ¼ rV1 þ ð1 rÞV2 The scalar ‘‘r’’ can be taken as the parametric representation of P since r ¼ 0 implies P ¼ B and r ¼ 1 implies P ¼ A (fig. 2). If P divides the line AB in the ratio r : s then r s V¼ V1 þ V2 rþs rþs

B(r = 0)

0>r

1>r>0

V1 A(r = 1)

V2 V1

r>1 V2

o

Figure 1.

Figure 2.

The vectors V1, V2, V3, . . ., Vn are said to be linearly dependent if there exist scalars r1, r2, r3, . . . , rn, not all zero, such that r1 V1 þ r2 V2 þ þ rn Vn ¼ 0 A vector V is linearly dependent upon the set of vectors V1, V2, V3, . . . ,Vn if V ¼ r1 V1 þ r2 V2 þ r3 V3 þ þ rn Vn Three vectors are linearly dependent if and only if they are co-planar. All points in space can be uniquely determined by linear dependence upon three base vectors i.e., three vectors any one of which is linearly independent of the other two. The simplest set of base vectors are the unit vectors along the coordinate Ox, Oy and Oz axes. These are usually designated by i, j and k respectively. If V is a vector in space, and a, b and c are the respective magnitudes of the projections of the vector along the axes then V ¼ ai þ bj þ ck and v¼

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 þ b2 þ c2

and the direction cosines of V are cos ¼ a=v,

cos ¼ b=v,

cos ¼ c=v:

The law of addition yields V1 þ V2 ¼ ða1 þ a2 Þi þ ðb1 þ b2 Þj þ ðc1 þ c2 Þk The Scalar, Dot, or Inner Product of Two Vectors V1 and V2 This product is represented as V1 V2 and is defined to be equal to v1 v2 cos , where is the angle from V1 to V2, i.e., V1 V2 ¼ v1 v2 cos The following rules apply for this product: V1 V2 ¼ a1 a2 þ b1 b2 þ c1 c2 ¼ V2 V1 It should be noted that this verifies that scalar multiplication is commutative. ðV1 þ V2 Þ V3 ¼ V1 V3 þ V2 V3 V1 ðV2 þ V3 Þ ¼ V1 V2 þ V1 V3 If V1 is perpendicular to V2 then V1 V2 ¼ 0, and if V1 is parallel to V2 then V1 V2 ¼ v1 v2 ¼ rw12 In particular i i ¼ j j ¼ k k ¼ 1, and i j¼j k¼k i¼0 The Vector or Cross Product of Vectors V1 and V2 This product is represented as V1 V2 and is defined to be equal to v1 v2 ðsin Þ1, where is the angle from V1 to V2 and 1 is a unit vector perpendicular to the plane of V1 and V2, and so directed that a right-handed screw driven in the direction of 1 would carry V1 into V2, i.e., V1 V2 ¼ v1 v2 ðsin Þ1 and tan ¼

jV1 V2 j V1 V2

The following rules apply for vector products: V1 V2 ¼ V2 V1 V1 ðV2 þ V3 Þ ¼ V1 V2 þ V1 V3 ðV1 þ V2 Þ V3 ¼ V1 V3 þ V2 V3 V1 ðV2 V3 Þ ¼ V2 ðV3 V1 Þ V3 ðV1 V2 Þ i i ¼ j j ¼ k k ¼ 0:1 ðzero vectorÞ ¼0 i j ¼ k,

j k ¼ i,

ki¼j

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If V1 ¼ a1 i þ b1 j þ c1 k, then

V2 ¼ a2 i þ b2 j þ c2 k,

V3 ¼ a3 i þ b3 j þ c3 k,

i j k V1 V2 ¼ a1 b1 c1 ¼ ðb1 c2 b2 c1 Þi þ ðc1 a2 c2 a1 Þj þ ða1 b2 a2 b1 Þk a2 b2 c2 It should be noted that, since V1 V2 ¼ V2 V1 , the vector product is not commutative.

Scalar Triple Product There is only one possible interpretation of the expression V1 V2 V3 and that is V1 ðV2 V3 Þ which is obviously a scalar. Further V1 ðV2 V3 Þ ¼ ðV1 V2 Þ V3 ¼ V2 ðV3 V1 Þ a1 b1 c1 ¼ a2 b2 c2 a3 b3 c3 ¼ r1 r2 r3 cos sin ,

Where is the angle between V2 and V3 and is the angle between V1 and the normal to the plane of V2 and V3. This product is called the scalar triple product and is written as [V1V2V3]. The determinant indicates that it can be considered as the volume of the parallelepiped whose three determining edges are V1, V2 and V3. It also follows that cyclic permutation of the subscripts does not change the value of the scalar triple product so that ½V1 V2 V3 ¼ ½V2 V3 V1 ¼ ½V3 V1 V2 and ½V1 V1 V2 0 etc: but ½V1 V2 V3 ¼ ½V2 V1 V3 etc: Given three non-coplanar reference vectors V1, V2 and V3, the reciprocal system is given by V1 , V2 and V3 , where 1 ¼ v1 v1 ¼ v2 v2 ¼ v3 v3 0 ¼ v1 v2 ¼ v1 v3 ¼ v2 v1 etc: V2 V3 V3 V1 V1 V2 , V2 ¼ , V3 ¼ V1 ¼ ½V1 V2 V3 ½V1 V2 V3 ½V1 V2 V3 The system i, j, k is its own reciprocal. Vector Triple Product The product V1 ðV2 V3 Þ defines the vector triple product. Obviously, in this case, the brackets are vital to the definition. V1 ðV2 V3 Þ ¼ ðV1 V3 ÞV2 ðV1 V2 ÞV3 i j k a1 b1 c1 ¼ b2 c2 c2 a2 a2 b2 b c c a a b 3

3

i.e. it is a vector, perpendicular to V1, lying in the plane of Similarly i b c 1 1 ðV1 V2 Þ V3 ¼ b2 c2 a3

3

3

3

3

V2, V3. c1 c 2

j

a1 a2

b3

a1 a 2

b1 b2 c3 k

V1 ðV2 V3 Þ þ V2 ðV3 V1 Þ þ V3 ðV1 V2 Þ 0

If V1 ðV2 V3 Þ ¼ ðV1 V2 Þ V3 then V1, V2, V3 form an orthogonal set. Thus i, j, k form an orthogonal set. Geometry of the Plane, Straight Line and Sphere The position vectors of the fixed points A, B, C, D relative to O are V1, V2, V3, V4 and the position vector of the variable point P is V. The vector form of the equation of the straight line through A parallel to V2 is V ¼ V1 þ rV2 or ðV V1 Þ ¼ rV2 or ðV V1 Þ V2 ¼ 0 while that of the plane through A perpendicular to V2 is ðV V1 Þ V2 ¼ 0 The equation of the line AB is V ¼ rV1 þ ð1 rÞV2

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and those of the bisectors of the angles between V1 and V2 are V1 V2 V¼r v1 v2 or V ¼ rð^v1 ^v2 Þ The perpendicular from C to the line through A parallel to V2 has as its equation V ¼ V1 V3 ^v2 ðV1 V3 Þ^v2 : The condition for the intersection of the two lines, V ¼ V1 þ rV3 and V ¼ V2 þ sV4 is ½ðV1 V2 ÞV3 V4 ¼ 0: The common perpendicular to the above two lines is the line of intersection of the two planes ½ðV V1 ÞV3 ðV3 V4 Þ ¼ 0 and ½ðV V2 ÞV4 ðV3 V4 Þ ¼ 0 and the length of this perpendicular is ½ðV1 V2 ÞV3 V4 : jV3 V4 j The equation of the line perpendicular to the plane ABC is V ¼ V1 V2 þ V2 V3 þ V3 V1 and the distance of the plane from the origin is ½V1 V2 V3 : jðV2 V1 Þ ðV3 V1 Þj In general the vector equation V V2 ¼ r defines the plane which is perpendicular to V2, and the perpendicular distance from A to this plane is r V1 V2 v2 The distance from A, measured along a line parallel to V3, is r V1 V2 r V1 V2 or V2 ^v3 v2 cos where is the angle between V2 and V3. (If this plane contains the point C then r ¼ V3 V2 and if it passes through the origin then r ¼ 0.) Given two planes V V1 ¼ r V V2 ¼ s then any plane through the line of intersection of these two planes is given by V ðV1 þ V2 Þ ¼ r þ s where is a scalar parameter. In particular ¼ v1 =v2 yields the equation of the two planes bisecting the angle between the given planes. The plane through A parallel to the plane of V2, V3 is V ¼ V1 þ rV2 þ sV3 or ðV V1 Þ V2 V3 ¼ 0 or ½VV2 V3 ½V1 V2 V3 ¼ 0 so that the expansion in rectangular Cartesian coordinates yields ðx a1 Þ ðy b1 Þ ðz c1 Þ ¼ 0 a2 b2 c2 a3 b3 c3 ðV xi þ yj þ zkÞ which is obviously the usual linear equation in x, y and z. The plane through AB parallel to V3 is given by ½ðV V1 ÞðV1 V2 ÞV3 ¼ 0 or ½VV2 V3 ½VV1 V3 ½V1 V2 V3 ¼ 0 The plane through the three points A, B and C is V ¼ V1 þ sðV2 V1 Þ þ tðV3 V1 Þ

A-90

or

V ¼ rV1 þ sV2 þ tV3

or

½ðV V1 ÞðV1 V2 ÞðV2 V3 Þ ¼ 0

ðr þ s þ t 1Þ

or

½VV1 V2 þ ½VV2 V3 þ ½VV3 V1 ½V1 V2 V3 ¼ 0

For four points A, B, C, D to be coplanar, then rV1 þ sV2 þ tV3 þ uV4 0 r þ s þ t þ u The following formulae relate to a sphere when the vectors are taken to lie in three dimensional space and to a circle when the space is two dimensional. For a circle in three dimensions take the intersection of the sphere with a plane. The equation of a sphere with center O and radius OA is V V ¼ v21 ðnot V ¼ V1 Þ ðV V1 Þ ðV þ V1 Þ ¼ 0

or while that of a sphere with center B radius v1 is

ðV V2 Þ ðV V2 Þ ¼ v21 V ðV 2V2 Þ ¼ v21 v22

or

If the above sphere passes through the origin then V ðV 2V2 Þ ¼ 0 (note that in two dimensional polar coordinates this is simply) r ¼ 2a cos while in three dimensional Cartesian coordinates it is x2 þ y2 þ z2 2 ða2 x þ b2 y þ c2 xÞ ¼ 0: The equation of a sphere having the points A and B as the extremities of a diameter is ðV V1 Þ ðV V2 Þ ¼ 0: The square of the length of the tangent from C to the sphere with center B and radius v1 is given by ðV3 V2 Þ ðV3 V2 Þ ¼ v21 The condition that the plane V V3 ¼ s is tangential to the sphere ðV V2 Þ ðV V2 Þ ¼ v21 is ðs V3 V2 Þ ðs V3 V2 Þ ¼ v21 v23 : The equation of the tangent plane at D, on the surface of sphere ðV V2 Þ ðV V2 Þ ¼ v21 , is ðV V4 Þ ðV4 V2 Þ ¼ 0 or

V V4 V2 ðV þ V4 Þ ¼ v21 v22

The condition that the two circles ðV V2 Þ ðV V2 Þ ¼ v21 and ðV V4 Þ ðV V4 Þ ¼ v23 intersect orthogonally is clearly ðV2 V4 Þ ðV2 V4 Þ ¼ v21 þ v23 The polar plane of D with respect to the circle ðV V2 Þ ðV V2 Þ ¼ v21

is

V V4 V2 ðV þ V4 Þ ¼ v21 v22 Any sphere through the intersection of the two spheres ðV V2 Þ ðV V2 Þ ¼ v21 and ðV V4 Þ ðV V4 Þ ¼ v23 is given by ðV V2 Þ ðV V2 Þ þ ðV V4 Þ ðV V4 Þ ¼ v21 þ v23 while the radical plane of two such spheres is V ðV2 V4 Þ ¼ 12ðv21 v22 v23 þ v24 Þ

Differentiation of Vectors If V1 ¼ a1 i þ b1 j þ c1 k, and V2 ¼ a2 i þ b2 j þ c2 k, and if V1 and V2 are functions of the scalar t, then d dV1 dV2 ðV1 þ V2 þ Þ ¼ þ þ

, dt dt dt dV1 da1 db1 dc1 ¼ iþ jþ k, etc: where dt dt dt dt d dV1 dV2 ðV1 V2 Þ ¼

V2 þ V1 dt dt dt d dV1 dV2 ðV1 V2 Þ ¼ V2 þ V1 dt dt dt dV dv ¼v V dt dt

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In particular, if V is a vector of constant length then the right hand side of the last equation is identically zero showing that V is perpendicular to its derivative.

The derivatives of the triple products are

and

d dV1 dV2 dV3 ½V1 V2 V3 ¼ V2 V3 þ V1 V3 þ V1 V2 dt dt dt dt d dV1 dV2 dV3 fV1 ðV2 V3 Þg ¼ ðV2 V3 Þ þ V1 V3 þ V1 V2 dt dt dt dt

Geometry of Curves in Space s ¼ the length of arc, measured from some fixed point on the curve (fig. 3). V1 ¼ the position vector of the point A on the curve V1 þ V1 ¼ the position vector of the point P in the neighborhood of A ^t ¼ the unit tangent to the curve at the point A, measured in the direction of s increasing. The normal plane is that plane which is perpendicular to the unit tangent. The principal normal is defined as the intersection of the normal plane with the plane defined by V1 and V1 þ V1 in the limit as V1 0. ^n ¼ the unit normal (principal) at the point A. The plane defined by ^t and ^ n is called the osculating plane (alternatively plane of curvature or local plane). ¼ the radius of curvature at A. ¼ the angle subtended at the origin by V1. ¼

d 1 ¼ ds

^b ¼ the unit binormal i.e. the unit vector which is parallel to ^t ^ n at the point A:

¼ the torsion of the curve at A

A

b t n

P

s increasing

ρ

s=0 V1

V1 + δV1

δθ

o

Figure 3. Frenet’s Formulae: d^t ¼ ^n ds d^n ¼ ^t þ ^b ds d ^b ¼ ^n ds

The following formulae are also applicable: ^t ¼ dV1 ds

Unit tangent

ðV V1 Þ ^t ¼ 0

Equation of the tangent or

V ¼ V1 þ q^t ^n ¼

Unit normal

1d 2 V1 ds2

ðV V1 Þ ^t ¼ 0

Equation of the normal plane Equation of the normal

ðV V1 Þ ^ n¼0 or

V ¼ V1 þ r^ n ^b ¼ ^t ^ n

Unit binormal

ðV V1 Þ ^ b¼0

Equation of the binormal or

V ¼ V 1 þ u^ b

or

V ¼ V1 þ w

Equation of the osculating plane:

dV1 d 2 V1 ds ds2

½ðV V1 Þ^t^ n ¼ 0

dV1 d 2 V1 ¼0 or ðV V1 Þ 2 ds ds

A geodetic line on a surface is a curve, the osculating plane of which is everywhere normal to the surface.

A-92

The differential equation of the geodetic is ½^ndV1 d 2 V1 ¼ 0 Differential Operators—Rectangular Coordinates dS ¼

@S @S @S

dx þ

dy þ

dz @x @y @z

By definition @ @ @ þj þk @x @y @z @2 @2 @2 r2 Laplacian 2 þ 2 þ 2 @x @y @z r del i

If S is a scalar function, then @S @S @S iþ jþ k dx dy dz Grad S defines both the direction and magnitude of the maximum rate of increase of S at any point. Hence the name gradient and also its vectorial nature. rS is independent of the choice of rectangular coordinates. rS grad S

S

S+d

S + dS

V +d

n

V

S S

V

o Figure 4. @S ^n @n where ^n is the unit normal to the surface S ¼ constant, in the direction of S increasing. The total derivative of S at a point having the position vector V is given by (fig. 4) @S ^n dV dS ¼ @n ¼ dV rS and the directional derivative of S in the direction of U is U rS ¼ U ðrSÞ ¼ ðU rÞS Similarly the directional derivative of the vector V in the direction of U is ðU rÞV The distributive law holds for finding a gradient. Thus if S and T are scalar functions rðS þ TÞ ¼ rS þ rT The associative law becomes the rule for differentiating a product: rðSTÞ ¼ SrT þ TrS If V is a vector function with the magnitudes of the components parallel to the three coordinate axes Vx, Vy, Vz, then @Vx @Vy @Vz þ þ r V div V @x @y @z The divergence obeys the distributive law. Thus, if V and U are vector functions, then r ðV þ UÞ ¼ r V þ r U rS ¼

r ðSVÞ ¼ ðrSÞ V þ Sðr VÞ r ðU VÞ ¼ V ðr UÞ U ðr VÞ As with the gradient of a scalar, the divergence of a vector is invariant under a transformation from one set of rectangular coordinates to another. r V curl V ðsometimes r V or rot VÞ @Vy @Vx @Vx @Vy @Vx @Vz iþ jþ k @y @z @z @x @x @y i j k @ @ @ ¼ @x @y @z V V V x

y

z

A-93

The curl (or rotation) of a vector is a vector which is invariant under a transformation from one set of rectangular coordinates to another. r ðU þ VÞ ¼ r U þ r V r ðSVÞ ¼ ðrSÞ V þ Sðr VÞ r ðU VÞ ¼ ðV rÞU ðU rÞV þ Uðr VÞ Vðr UÞ grad ðU VÞ ¼ rðU VÞ ¼ ðV rÞU þ ðU rÞV þ V ðr UÞ þ U ðr VÞ If

V ¼ V x i þ Vy j þ Vz k r V ¼ rVx i þ rVy j þ rVz k and r V ¼ rVx i þ rVy j þ rVz k

The operator r can be used more than once. The number of possibilities where r is used twice are r ðrÞ div grad r ðrÞ curl grad rðr VÞ grad div V r ðr VÞ div curl V r ðr VÞ curl curl V Thus:

div grad S r ðr SÞ Laplacian S r2 S

@2 S @2 S @2 S þ þ @x2 @y2 @z2

curl grad S 0; curl curl V grad div V r2 V; div curl V 0 Taylor’s expansion in three dimensions can be written where and

f ðV þ "Þ ¼ e" r f ðVÞ V ¼ xi þ yj þ zk " ¼ hi þ lj þ mk

(note the analogy with fp ¼ ephDf0 in finite difference methods). Orthogonal Curvilinear Coordinates If at a point P there exist three uniform point functions u, v and w so that the surfaces u ¼ const., v ¼ const., and w ¼ const., intersect in three distinct curves through P then the surfaces are called the coordinate surfaces through P. The three lines of intersection are referred to as the coordinate lines and their tangents a, b, and c as the coordinate axes. When the coordinate axes form an orthogonal set the system is said to define orthogonal curvilinear coordinates at P. Consider an infinitesimal volume enclosed by the surfaces u, v, w, u þ du, v þ dv, and w þ dw (fig. 5). w

S

h3dw

P

h2 dv

u h 1d

R v

Q u Figure 5.

The surface PRS u ¼ const., and the face of the curvilinear figure immediately opposite this is u þ du ¼ const. etc. In terms of these surface constants P ¼ Pðu, v, wÞ Q ¼ Qðu þ du, v, wÞ

and PQ ¼ h1 du

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R ¼ Rðu, v þ dv, wÞ S ¼ Sðu, v, w þ dwÞ

PR ¼ h2 dv PS ¼ h3 dw

where h1, h2, and h3 are functions of u, v, and w. In rectangular Cartesians i, j, k h1 ¼ 1,

h2 ¼ 1, ^b @ @ ¼j , @y h2 @v

â @ @ ¼i , @x h1 @u ^ , k^ In cylindrical coordinates ^r, f h1 ¼ 1, â @ @ ¼ ^r , @r h1 @u

h3 ¼ 1: ^c @ ^ @ ¼k @z h3 @w h3 ¼ 1:

h2 ¼ r, ^ @ b^ @ f ¼ , h2 @v r @

^c @ ^ @ ¼k @z h3 @w

^ In spherical coordinates ^r,û, f h1 ¼ 1,

h3 ¼ r sin

h2 ¼ r, ^ @ b @ f ¼ , h2 @v r @

â @ @ ¼ ^r , @r h1 @u

^ @ ^c @ f ¼ h3 @w r sin @

The general expressions for grad, div and curl together with those for r2 and the directional derivative are, in orthogonal curvilinear coordinates, given by â @S ^b @S ^c @S þ þ h1 @u h2 @v h3 @w V1 @S V2 @S V3 @S þ þ ðV rÞS ¼ h1 @u h2 @v h3 @w , 1 @ @ @ ðh2 h3 V1 Þ þ ðh3 h1 V2 Þ þ ðh1 h2 V3 Þ r V¼ h1 h2 h3 @u @v @w , , â b^ @ @ @ @ ðh3 V3 Þ ðh2 V2 Þ þ ðh1 V1 Þ ðh3 V3 Þ rV¼ h3 h1 @w @w @u h2 h3 @v , ^c @ @ þ ðh2 V2 Þ ðh1 V1 Þ @v h1 h2 @u , 1 @ h2 h3 @S @ h3 h1 @S @ h1 h2 @S 2 þ þ r S¼ h1 h2 h3 @u h1 @u @v h2 @v @w h3 @w rS ¼

FORMULAS OF VECTOR ANALYSIS Rectangular coordinates

Cylindrical coordinates

Conversion to rectangular coordinates

x ¼ r cos ’ @ @ @ iþ jþ k @x @y @z

Gradient .......

J ¼

Divergence....

J A¼

Curl ..............

@Ax @Ay @Az þ þ @x @y @z

i @ JA¼ @x A

x

j @ @y Ay

k @ @z A z

J ¼

y ¼ r sin ’ z ¼ z

@ 1 @ @ rþ fþ k @r r @’ @z

J A¼

1 @ðrAr Þ 1 @A’ þ r @r r @’ @Az þ @z

1 r r JA¼ @ @r A r

Laplacian......

2

r2 ¼

2

2

@ @ @ þ þ @x2 @y2 @z2

f @ @’ rA’

1 k r @ @z A z

1@ @ 1 @2 r þ 2 2 r2 ¼ r @r @r r @’ þ

@2 @z2

A-95

Spherical coordinates

x ¼ r cos ’ sin y ¼ r sin ’ sin z ¼ r cos @ 1 @ 1 @ rþ uþ f @r r @ r sin @’

J ¼

J A¼

1 @ðr2 Ar Þ 1 @ðA sin Þ þ r sin @ r2 @r 1 @A’ þ r sin @’

r 2 r sin JA¼ @ @r A r

r2 ¼

rA

rA’ sin f r @ @’

1 @ 2 @ 1 @ @ r þ sin @r @ r2 @r r2 sin @ þ

r2

u r sin @ @

1 @2 2 @’2 sin

Transformation of Integrals s ¼ the distance along some curve ‘‘C’’ in space and is measured from some fixed point. S ¼ a surface area V ¼ a volume contained by a specified surface ^t ¼ the unit tangent to C at the point P ^n ¼ the unit outward pointing normal F ¼ some vector function ds ¼ the vector element of curve (¼ ^t ds) dS ¼ the vector element of surface (¼ ^ n dS) Z

F ^t ds ¼

Then

Z

ðcÞ

and when

F ds ðcÞ

F ¼ r Z

ðrÞ ^t ds ¼

Z

ðcÞ

Gauss’ Theorem (Green’s Theorem) When S defines a closed region having a volume V ZZZ

also and

ZZ

ZZ ðF ^nÞ dS ¼

ðr FÞ dV ¼

ZZZ

ZZ

ðvÞ

d ðcÞ

ðsÞ

ðrÞ dV ¼ ^ n dS Z Z Z ðvÞ ZðsÞZ ðr FÞ dV ¼ ð^ n FÞ dS ðvÞ

ðsÞ

Stokes’ Theorem When C is closed and bounds the open surface S. ZZ

Z

^n ðr FÞ dS ¼ ðsÞ

Z

ZZ also

ð^n rÞ dS ¼ ðsÞ

Green’s Theorem

F dS ðsÞ

F ds ðcÞ

ds ðcÞ

ZZ

ZZ

ZZZ

ðsÞ

ðr2 Þ dV

^n ðrÞ dS ¼

ðr rÞ dS ¼ ZZ

ðsÞ

ZZZ

ðvÞ

ðr2 Þ dV

^nðrÞ dS ¼

¼ ðsÞ

ðvÞ

A-96

t ^n

^n

MOMENT OF INERTIA FOR VARIOUS BODIES OF MASS The mass of the body is indicated by m Body

Axis

Moment of inertia

Normal to the length, at one l2 m end 3 Uniform thin rod Normal to the length, at the l2 m center 12 The rectangular sheet, sides Through the center parallel a2 m a and b to b 12 Uniform thin rod

Thin rectangular sheet, sides Through the center perpena2 þ b2 m a and b dicular to the sheet 12 Thin circular sheet of radius Normal to the plate through r2 m r the center 2 Thin circular sheet of radius Along any diameter r2 m r 4 Thin circular ring. Radii r1 Through center normal to r 2 þ r2 2 m 1 and r2 plane of ring 2 Thin circular ring. Radii r1 Any diameter and r2

m

r1 2 þ r2 2 4

Rectangular parallelopiped, Through center perpendicua2 þ b2 edges a, b, and c lar to face ab, (parallel to m 12 edge c) Sphere, radius r

Any diameter

Spherical shell, external Any diameter radius, r1, internal radius r2

2 m r2 5 m

2 ðr1 5 r2 5 Þ 5 ðr1 3 r2 3 Þ

Body

Axis

Spherical shell, very thin, Any diameter mean radius, r

Moment of inertia

2 m r2 3

Right circular cylinder of The longitudinal axis of r2 m radius r, length l the solid 2 ! Right circular cylinder of Transverse diameter r2 l 2 þ m radius r, length l 4 12 Hollow circular cylinder, The longitudinal axis of ðr 2 þ r2 2 Þ m 1 the figure length l, radii r1 and r2 2 Thin cylindrical shell, length The longitudinal axis of mr2 l, mean radius, r the figure " # Hollow circular cylinder, Transverse diameter r 2 þ r2 2 l 2 þ m 1 length l, radii r1 and r2 4 12 ! Hollow circular cylinder, Transverse diameter r2 l 2 þ m length l, very thin, mean 2 12 radius ! Elliptic cylinder, length l, Longitudinal axis a2 þ b2 m transverse semiaxes a and b 4 Right cone, altitude h, radius Axis of the figure of base r

m

3 2 r 10

Spheroid of revolution, Polar axis equatorial radius r

m

2r2 5

m

ðb2 þ c2 Þ 5

Ellipsoid, axes 2a, 2b, 2c

A-96

Axis 2a

Bessel Functions* 1. Bessel’s differential equation for a real variable x is d2y dy x2 2 þ x þ ðx2 n2 Þy ¼ 0 dx dx * From Beyer, W. H., Ed., CRC Handbook of Mathematical Sciences, 5th ed., CRC Press, Boca Raton, 1978, 500–503. With permission.

2. When n is not an integer, two independent solutions of the equation are Jn(x), Jn(x), where Jn ðxÞ ¼

xnþ2k ð1Þk k!ðn þ k þ 1Þ 2 k¼0

1 X

3. If n is an integer Jn ðxÞ ¼ ð1Þn Jn ðxÞ, where Jn ðxÞ ¼

, xn x2 x4 x6 1 þ þ þ

2n n! 22 1!ðn þ 1Þ 24 2!ðn þ 1Þ ðn þ 2Þ 26 3!ðn þ 1Þ ðn þ 2Þ ðn þ 3Þ

4. For n ¼ 0 and n ¼ 1, this formula becomes x2 x4 x6 x8 þ þ

22 ð1!Þ2 24 ð2!Þ2 26 ð3!Þ2 28 ð4!Þ2 3 5 7 9 x x x x x þ þ

J1 ðxÞ ¼ 3 2 2 1!2! 25 2!3! 27 3!4! 29 4!5!

J0 ðxÞ ¼ 1

5. When x is large and positive, the following asymptotic series may be used

1 2 2n p po Q0 ðxÞ sin x P0 ðxÞ cos x px 4 4 1 , 2 2 3p 3p Q1 ðxÞ sin x J1 ðxÞ ¼ P1 ðxÞ cos x px 4 4

J0 ðxÞ ¼

where 12 32 12 32 52 72 12 32 52 72 92 112 þ þ

2!ð8xÞ2 4!ð8xÞ4 6!ð8xÞ6 2 2 2 2 2 2 2 2 2 1 1 3 5 1 3 5 7 9 þ Q0 ðxÞ þ

1!8x 5!ð8xÞ5 3!ð8xÞ3 12 3 5 12 32 52 7 9 12 32 52 72 92 11 13 P1 ðxÞ 1 þ þ þ

2!ð8xÞ2 4!ð8xÞ4 6!ð8xÞ6 1 3 12 32 5 7 12 32 52 72 9 11 Q1 ðxÞ þ

1!8x 5!ð8xÞ5 3!ð8xÞ3 P0 ðxÞ 1

[In P1(x) the signs alternate from þ to after the first term] 6. If x > 25, it is convenient to use the formulas J0 ðxÞ ¼ A0 ðxÞ sin x þ B0 ðxÞ cos x J1 ðxÞ ¼ B1 ðxÞ sin x A1 ðxÞ cos x where A0 ðxÞ ¼ B0 ðxÞ ¼

P0 ðxÞ Q0 ðxÞ 1

and A1 ðxÞ ¼

1

and B1 ðxÞ ¼

ðpxÞ2 P0 ðxÞ þ Q0 ðxÞ ðpxÞ2

P1 ðxÞ Q1 ðxÞ 1

ðpxÞ2 P1 ðxÞ þ Q1 ðxÞ 1

ðpxÞ2

7. The zeros of J0(x) and J1(x) If j0s and j1s are the sth zeros of J0(x) and J1(x) respectively, and if a ¼ 4s 1, b ¼ 4s þ 1 , 1 2 62 15,116 12,554,474 8,368,654,292 þ þ

j0, s pa 1 þ 2 2 4 4 þ 4 p a 3p a 15p6 a6 105p8 a8 315p10 a10 , 1 6 6 4716 3,902,418 895,167,324 j1, s pb 1 2 2 þ 4 4 6 6 þ þ

8 8 10 10 4 p b p b 5p b 35p b 35p b 3 sþ1 2 , ð1Þ 2 56 9664 7,381,280 J1 ð j0, s Þ 1 þ þ

1 3p4 a4 5p6 a6 21p8 a8 pa2 3, ð1Þs 22 24 19,584 2,466,720 1þ 4 4 þ

J0 ð j1, s Þ 1 6 6 8 8 p b 10p b 7p b pb2

A-97

8. Table of zeros for J0(x) and J1(x) J1 ðn Þ ¼ 0 Roots n

J0 ð n Þ ¼ 0 Roots n

J1 ðn Þ

J0 ð n Þ

2.4048

0.5191

0.0000

1.0000

5.5201

0.3403

3.8317

0.4028

8.6537

0.2715

7.0156

0.3001

11.7915

0.2325

10.1735

0.2497

14.9309

0.2065

13.3237

0.2184

18.0711

0.1877

16.4706

0.1965

21.2116

0.1733

19.6159

0.1801

9. Recurrence formulas 2n Jn ðxÞ x Jn1 ðxÞ Jnþ1 ðxÞ ¼ 2Jn0 ðxÞ

nJn ðxÞ þ xJn0 ðxÞ ¼ xJn1 ðxÞ

Jn1 ðxÞ þ Jnþ1 ðxÞ ¼

10. If Jn is written for Jn(x) and important

JnðkÞ

nJn ðxÞ xJn0 ðxÞ ¼ xJnþ1 ðxÞ dk is written for k fJn ðxÞg, then the following derivative relationships are dx

J0ðrÞ ¼ J1ðr1Þ 1 1 J0ð2Þ ¼ J0 þ J1 ¼ ðJ2 J0 Þ x 2 1 2 1 ð3Þ J0 ¼ J0 þ 1 2 J1 ¼ ðJ3 þ 3J1 Þ x x 4 3 2 6 1 ð4Þ 3 J1 ¼ ðJ4 4J2 þ 3J0 Þ, etc: J0 ¼ 1 2 J0 x x x 8 11. Half order Bessel functions

rffiffiffiffiffiffi 2 sin x px 2 rffiffiffiffiffiffi 2 cos x J1 ðxÞ ¼ px 2 1 1 d J 3 ðxÞ ¼ xnþ2 fx nþ2 J 1 ðxÞg nþ2 nþ2 dx 1 1 nþ2 d nþ2 fx Jnþ1 ðxÞg Jn1 ðxÞ ¼ x dx 2 2 px1 px1 2 2 Jnþ1 ðxÞ Jnþ1 ðxÞ 2 2 2 2 sin x cos x cos x sin x sin x cos x x x 3 3 3 3 1 sin x cos x 1 cos x þ sin x x x x2 x2 15 6 15 15 6 15 sin x 2 1 cos x 3 cos x 2 1 sin x 3 x x x x x x etc: J1 ðxÞ ¼

n 0 1 2 3

12. Additional solutions to Bessel’s equation are Yn ðxÞ ðalso called Weber’s function, and sometimes denoted by Nn ðxÞÞ Hnð1Þ ðxÞ

and Hnð2Þ ðxÞ ðalso called Hankel functions)

These solutions are defined as follows 8 Jn ðxÞ cos ðnpÞ Jn ðxÞ > > n not an integer > < sin ðnpÞ Yn ðxÞ ¼ > > > limv!n Jv ðxÞ cos ðvpÞ Jv ðxÞ n an integer : sin ðvpÞ

A-98

Hnð1Þ ðxÞ ¼ Jn ðxÞ þ iYn ðxÞ Hnð2Þ ðxÞ ¼ Jn ðxÞ iYn ðxÞ

The additional properties of these functions may all be derived from the above relations and the known properties of Jn(x). 13. Complete solutions to Bessel’s equation may be written as or or

if n is not an integer 9 c1 Jn ðxÞ þ c2 Yn ðxÞ = for any value of n c H ð1Þ x þ c H ð2Þ ðxÞ ;

c1 Jn ðxÞ þ c2 Jn ðxÞ

1

n

2

n

14. The modified (or hyperbolic) Bessel’s differential equation is d2y dy þ x ðx2 þ n2 Þy ¼ 0 dx2 dx 15. When n is not an integer, two independent solutions of the equation are In(x) and In(x), where 1 xnþ2k X 1 In ðxÞ ¼ k!ðn þ k þ 1Þ 2 k¼0 x2

16. If n is an integer, In ðxÞ ¼ In ðxÞ ¼

, xn x2 x4 1þ 2 þ n 2 n! 2 1!ðn þ 1Þ 24 2!ðn þ 1Þðn þ 2Þ x6 þ

þ 6 2 3!ðn þ 1Þ ðn þ 2Þ ðn þ 3Þ

17. For N ¼ 0 and n ¼ 1, this formula becomes x2 x4 x6 x8 þ þ þ þ

2 2 2 22 ð1!Þ 24 ð2!Þ 26 ð3!Þ 28 ð4!Þ2 x x3 x5 x7 x9 þ 5 þ 7 þ 9 þ

I1 ðxÞ ¼ þ 3 2 2 1!2! 2 2!3! 2 3!4! 2 4!5! 18. Another solution to the modified Bessel’s equation is 8 1 In ðxÞ In ðxÞ > > n not an integer > p < 2 sin ðnpÞ Kn ðxÞ ¼ > 1 Iv ðxÞ Iv ðxÞ > > n an integer : lim p v!n 2 sin ðvpÞ I0 ðxÞ ¼ 1 þ

This function is linearly independent of In(x) for all values of n. Thus the complete solution to the modified Bessel’s equation may be written as c1 In ðxÞ þ c2 In ðxÞ

n not an integer

c1 In ðxÞ þ c2 Kn ðxÞ

any n

or 19. The following relations hold among the various Bessel functions: In ðzÞ ¼ im Jm ðizÞ 2 Yn ðizÞ ¼ ðiÞnþ1 In ðzÞ in Kn ðzÞ p Most of the properties of the modified Bessel function may be deduced from the known properties of Jn(x) by use of these relations and those previously given. 20. Recurrence formulas In1 ðxÞ Inþ1 ðxÞ ¼

2n In ðxÞ x

n In1 ðxÞ In ðxÞ ¼ In0 ðxÞ x

A-99

In1 ðxÞ þ Inþ1 ðxÞ ¼ 2In0 ðxÞ n In0 ðxÞ ¼ Inþ1 ðxÞ þ In ðzÞ x

Z Definition: ðnÞ ¼

The Gamma Function*

1

t

n1 t

n>0

e dt

0

Recursion Formula:

ðn þ 1Þ ¼ nðnÞ ðn þ 1Þ ¼ n!; if n ¼ 0, 1, 2, . . . where 0! ¼ 1 For n < 0 the gamma function can be defined by using ðn þ 1Þ ðnÞ ¼ n

* From Beyer, W. H., Ed., CRC Handbook of Mathematical Sciences, 5th ed., CRC Press, Boca Raton, 1978, 484–485. With permission. ðnÞ ¼

ðn þ 1Þ n

Graph:

5 Γ(n) 4 3 2 1 −5 −4

−3 −2 −1

ð12Þ ¼

Special Values:

1 2 3 4 5 −1 −2 −3 −4 −5 pffiffiffi p

1 3 5 ð2m 1Þ pffiffiffi p 2m m m pffiffiffi ð1Þ 2 p ðm þ 12Þ ¼ 1 3 5 ð2m 1Þ ðm þ 12Þ ¼

n

m ¼ 1, 2, 3, . . . m ¼ 1, 2, 3, . . .

Definition: 1 2 3

k kx þ 1Þ ðx þ 2Þ ðx þ kÞ 1 n Y 1 x x=m o ¼ xex e 1þ ðxÞ m m¼1

ðx þ 1Þ ¼ lim

k!1 ðx

This is an infinite product representation for the gamma function where is Euler’s constant. Properties: Z

1

ex ln x dx ¼ 0 ðxÞ 1 1 1 1 1 1 ¼ þ þ þ

þ þ

ðxÞ 1 x 2 xþ1 n xþn1 , pffiffiffiffiffiffiffiffi 1 1 139 þ þ

ðx þ 1Þ ¼ 2px xx ex 1 þ 12x 288x2 51, 840x3 0 ð1Þ ¼

0

This is called Stirling’s asymptotic series. If we let x ¼ n a positive integer, then a useful approximation for n! where n is large (e.g., n > 10) is given by Stirling’s formula pffiffiffiffiffiffiffiffi n! 2pn nn en

A-100

The Gamma Function* Z 1

ex xn1 dx; ðn þ 1Þ ¼ nðnÞ

Values of ðnÞ ¼ 0

n

(n)

n

(n)

n

(n)

n

(n)

1.00 1.01 1.02 1.03 1.04 1.05 1.06 1.07 1.08 1.09 1.10 1.11 1.12 1.13 1.14 1.15 1.16 1.17 1.18 1.19 1.20 1.21 1.22 1.23 1.24

1.00000 .99433 .98884 .98355 .97844 .97350 .96874 .96415 .95973 .95546 .95135 .94740 .94359 .93993 .93642 .93304 .92980 .92670 .92373 .92089 .91817 .91558 .91311 .91075 .90852

1.25 1.26 1.27 1.28 1.29 1.30 1.31 1.32 1.33 1.34 1.35 1.36 1.37 1.38 1.39 1.40 1.41 1.42 1.43 1.44 1.45 1.46 1.47 1.48 1.49

0.90640 .90440 .90250 .90072 .89904 .89747 .89600 .89464 .89338 .89222 .89115 .89018 .88931 .88854 .88785 .88726 .88676 .88636 .88604 .88581 .88566 .88560 .88563 .88575 .88595

1.50 1.51 1.52 1.53 1.54 1.55 1.56 1.57 1.58 1.59 1.60 1.61 1.62 1.63 1.64 1.65 1.66 1.67 1.68 1.69 1.70 1.71 1.72 1.73 1.74

.88623 .88659 .88704 .88757 .88818 .88887 .88964 .89049 .89142 .89243 .89352 .89468 .89592 .89724 .89864 .90012 .90167 .90330 .90500 .90678 .90864 .91057 .91258 .91466 .91683

1.75 1.76 1.77 1.78 1.79 1.80 1.81 1.82 1.83 1.84 1.85 1.86 1.87 1.88 1.89 1.90 1.91 1.92 1.93 1.94 1.95 1.96 1.97 1.98 1.99 2.00

.91906 .92137 .92376 .92623 .92877 .93138 .93408 .93685 .93969 .94261 .94561 .94869 .95184 .95507 .95838 .96177 .96523 .96877 .97240 .97610 .97988 .98374 .98768 .99171 .99581 1.00000

* For large positive values of x, (x) approximates Stirling’s asymptotic series

xx ex

rffiffiffiffiffiffi

2p 1 1 139 571 1þ þ þ

x 12x 288x2 51840x3 2488320x4

Z Definition: Bðm, nÞ ¼

The Beta Function*

1

t

m1

ð1 tÞ

m1

dt

m > 0, n > 0

0

Relationship with Gamma Function: Bðm, nÞ ¼ Properties:

ðmÞðnÞ ðm þ nÞ Bðm, nÞ ¼ Bðn, mÞ Z p=2 Bðm, nÞ ¼ 2 sin2m1 cos2n1 d 0 Z1 tm1 Bðm, nÞ ¼ dt ð1 þ tÞmþn 0 Z 1 m1 t ð1 tÞn1 Bðm, nÞ ¼ rn ðr þ 1Þm dt ðr þ tÞmþn 0

The Error Function Z 2 x t2 Definition: erf(x) ¼ pffiffiffi e dt p 0 3 2 x 1 x5 1 x7 þ

Series: erf(x) ¼ pffiffiffi x þ 3 2! 5 3! 7 p Property: erf(x) ¼ erf ðxÞ

1 x f ðtÞ dt ¼ erf pffiffiffi 2 2 0 pffiffiffi x To evaluate erf (2.3), one proceeds as follows: Since pffiffiffi ¼ 2:3, one finds x ¼ ð2:3Þ ð 2Þ ¼ 3:25. In the normal 2 probability function table (page A-104), one finds the entry 0.4994 opposite the value 3.25. Thus erf (2.3) ¼ 2(0.4994) ¼ 0.9988. Z 2 1 t2 erfc (z) ¼ 1 erf(z) ¼pffiffiffi e dt p z

Relationship with Normal Probability Function f(t):

is known as the complementary error function.

Z

x

Orthogonal Polynomials* I Name: Legendre Symbol: Pn(x) Interval: [ 1, 1] Differential Equation: ð1 x2 Þy00 2 xy0 þ nðn þ 1Þy ¼ 0 y ¼ Pn ðxÞ Explicit Expression: Pn ðxÞ ¼

½n=2 1X n 2n 2m n2m x ð1Þm n m n 2 m¼0

Recurrence Relation: ðn þ 1ÞPnþ1 ðxÞ ¼ ð2n þ 1ÞxPn ðxÞ nPn1 ðxÞ Weight: 1 Standardization: Pn(1) ¼ 1 Z þ1 2 ½Pn ðxÞ 2 dx ¼ Norm: 2n þ1 1 ð1Þn d n Rodrigues’ Formula: Pn ðxÞ ¼ n fð1 x2 Þn g 2 n! dxn R1 ¼

Generating Function:

1 X

Pn ðxÞzn ; 1 < x < 1,

jzj < 1,

n¼0

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R ¼ 1 2xz þ z2 Inequality: jPn ðxÞj 1, 1 x 1: II Name: Tschebysheff, First Kind Symbol: Tn(x) Differential Equation:

2

0

Interval: [ 1, 1]

2

ð1 x Þy xy þ n y ¼ 0 y ¼ Tn ðxÞ

nX ðn m 1Þ! ð2xÞn2m ¼ cos ðn arc cos xÞ ¼ Tn ðxÞ ð1Þm 2 m¼0 m!ðn 2mÞ! ½n=2

Explicit Expression:

Recurrence Relation: Tnþ1 ðxÞ ¼ 2xTn ðxÞ Tn1 ðxÞ Weight: ð1 x2 Þ1=2 Standardization: Tn ð1Þ ¼ 1 , Z þ1 p=2, n 6¼ 0 ð1 x2 Þ1=2 ½Tn ðxÞ 2 dx ¼ Norm: p, n¼0 1 pffiffiffi n n 2 1=2 p d ð1Þ ð1 x Þ fð1 x2 Þnð1=2Þ g ¼ Tn ðxÞ Rodrigues’ Formula: dxn 2nþ1 ðn þ 12Þ 1 X 1 xz ¼ Tn ðxÞ zn , 1 < x < 1, jzj < 1 Generating Function: 1 2xz z2 n¼0 Inequality: jTn ðxÞj 1, 1 x 1: III Name: Tschebysheff, Second Kind Symbol: Un(x) Interval: [ 1, 1] Differential Equation: ð1 x2 Þy00 3 xy0 þ nðn þ 2Þy ¼ 0 y ¼ Un ðxÞ Explicit Expression:

Un ðxÞ ¼

½n=2 X m¼0

Un ðcos Þ ¼

ð1Þm

ðm nÞ! ð2xÞn2m m!ðn 2mÞ!

sin½ðn þ 1Þ sin

* From Beyer, W. H., Ed., CRC Handbook of Mathematical Sciences, 5th ed., CRC Press, Boca Raton, 1978, 557– 560. With permission.

A-102

Recurrence Relation: Unþ1 ðxÞ ¼ 2xUn ðxÞ Un1 ðxÞ Weight: (1 x2)1/2 Standardization: Un ð1Þ ¼ n þ 1 Z þ1 p ð1 x2 Þ1=2 ½Un ðxÞ 2 dx ¼ Norm: 2 1 pffiffiffi ð1Þn ðn þ 1Þ p dn Rodrigues’ Formula: Un ðxÞ ¼ fð1 x2 Þnþð1=2Þ g 1=2 nþ1 3 dxn 2 ð1 x Þ 2 ðn þ 2Þ 1 X 1 ¼ Un ðxÞ zn , 1; < x < 1, jzj < 1 Generating Function: 2 1 2xz þ z n¼0 Inequality: jUn ðxÞj n þ 1, 1 x 1: IV Þ Name: Jacobi Symbol: Pð, ðxÞ n Differential Equation:

Interval: [ 1, 1]

ð1 x2 Þy00 þ ½ ð þ þ 2Þx y0 þ nðn þ þ þ 1Þy ¼ 0 Þ ðxÞ y ¼ Pð, n Þ ðxÞ ¼ Explicit Expression: Pð, n

Recurrence Relation:

n 1X nþ nþ ðx 1Þnm ðx þ 1Þm m nm 2n m¼0

Þ 2ðn þ 1Þ ðn þ þ þ 1Þ ð2n þ þ ÞPð, nþ1 ðxÞ

¼ ð2n þ þ þ 1Þ½ð2 2 Þ þ ð2n þ þ þ 2Þ Þ ðxÞ ð2n þ þ Þx Pð, n Þ 2ðn þ Þ ðn þ Þ ð2n þ þ þ 2ÞPð, ðxÞ n1 nþ ð, Þ Weight: ð1 xÞ ð1 þ xÞ ; , > 1 Standardization: Pn ðxÞ ¼ n Z þ1 þ þ1 2 ðn þ þ 1Þðn þ þ 1Þ ð, Þ ð1 xÞ ð1 þ xÞ ½Pn ðxÞ 2 dx ¼ Norm: ð2n þ þ þ 1Þn!ðn þ þ þ 1Þ 1 ð1Þn dn Þ Rodrigues’ Formula: Pð, ðxÞ ¼ n fð1 xÞnþ ð1 þ xÞnþ g n 2 n!ð1 xÞ ð1 þ xÞ dxn

Generating Function:

R1 ð1 z þ RÞ ð1 þ z þ RÞ ¼ R¼

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2xz þ z2 ,

1 X

Þ 2 Pð, ðxÞzn , n

n¼0

jzj < 1

8 nþq > > nq if q ¼ max ð, Þ 12 > > n > > > > > < jPð, Þ ðx0 Þj n1=2 if q < 1 n Þ 2 ðxÞj ¼ Inequality: max jPð, n > x0 is one of the two maximum points nearest 1 x 1 > > > > > > > > : þ þ1 V Name: Generalized Laguerre Differential Equation:

Symbol: LðÞ Interval: ½0, 1 n ðxÞ xy00 þ ð þ 1 xÞy0 þ ny ¼ 0

y ¼ LðÞ n ðxÞ 1 m m nþ ðÞ x ð1Þ Explicit Expression: Ln ðxÞ ¼ n m m! m¼0 n X

Recurrence Relation: ðn þ 1ÞLnðÞ þ 1ðxÞ ¼ ½ð2n þ þ 1Þ x LnðÞ ðxÞ ðn þ ÞLnðÞ 1ðxÞ Weight: x ex , > 1 Standardization: LnðÞ ðxÞ ¼ Z1 ðn þ þ 1Þ 2 x ex ½LðÞ Norm: n ðxÞ dx ¼ n! 0 1 d n nþ x Rodrigues’ Formula: LnðÞ ðxÞ ¼ fx e g x n!x e dxn

ð1Þn n x þ

n!

A-103

1 xz X n ¼ Generating Function: ð1 zÞ1 exp LðÞ n ðxÞz z1 n¼0

Inequality:

x0 ðn þ þ 1Þ x=2 e ; n!ð þ 1Þ >0

ðaÞ L ðxÞ 2 ð þ n þ 1Þ ex=2 ; x 0 n n!ð þ 1Þ 1 < < 0 jLðÞ n ðxÞj

Orthogonal Polynomials Name: Hermite Symbol: Hn ðxÞ Interval: ½1, 1 Differential Equation: y00 2xy0 þ 2ny ¼ 0 ½n=2 X ð1Þm n!ð2xÞn2m Explicit Expression: Hn ðxÞ ¼ m!ðn 2mÞ! m¼0 Recurrence Relation: Hnþ1 ðxÞ ¼ 2xHn ðxÞ 2nHn1 ðxÞ 2 Weight: ex Standardization: Hn ð1Þ ¼ 2n xn þ Z1 pffiffiffi 2 2 ex ½Hn ðxÞ dx ¼ 2n n! p Norm: 1 n 2 d 2 ðex Þ Rodriques’ Formula: Hn ðxÞ ¼ ð1Þn ex n dx 1 n X z 2 Hn ðxÞ Generating Function: ez þ2zx ¼ n! pn¼0 ffiffiffiffi 2 Inequality: jHn ðxÞj < ex =2 k2n=2 n! k 1:086435

NORMAL PROBABILITY FUNCTION

Areas under the Standard Normal Curve from 0 to z

z

0

z

0

1

2

3

4

5

6

7

8

9

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9

.0000 .0398 .0793 .1179 .1554 .1915 .2258 .2580 .2881 .3159 .3413 .3643 .3849 .4032 .4192 .4332 .4452 .4554 .4641 .4713 .4772 .4821 .4861 .4893 .4918 .4938 .4953 .4965 .4974 .4981 .4987 4990 4993 4995 4997 4998 4998 4999 4999 5000

.0040 .0438 .0832 .1217 .1591 .1950 .2291 .2612 .2910 .3186 .3438 .3665 .3869 .4049 .4207 .4345 .4463 .4564 .4649 .4719 .4778 .4826 .4864 .4896 .4920 .4940 .4955 .4966 .4975 .4982 .4987 .4991 .4993 .4995 .4997 .4998 .4998 .4999 .4999 .5000

.0080 .0478 .0871 .1255 .1628 .1985 .2324 .2652 .2939 .3212 .3461 .3686 .3888 .4066 .4222 .4357 .4474 .4573 .4656 .4726 .4783 .4830 .4868 .4898 .4922 .4941 .4956 .4967 .4976 .4982 .4987 .4991 .4994 .4995 .4997 .4998 .4999 .4999 .4999 .5000

.0120 .0517 .0910 .1293 .1664 .2019 .2357 .2673 .2967 .3238 .3485 .3708 .3907 .4082 .4236 .4370 .4484 .4582 .4664 .4732 .4788 .4834 .4871 .4901 .4925 .4943 .4957 .4968 .4977 .4983 .4988 .4991 .4994 .4996 .4997 .4998 .4999 .4999 .4999 .5000

.0160 .0557 .0948 .1331 .1700 .2054 .2389 .2704 .2996 .3264 .3508 .3729 .3925 .4099 .4251 .4382 .4495 .4591 .4671 .4738 .4793 .4838 .4875 .4904 .4927 .4945 .4959 .4969 .4977 .4984 .4988 .4992 .4994 .4996 .4997 .4998 .4999 .4999 .4999 .5000

.0199 .0596 .0987 .1368 .1736 .2088 .2422 .2734 .3023 .3289 .3531 .3749 .3944 .4115 .4265 .4394 .4505 .4599 .4678 .4744 .4798 .4842 .4878 .4906 .4929 .4946 .4960 .4970 .4978 .4984 .4989 .4992 .4994 .4996 .4997 .4998 .4999 .4999 .4999 .5000

.0239 .0636 .1026 .1406 .1772 .2123 .2454 .2764 .3051 .3315 .3554 .3770 .3962 .4131 .4279 .4406 .4515 .4608 .4686 .4750 .4803 .4846 .4881 .4909 .4931 .4948 .4961 .4971 .4979 .4985 .4989 .4992 .4994 .4996 .4997 .4998 .4999 .4999 .4999 .5000

.0279 .0675 .1064 .1443 .1808 .2157 .2486 .2794 .3078 .3340 .3577 .3790 .3980 .4147 .4292 .4418 .4525 .4616 .4693 .4756 .4808 .4850 .4884 .4911 .4932 .4949 .4962 .4972 .4979 .4985 .4989 .4992 .4995 .4996 .4997 .4998 .4999 .4999 .4999 .5000

.0319 .0714 .1103 .1480 .1844 .2190 .2518 .2823 .3106 .3365 .3599 .3810 .3997 .4162 .4306 .4429 .4535 .4625 .4699 .4761 .4812 .4854 .4887 .4913 .4934 .4951 .4963 .4973 .4980 .4986 .4990 .4993 .4995 .4996 .4997 .4998 .4999 .4999 .4999 .5000

.0359 .0754 .1141 .1517 .1879 .2224 .2549 .2852 .3133 .3389 .3621 .3830 .4015 .4177 .4319 .4441 .4545 .4633 .4706 .4767 .4817 .4857 .4890 .4916 .4936 .4952 .4964 .4974 .4981 .4986 .4990 .4993 .4995 .4997 .4998 .4998 .4999 .4999 .4999 .5000

F(z) below refers to area under Standard Normal Curve from 1 to z z F(z) 2[1 F(z)]

1.282 .90 .20

1.645 .95 .10

1.960 .975 .05

A-104

2.326 .99 .02

2.576 .995 .01

3.090 .999 .002

PERCENTAGE POINTS, STUDENT’S t-DISTRIBUTION This table gives values of t such that

nþ1 x2 nþ1 2 FðtÞ ¼ pffiffiffiffiffiffi n 1 þ n 2 dx 1 np 2 for n, the number of degrees of freedom, equal to 1, 2, . . . , 30, 40, 60, 120, 1; and for F(t) ¼ 0.60, 0.75, 0.90, 0.95, 0.975, 0.99, 0.995, and 0.9995. The t-distribution is symmetrical, so that FðtÞ ¼ 1 FðtÞ Z

t

n nF

.60

.75

.90

.95

.975

.99

.995

.9995

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 40 60 120 1

.325 .289 .277 .271 .267 .265 .263 .262 .261 .260 .260 .259 .259 .258 .258 .258 .257 .257 .257 .257 .257 .256 .256 .256 .256 .256 .256 .256 .256 .256 .255 .254 .254 .253

1.000 .816 .765 .741 .727 .718 .711 .706 .703 .700 .697 .695 .694 .692 .691 .690 .689 .688 .688 .687 .686 .686 .685 .685 .684 .684 .684 .683 .683 .683 .681 .679 .677 .674

3.078 1.886 1.638 1.533 1.476 1.440 1.415 1.397 1.383 1.372 1.363 1.356 1.350 1.345 1.341 1.337 1.333 1.330 1.328 1.325 1.323 1.321 1.319 1.318 1.316 1.315 1.314 1.313 1.311 1.310 1.303 1.296 1.289 1.282

6.314 2.920 2.353 2.132 2.015 1.943 1.895 1.860 1.833 1.812 1.796 1.782 1.771 1.761 1.753 1.746 1.740 1.734 1.729 1.725 1.721 1.717 1.714 1.711 1.708 1.706 1.703 1.701 1.699 1.697 1.684 1.671 1.658 1.645

12.706 4.303 3.182 2.776 2.571 2.447 2.365 2.306 2.262 2.228 2.201 2.179 2.160 2.145 2.131 2.120 2.110 2.101 2.093 2.086 2.080 2.074 2.069 2.064 2.060 2.056 2.052 2.048 2.045 2.042 2.021 2.000 1.980 1.960

31.821 6.965 4.541 3.747 3.365 3.143 2.998 2.896 2.821 2.764 2.718 2.681 2.650 2.624 2.602 2.583 2.567 2.552 2.539 2.528 2.518 2.508 2.500 2.492 2.485 2.479 2.473 2.467 2.462 2.457 2.423 2.390 2.358 2.326

63.657 9.925 5.841 4.604 4.032 3.707 3.499 3.355 3.250 3.169 3.106 3.055 3.012 2.977 2.947 2.921 2.898 2.878 2.861 2.845 2.831 2.819 2.807 2.797 2.787 2.779 2.771 2.763 2.756 2.750 2.704 2.660 2.617 2.576

636.619 31.598 12.924 8.610 6.869 5.959 5.408 5.041 4.781 4.587 4.437 4.318 4.221 4.140 4.073 4.015 3.965 3.922 3.883 3.850 3.819 3.792 3.767 3.745 3.725 3.707 3.690 3.674 3.659 3.646 3.551 3.460 3.373 3.291

*

This table is abridged from the ‘‘Statistical Tables’’ of R. A. Fisher and Frank Yates published by Oliver & Boyd. Ltd., Edinburgh and London, 1938. It is here published with the kind permission of the authors and their publishers.

PERCENTAGE POINTS, CHI-SQUARE DISTRIBUTION This table gives values of 2 such that FðÞ2 ¼

Z 0

2

1 xðn2Þ=2 eðx=2Þ dx n 2 2 ðn=2Þ

for n, the number of ffidegrees of freedom, equal to 1, 2, . . . , 30. For n > 30, a normal approximation is quite accurate. pffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi ffi The expression 2x2 2n 1 is approximately normally distributed as the standard normal distribution. Thus 2 , the -point of the distribution, may be computed by the formula pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ¼ 12½x þ 2n 1 2 ; where x is the -point of the cumulative normal distribution. For even values of n, F(2) can be written as 1 Fð2 Þ ¼

0 x 1 x X

e x! x¼0

with ¼ 122 and x0 ¼ 12n. Thus the cumulative Chi-Square distribution is related to the cumulative Poisson distribution.

A-105

Another approximate formula for large n 2 ¼ n 1

rffiffiffiffiffi!3 2 2 þ z 9n 9n

n ¼ degrees of freedom z ¼ the normal deviate (the value of x for which F(x) ¼ the desired percentile). x

1.282

1.645

1.960

2.326

2.576

3.090

F(x)

.90

.95

.975

.99

.995

.999

2:99 ¼ 60½1 0:00370 þ 2:326ð0:06086Þ 3 ¼ 88:4 is the 99th percentile for 60 degrees of freedom.

Fð2 Þ ¼

Z

2

0

F

n 1 2 3 4 5

.005 .0000393 .0100 .0717 .207 .412

.010

.025

.000157 .0201 .115 .297 .554

.000982 .0506 .216 .484 .831

1 2n=2

n xðn2Þ=2 ex=2 dx 2

.050

.100

.250

.500

.750

.900

.950

.975

.990

.995

.00393 .103 .352 .711 1.15

.0158 .211 .584 1.06 1.61

.102 .575 1.21 1.92 2.67

.455 1.39 2.37 3.36 4.35

1.32 2.77 4.11 5.39 6.63

2.71 4.61 6.25 7.78 9.24

3.84 5.99 7.81 9.49 11.1

5.02 7.38 9.35 11.1 12.8

6.63 9.21 11.3 13.3 15.1

7.88 10.6 12.8 14.9 16.7

5.35 6.35 7.34 8.34 9.34

7.84 9.04 10.2 11.4 12.5

10.6 12.0 13.4 14.7 16.0

12.6 14.1 15.5 16.9 18.3

14.4 16.0 17.5 19.0 20.5

16.8 18.5 20.1 21.7 23.2

18.5 20.3 22.0 23.6 25.2

6 7 8 9 10

.676 .989 1.34 1.73 2.16

.872 1.24 1.65 2.09 2.56

1.24 1.69 2.18 2.70 3.25

1.64 2.17 2.73 3.33 3.94

2.20 2.83 3.49 4.17 4.87

3.45 4.25 5.07 5.90 6.74

11 12 13 14 15

2.60 3.07 3.57 4.07 4.60

3.05 3.57 4.11 4.66 5.23

3.82 4.40 5.01 5.63 6.26

4.57 5.23 5.89 6.57 7.26

5.58 6.30 7.04 7.79 8.55

7.58 8.44 9.30 10.2 11.0

10.3 11.3 12.3 13.3 14.3

13.7 14.8 16.0 17.1 18.2

17.3 18.5 19.8 21.1 22.3

19.7 21.0 22.4 23.7 25.0

21.9 23.3 24.7 26.1 27.5

24.7 26.2 27.7 29.1 30.6

26.8 28.3 29.8 31.3 32.8

16 17 18 19 20

5.14 5.70 6.26 6.84 7.43

5.81 6.41 7.01 7.63 8.26

6.91 7.56 8.23 8.91 9.59

7.96 8.67 9.39 10.1 10.9

9.31 10.1 10.9 11.7 12.4

11.9 12.8 13.7 14.6 15.5

15.3 16.3 17.3 18.3 19.3

19.4 20.5 21.6 22.7 23.8

23.5 24.8 26.0 27.2 28.4

26.3 27.6 28.9 30.1 31.4

28.8 30.2 31.5 32.9 34.2

32.0 33.4 34.8 36.2 37.6

34.3 35.7 37.2 38.6 40.0

21 22 23 24 25

8.03 8.64 9.26 9.89 10.5

8.90 9.54 10.2 10.9 11.5

10.3 11.0 11.7 12.4 13.1

11.6 12.3 13.1 13.8 14.6

13.2 14.0 14.8 15.7 16.5

16.3 17.2 18.1 19.0 19.9

20.3 21.3 22.3 23.3 24.3

24.9 26.0 27.1 28.2 29.3

29.6 30.8 32.0 33.2 34.4

32.7 33.9 35.2 36.4 37.7

35.5 36.8 38.1 39.4 40.6

38.9 40.3 41.6 43.0 44.3

41.4 42.8 44.2 45.6 46.9

26 27 28 29 30

11.2 11.8 12.5 13.1 13.8

12.2 12.9 13.6 14.3 15.0

13.8 14.6 15.3 16.0 16.8

15.4 16.2 16.9 17.7 18.5

17.3 18.1 18.9 19.8 20.6

20.8 21.7 22.7 23.6 24.5

25.3 26.3 27.3 28.3 29.3

30.4 31.5 32.6 33.7 34.8

35.6 36.7 37.9 39.1 40.3

38.9 40.1 41.3 42.6 43.8

41.9 43.2 44.5 45.7 47.0

45.6 47.0 48.3 49.6 50.9

48.3 49.6 51.0 52.3 53.7

A-106

PERCENTAGE POINTS, F-DISTRIBUTION This table gives values of F such that Z FðFÞ ¼ 0

F

m þ n m2 n mm=2 nn=2 xðm2Þ=2 ðn þ mxÞðmþnÞ=2 dx 2 2

for selected values of m, the number of degrees of freedom of the numerator of F; and for selected values of n, the number of degrees of freedom of the denominator of F. The table also provides values corresponding to F(F) ¼ .10, .05, .025, .01, .005, .001 since F1 for m and n degrees of freedom is the reciprocal of F for n and m degrees of freedom. Thus F:05 ð4, 7Þ ¼

1 1 ¼ :164 ¼ F:95 ð7, 4Þ 6:09

A-106

Z

F

FðFÞ ¼ 0 n

m

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 40 60 120 1

m þ n m2 n mm=2 nn=2 xm=21 ðn þ mxÞðmþnÞ=2 dx ¼ :90 2 2

1

2

3

4

5

6

7

8

9

10

12

15

20

24

30

40

60

120

1

39.86 8.53 5.54 4.54 4.06 3.78 3.59 3.46 3.36 3.29 3.23 3.18 3.14 3.10 3.07 3.05 3.03 3.01 2.99 2.97 2.96 2.95 2.94 2.93 2.92 2.91 2.90 2.89 2.89 2.88 2.84 2.79 2.75 2.71

49.50 9.00 5.46 4.32 3.78 3.46 3.26 3.11 3.01 2.92 2.86 2.81 2.76 2.73 2.70 2.67 2.64 2.62 2.61 2.59 2.57 2.56 2.55 2.54 2.53 2.52 2.51 2.50 2.50 2.49 2.44 2.39 2.35 2.30

53.59 9.16 5.39 4.19 3.62 3.29 3.07 2.92 2.81 2.73 2.66 2.61 2.56 2.52 2.49 2.46 2.44 2.42 2.40 2.38 2.36 2.35 2.34 2.33 2.32 2.31 2.30 2.29 2.28 2.28 2.23 2.18 2.13 2.08

55.83 9.24 5.34 4.11 3.52 3.18 2.96 2.81 2.69 2.61 2.54 2.48 2.43 2.39 2.36 2.33 2.31 2.29 2.27 2.25 2.23 2.22 2.21 2.19 2.18 2.17 2.17 2.16 2.15 2.14 2.09 2.04 1.99 1.94

57.24 9.29 5.31 4.05 3.45 3.11 2.88 2.73 2.61 2.52 2.45 2.39 2.35 2.31 2.27 2.24 2.22 2.20 2.18 2.16 2.14 2.13 2.11 2.10 2.09 2.08 2.07 2.06 2.06 2.05 2.00 1.95 1.90 1.85

58.20 9.33 5.28 4.01 3.40 3.05 2.83 2.67 2.55 2.46 2.39 2.33 2.28 2.24 2.21 2.18 2.15 2.13 2.11 2.09 2.08 2.06 2.05 2.04 2.02 2.01 2.00 2.00 1.99 1.98 1.93 1.87 1.82 1.77

58.91 9.35 5.27 3.98 3.37 3.01 2.78 2.62 2.51 2.41 2.34 2.28 2.23 2.19 2.16 2.13 2.10 2.08 2.06 2.04 2.02 2.01 1.99 1.98 1.97 1.96 1.95 1.94 1.93 1.93 1.87 1.82 1.77 1.72

59.44 9.37 5.25 3.95 3.34 2.98 2.75 2.59 2.47 2.38 2.30 2.24 2.20 2.15 2.12 2.09 2.06 2.04 2.02 2.00 1.98 1.97 1.95 1.94 1.93 1.92 1.91 1.90 1.89 1.88 1.83 1.77 1.72 1.67

59.86 9.38 5.24 3.94 3.32 2.96 2.72 2.56 2.44 2.35 2.27 2.21 2.16 2.12 2.09 2.06 2.03 2.00 1.98 1.96 1.95 1.93 1.92 1.91 1.89 1.88 1.87 1.87 1.86 1.85 1.79 1.74 1.68 1.63

60.19 9.39 5.23 3.92 3.30 2.94 2.70 2.54 2.42 2.32 2.25 2.19 2.14 2.10 2.06 2.03 2.00 1.98 1.96 1.94 1.92 1.90 1.89 1.88 1.87 1.86 1.85 1.84 1.83 1.82 1.76 1.71 1.65 1.60

60.71 9.41 5.22 3.90 3.27 2.90 2.67 2.50 2.38 2.28 2.21 2.15 2.10 2.05 2.02 1.99 1.96 1.93 1.91 1.89 1.87 1.86 1.84 1.83 1.82 1.81 1.80 1.79 1.78 1.77 1.71 1.66 1.60 1.55

61.22 9.42 5.20 3.87 3.24 2.87 2.63 2.46 2.34 2.24 2.17 2.10 2.05 2.01 1.97 1.94 1.91 1.89 1.86 1.84 1.83 1.81 1.80 1.78 1.77 1.76 1.75 1.74 1.73 1.72 1.66 1.60 1.55 1.49

61.74 9.44 5.18 3.84 3.21 2.84 2.59 2.42 2.30 2.20 2.12 2.06 2.01 1.96 1.92 1.89 1.86 1.84 1.81 1.79 1.78 1.76 1.74 1.73 1.72 1.71 1.70 1.69 1.68 1.67 1.61 1.54 1.48 1.42

62.00 9.45 5.18 3.83 3.19 2.82 2.58 2.40 2.28 2.18 2.10 2.04 1.98 1.94 1.90 1.87 1.84 1.81 1.79 1.77 1.75 1.73 1.72 1.70 1.69 1.68 1.67 1.66 1.65 1.64 1.57 1.51 1.45 1.38

62.26 9.46 5.17 3.82 3.17 2.80 2.56 2.38 2.25 2.16 2.08 2.01 1.96 1.91 1.87 1.84 1.81 1.78 1.76 1.74 1.72 1.70 1.69 1.67 1.66 1.65 1.64 1.63 1.62 1.61 1.54 1.48 1.41 1.34

62.53 9.47 5.16 3.80 3.16 2.78 2.54 2.36 2.23 2.13 2.05 1.99 1.93 1.89 1.85 1.81 1.78 1.75 1.73 1.71 1.69 1.67 1.66 1.64 1.63 1.61 1.60 1.59 1.58 1.57 1.51 1.44 1.37 1.30

62.79 9.47 5.15 3.79 3.14 2.76 2.51 2.34 2.21 2.11 2.03 1.96 1.90 1.86 1.82 1.78 1.75 1.72 1.70 1.68 1.66 1.64 1.62 1.61 1.59 1.58 1.57 1.56 1.55 1.54 1.47 1.40 1.32 1.24

63.06 9.48 5.14 3.78 3.12 2.74 2.49 2.32 2.18 2.08 2.00 1.93 1.88 1.83 1.79 1.75 1.72 1.69 1.67 1.64 1.62 1.60 1.59 1.57 1.56 1.54 1.53 1.52 1.51 1.50 1.42 1.35 1.26 1.17

63.33 9.49 5.13 3.76 3.10 2.72 2.47 2.29 2.16 2.06 1.97 1.90 1.85 1.80 1.76 1.72 1.69 1.66 1.63 1.61 1.59 1.57 1.55 1.53 1.52 1.50 1.49 1.48 1.47 1.46 1.38 1.29 1.19 1.00

s21 S1 . S2 , where s21 ¼ S1 =m and s22 ¼ S2 =n are independent mean squares estimating a common variance 2 and based on m ¼ m n s22 and n degrees of freedom, respectively. F¼

Z

F

FðFÞ ¼ 0

n

m 1 2 3 4

1

2

3

4

5

6

m þ n m2 nmm=2 nn=2 xm=21 ðn þ mxÞðmþnÞ=2 dx ¼ :95 2 2 7

8

9

10

12

15

20

24

30

40

60

120

1

161.4 199.5 215.7 224.6 230.2 234.0 236.8 238.9 240.5 241.9 243.9 245.9 248.0 249.1 250.1 251.1 252.2 253.3 254.3 18.51 19.00 19.16 19.25 19.30 19.33 19.35 19.37 19.38 19.40 19.41 19.43 19.45 19.45 19.46 19.47 19.48 19.49 19.50 10.13 9.55 9.28 9.12 9.01 8.94 8.89 8.85 8.81 8.79 8.74 8.70 8.66 8.64 8.62 8.59 8.57 8.55 8.53 7.71 6.94 6.59 6.39 6.26 6.16 6.09 6.04 6.00 5.96 5.91 5.86 5.80 5.77 5.75 5.72 5.69 5.66 5.63

5 6 7 8 9

6.61 5.99 5.59 5.32 5.12

5.79 5.14 4.74 4.46 4.26

5.41 4.76 4.35 4.07 3.86

5.19 4.53 4.12 3.84 3.63

5.05 4.39 3.97 3.69 3.48

4.95 4.28 3.87 3.58 3.37

4.88 4.21 3.79 3.50 3.29

4.82 4.15 3.73 3.44 3.23

4.77 4.10 3.68 3.39 3.18

4.74 4.06 3.64 3.35 3.14

4.68 4.00 3.57 3.28 3.07

4.62 3.94 3.51 3.22 3.01

4.56 3.87 3.44 3.15 2.94

4.53 3.84 3.41 3.12 2.90

4.50 3.81 3.38 3.08 2.86

4.46 3.77 3.34 3.04 2.83

4.43 3.74 3.30 3.01 2.79

4.40 3.70 3.27 2.97 2.75

4.36 3.67 3.23 2.93 2.71

10 11 12 13 14

4.96 4.84 4.75 4.67 4.60

4.10 3.98 3.89 3.81 3.74

3.71 3.59 3.49 3.41 3.34

3.48 3.36 3.26 3.18 3.11

3.33 3.20 3.11 3.03 2.96

3.22 3.09 3.00 2.92 2.85

3.14 3.01 2.91 2.83 2.76

3.07 2.95 2.85 2.77 2.70

3.02 2.90 2.80 2.71 2.65

2.98 2.85 2.75 2.67 2.60

2.91 2.79 2.69 2.60 2.53

2.85 2.72 2.62 2.53 2.46

2.77 2.65 2.54 2.46 2.39

2.74 2.61 2.51 2.42 2.35

2.70 2.57 2.47 2.38 2.31

2.66 2.53 2.43 2.34 2.27

2.62 2.49 2.38 2.30 2.22

2.58 2.45 2.34 2.25 2.18

2.54 2.40 2.30 2.21 2.13

15 16 17 18 19

4.54 4.49 4.45 4.41 4.38

3.68 3.63 3.59 3.55 3.52

3.29 3.24 3.20 3.16 3.13

3.06 3.01 2.96 2.93 2.90

2.90 2.85 2.81 2.77 2.74

2.79 2.74 2.70 2.66 2.63

2.71 2.66 2.61 2.58 2.54

2.64 2.59 2.55 2.51 2.48

2.59 2.54 2.49 2.46 2.42

2.54 2.49 2.45 2.41 2.38

2.48 2.42 2.38 2.34 2.31

2.40 2.35 2.31 2.27 2.23

2.33 2.28 2.23 2.19 2.16

2.29 2.24 2.19 2.15 2.11

2.25 2.19 2.15 2.11 2.07

2.20 2.15 2.10 2.06 2.03

2.16 2.11 2.06 2.02 1.98

2.11 2.06 2.01 1.97 1.93

2.07 2.01 1.96 1.92 1.88

20 21 22 23 24

4.35 4.32 4.30 4.28 4.26

3.49 3.47 3.44 3.42 3.40

3.10 3.07 3.05 3.03 3.01

2.87 2.84 2.82 2.80 2.78

2.71 2.68 2.66 2.64 2.62

2.60 2.57 2.55 2.53 2.51

2.51 2.49 2.46 2.44 2.42

2.45 2.42 2.40 2.37 2.36

2.39 2.37 2.34 2.32 2.30

2.35 2.32 2.30 2.27 2.25

2.28 2.25 2.23 2.20 2.18

2.20 2.18 2.15 2.13 2.11

2.12 2.10 2.07 2.05 2.03

2.08 2.05 2.03 2.01 1.98

2.04 2.01 1.98 1.96 1.94

1.99 1.96 1.94 1.91 1.89

1.95 1.92 1.89 1.86 1.84

1.90 1.87 1.84 1.81 1.79

1.84 1.81 1.78 1.76 1.73

25 26 27 28 29

4.24 4.23 4.21 4.20 4.18

3.39 3.37 3.35 3.34 3.33

2.99 2.98 2.96 2.95 2.93

2.76 2.74 2.73 2.71 2.70

2.60 2.59 2.57 2.56 2.55

2.49 2.47 2.46 2.45 2.43

2.40 2.39 2.37 2.36 2.35

2.34 2.32 2.31 2.29 2.28

2.28 2.27 2.25 2.24 2.22

2.24 2.22 2.20 2.19 2.18

2.16 2.15 2.13 2.12 2.10

2.09 2.07 2.06 2.04 2.03

2.01 1.99 1.97 1.96 1.94

1.96 1.95 1.93 1.91 1.90

1.92 1.90 1.88 1.87 1.85

1.87 1.85 1.84 1.82 1.81

1.82 1.80 1.79 1.77 1.75

1.77 1.75 1.73 1.71 1.70

1.71 1.69 1.67 1.65 1.64

30 40 60 120 1

4.17 4.08 4.00 3.92 3.84

3.32 3.23 3.15 3.07 3.00

2.92 2.84 2.76 2.68 2.60

2.69 2.61 2.53 2.45 2.37

2.53 2.45 2.37 2.29 2.21

2.42 2.34 2.25 2.17 2.10

2.33 2.25 2.17 2.09 2.01

2.27 2.18 2.10 2.02. 1.94

2.21 2.12 2.04 1.96 1.88

2.16 2.08 1.99 1.91 1.83

2.09 2.00 1.92 1.83 1.75

2.01 1.92 1.84 1.75 1.67

1.93 1.84 1.75 1.66 1.57

1.89 1.79 1.70 1.61 1.52

1.84 1.74 1.65 1.55 1.46

1.79 1.69 1.59 1.50 1.39

1.74 1.64 1.53 1.43 1.32

1.68 1.58 1.47 1.35 1.22

1.62 1.51 1.39 1.25 1.00

s21 S1 . S2 , where s21 ¼ S1 =m and s22 ¼ S2 =n are independent mean squares estimating a common variance 2 ¼ m n s22 and based on m and n degrees of freedom, respectively. F¼

A-107

Z

F

FðFÞ ¼ 0

n

m

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 40 60 120 1

1

2

3

4

5

6

m þ n m2 n mm=2 nn=2 xm=21 ðn þ mxÞðmþnÞ=2 dx ¼ :975 2 2 7

8

9

10

12

15

20

24

30

40

60

120

1

647.8 799.5 864.2 899.6 921.8 937.1 948.2 956.7 963.3 968.6 976.7 984.9 993.1 997.2 1001 1006 1010 1014 1018 38.51 39.00 39.17 39.25 39.30 39.33 39.36 39.37 39.39 39.40 39.41 39.43 39.45 39.46 39.46 39.47 39.48 39.49 39.50 17.44 16.04 15.44 15.10 14.88 14.73 14.62 14.54 14.47 14.42 14.34 14.25 14.17 14.12 14.08 14.04 13.99 13.95 13.90 12.22 10.65 9.98 9.60 9.36 9.20 9.07 8.98 8.90 8.84 8.75 8.66 8.56 8.51 8.46 8.41 8.36 8.31 8.26 10.01 8.43 7.76 7.39 7.15 6.98 6.85 6.76 6.68 6.62 6.52 6.43 6.33 6.28 6.23 6.18 6.12 6.07 6.02 8.81 7.26 6.60 6.23 5.99 5.82 5.70 5.60 5.52 5.46 5.37 5.27 5.17 5.12 5.07 5.01 4.96 4.90 4.85 8.07 6.54 5.89 5.52 5.29 5.12 4.99 4.90 4.82 4.76 4.67 4.57 4.47 4.42 4.36 4.31 4.25 4.20 4.14 7.57 6.06 5.42 5.05 4.82 4.65 4.53 4.43 4.36 4.30 4.20 4.10 4.00 3.95 3.89 3.84 3.78 3.73 3.67 7.21 5.71 5.08 4.72 4.48 4.32 4.20 4.10 4.03 3.96 3.87 3.77 3.67 3.61 3.56 3.51 3.45 3.39 3.33 6.94 5.46 4.83 4.47 4.24 4.07 3.95 3.85 3.78 3.72 3.62 3.52 3.42 3.37 3.31 3.26 3.20 3.14 3.08 6.72 5.26 4.63 4.28 4.04 3.88 3.76 3.66 3.59 3.53 3.43 3.33 3.23 3.17 3.12 3.06 3.00 2.94 2.88 6.55 5.10 4.47 4.12 3.89 3.73 3.61 3.51 3.44 3.37 3.28 3.18 3.07 3.02 2.96 2.91 2.85 2.79 2.72 6.41 4.97 4.35 4.00 3.77 3.60 3.48 3.39 3.31 3.25 3.15 3.05 2.95 2.89 2.84 2.78 2.72 2.66 2.60 6.30 4.86 4.24 3.89 3.66 3.50 3.38 3.29 3.21 3.15 3.05 2.95 2.84 2.79 2.73 2.67 2.61 2.55 2.49 6.20 4.77 4.15 3.80 3.58 3.41 3.29 3.20 3.12 3.06 2.96 2.86 2.76 2.70 2.64 2.59 2.52 2.46 2.40 6.12 4.69 4.08 3.73 3.50 3.34 3.22 3.12 3.05 2.99 2.89 2.79 2.68 2.63 2.57 2.51 2.45 2.38 2.32 6.04 4.62 4.01 3.66 3.44 3.28 3.16 3.06 2.98 2.92 2.82 2.72 2.62 2.56 2.50 2.44 2.38 2.32 2.25 5.98 4.56 3.95 3.61 3.38 3.22 3.10 3.01 2.93 2.87 2.77 2.67 2.56 2.50 2.44 2.38 2.32 2.26 2.19 5.92 4.51 3.90 3.56 3.33 3.17 3.05 2.96 2.88 2.82 2.72 2.62 2.51 2.45 2.39 2.33 2.27 2.20 2.13 5.87 4.46 3.86 3.51 3.29 3.13 3.01 2.91 2.84 2.77 2.68 2.57 2.46 2.41 2.35 2.29 2.22 2.16 2.09 5.83 4.42 3.82 3.48 3.25 3.09 2.97 2.87 2.80 2.73 2.64 2.53 2.42 2.37 2.31 2.25 2.18 2.11 2.04 5.79 4.38 3.78 3.44 3.22 3.05 2.93 2.84 2.76 2.70 2.60 2.50 2.39 2.33 2.27 2.21 2.14 2.08 2.00 5.75 4.35 3.75 3.41 3.18 3.02 2.90 2.81 2.73 2.67 2.57 2.47 2.36 2.30 2.24 2.18 2.11 2.04 1.97 5.72 4.32 3.72 3.38 3.15 2.99 2.87 2.78 2.70 2.64 2.54 2.44 2.33 2.27 2.21 2.15 2.08 2.01 1.94 5.69 4.29 3.69 3.35 3.13 2.97 2.85 2.75 2.68 2.61 2.51 2.41 2.30 2.24 2.18 2.12 2.05 1.98 1.91 5.66 4.27 3.67 3.33 3.10 2.94 2.82 2.73 2.65 2.59 2.49 2.39 2.28 2.22 2.16 2.09 2.03 1.95 1.88 5.63 4.24 3.65 3.31 3.08 2.92 2.80 2.71 2.63 2.57 2.47 2.36 2.25 2.19 2.13 2.03 2.00 1.93 1.85 5.61 4.22 3.63 3.29 3.06 2.90 2.78 2.69 2.61 2.55 2.45 2.34 2.23 2.17 2.11 2.05 1.98 1.91 1.83 5.59 4.20 3.61 3.27 3.04 2.88 2.76 2.67 2.59 2.53 2.43 2.32 2.21 2.15 2.09 2.03 1.96 1.89 1.81 5.57 4.18 3.59 3.25 3.03 2.87 2.75 2.65 2.57 2.51 2.41 2.31 2.20 2.14 2.07 2.01 1.94 1.87 1.79 5.42 4.05 3.46 3.13 2.90 2.74 2.62 2.53 2.45 2.39 2.29 2.18 2.07 2.01 1.94 1.88 1.80 1.72 1.64 5.29 3.93 3.34 3.01 2.79 2.63 2.51 2.41 2.33 2.27 2.17 2.06 1.94 1.88 1.82 1.74 1.67 1.58 1.48 5.15 3.80 3.23 2.89 2.67 2.52 2.39 2.30 2.22 2.16 2.05 1.94 1.82 1.76 1.69 1.64 1.53 1.43 1.31 5.02 3.69 3.12 2.79 2.57 2.41 2.29 2.19 2.11 2.05 1.94 1.83 1.71 1.64 1.57 1.48 1.39 1.24 1.00

s21 S1 . S2 , where s21 ¼ S1 =m and s22 ¼ S2 =n are independent mean squares estimating a common variance 2 ¼ m n s22 and based on m and n degrees of freedom, respectively. Z F mþn m2 nmm=2 nn=2 xm=21 ðn þ mxÞðmþnÞ=2 dx ¼ :99 FðFÞ ¼ 0 2 2 F¼

n

m

1

2

3

4

5

6

7

8

9

10

12

15

20

24

30

40

60

120

1

1 4052 4999.5 5403 5625 5764 5859 5928 5982 6022 6056 6106 6157 6209 6235 6261 6287 6313 6339 6366 2 98.50 99.00 99.17 99.25 99.30 99.33 99.36 99.37 99.39 99.40 99.42 99.43 99.45 99.46 99.47 99.47 99.48 99.49 99.50 3 34.12 30.82 29.46 28.71 28.24 27.91 27.67 27.49 27.35 27.23 27.05 26.87 26.69 26.60 26.50 26.41 26.32 26.22 26.13 4 21.20 18.00 16.69 15.98 15.52 15.21 14.98 14.80 14.66 14.55 14.37 14.20 14.02 13.93 13.84 13.75 13.65 13.56 13.46 5 16.26 13.27 12.06 11.39 10.97 10.67 10.46 10.29 10.16 10.05 9.89 9.72 9.55 9.47 9.38 9.29 9.20 9.11 9.02 6 13.75 10.92 9.78 9.15 8.75 8.47 8.26 8.10 7.98 7.87 7.72 7.56 7.40 7.31 7.23 7.14 7.06 6.97 6.88 7 12.25 9.55 8.45 7.85 7.46 7.19 6.99 6.84 6.72 6.62 6.47 6.31 6.16 6.07 5.99 5.91 5.82 5.74 5.65 8 11.26 8.65 7.59 7.01 6.63 6.37 6.18 6.03 5.91 5.81 5.67 5.52 5.36 5.28 5.20 5.12 5.03 4.95 4.86 9 10.56 8.02 6.99 6.42 6.06 5.80 5.61 5.47 5.35 5.26 5.11 4.96 4.81 4.73 4.65 4.57 4.48 4.40 4.31 10 10.04 7.56 6.55 5.99 5.64 5.39 5.20 5.06 4.94 4.85 4.71 4.56 4.41 4.33 4.25 4.17 4.08 4.00 3.91 11 9.65 7.21 6.22 5.67 5.32 5.07 4.89 4.74 4.63 4.54 4.40 4.25 4.10 4.02 3.94 3.86 3.78 3.69 3.60 12 9.33 6.93 5.95 5.41 5.06 4.82 4.64 4.50 4.39 4.30 4.16 4.01 3.86 3.78 3.70 3.62 3.54 3.45 3.36 13 9.07 6.70 5.74 5.21 4.86 4.62 4.44 4.30 4.19 4.10 3.96 3.82 3.66 3.59 3.51 3.43 3.34 3.25 3.17 14 8.86 6.51 5.56 5.04 4.69 4.46 4.28 4.14 4.03 3.94 3.80 3.66 3.51 3.43 3.35 3.27 3.18 3.09 3.00 15 8.68 6.36 5.42 4.89 4.56 4.32 4.14 4.00 3.89 3.80 3.67 3.52 3.37 3.29 3.21 3.13 3.05 2.96 2.87 16 8.53 6.23 5.29 4.77 4.44 4.20 4.03 3.89 3.78 3.69 3.55 3.41 3.26 3.18 3.10 3.02 2.93 2.84 2.75 17 8.40 6.11 5.18 4.67 4.34 4.10 3.93 3.79 3.68 3.59 3.46 3.31 3.16 3.08 3.00 2.92 2.83 2.75 2.65 18 8.29 6.01 5.09 4.58 4.25 4.01 3.84 3.71 3.60 3.51 3.37 3.23 3.08 3.00 2.92 2.84 2.75 2.66 2.57 19 8.18 5.93 5.01 4.50 4.17 3.94 3.77 3.63 3.52 3.43 3.30 3.15 3.00 2.92 2.84 2.76 2.67 2.58 2.49 20 8.10 5.85 4.94 4.43 4.10 3.87 3.70 3.56 3.46 3.37 3.23 3.09 2.94 2.86 2.78 2.69 2.61 2.52 2.42 21 8.02 5.78 4.87 4.37 4.04 3.81 3.64 3.51 3.40 3.31 3.17 3.03 2.88 2.80 2.72 2.64 2.55 2.46 2.36 22 7.95 5.72 4.82 4.31 3.99 3.76 3.59 3.45 3.35 3.26 3.12 2.98 2.83 2.75 2.67 2.58 2.50 2.40 2.31 23 7.88 5.66 4.76 4.26 3.94 3.71 3.54 3.41 3.30 3.21 3.07 2.93 2.78 2.70 2.62 2.54 2.45 2.35 2.26 24 7.82 5.61 4.72 4.22 3.90 3.67 3.50 3.36 3.26 3.17 3.03 2.89 2.74 2.66 2.58 2.49 2.40 2.31 2.21 25 7.77 5.57 4.68 4.18 3.85 3.63 3.46 3.32 3.22 3.13 2.99 2.85 2.70 2.62 2.54 2.45 2.36 2.27 2.17 26 7.72 5.53 4.64 4.14 3.82 3.59 3.42 3.29 3.18 3.09 2.96 2.81 2.66 2.58 2.50 2.42 2.33 2.23 2.13 27 7.68 5.49 4.60 4.11 3.78 3.56 3.39 3.26 3.15 3.06 2.93 2.78 2.63 2.55 2.47 2.38 2.29 2.20 2.10 28 7.64 5.45 4.57 4.07 3.75 3.53 3.36 3.23 3.12 3.03 2.90 2.75 2.60 2.52 2.44 2.35 2.26 2.17 2.06 29 7.60 5.42 4.54 4.04 3.73 3.50 3.33 3.20 3.09 3.00 2.87 2.73 2.57 2.49 2.41 2.33 2.23 2.14 2.03 30 7.56 5.39 4.51 4.02 3.70 3.47 3.30 3.17 3.07 2.98 2.84 2.70 2.55 2.47 2.39 2.30 2.21 2.11 2.01 40 7.31 5.18 4.31 3.83 3.51 3.29 3.12 2.99 2.89 2.80 2.66 2.52 2.37 2.29 2.20 2.11 2.02 1.92 1.80 60 7.08 4.98 4.13 3.65 3.34 3.12 2.95 2.82 2.72 2.63 2.50 2.35 2.20 2.12 2.03 1.94 1.84 1.73 1.60 120 6.85 4.79 3.95 3.48 3.17 2.96 2.79 2.66 2.56 2.47 2.34 2.19 2.03 1.95 1.86 1.76 1.66 1.53 1.38 1 6.63 4.61 3.78 3.32 3.02 2.80 2.64 2.51 2.41 2.32 2.18 2.04 1.88 1.79 1.70 1.59 1.47 1.32 1.00

s21 S1 . S2 , where s21 ¼ S1 =m and s22 ¼ S2 =n are independent mean squares estimating a common variance 2 ¼ m n s22 and based on m and n degrees of freedom, respectively. F¼

A-108

Z

F

FðFÞ ¼ 0

n

m

1

2

3

4

5

6

m þ n m2 n mm=2 nn=2 xm=21 ðn þ mxÞðmþnÞ=2 dx ¼ :995 2 2 7

8

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1

1 16211 2000 21615 22500 23056 23437 23715 23925 24091 24224 24426 24630 24836 24940 25044 25148 25253 25359 25465 2 198.5 199.0 199.2 199.2 199.3 199.3 199.4 199.4 199.4 199.4 199.4 199.4 199.4 199.5 199.5 199.5 199.5 199.5 199.5 3 55.55 49.80 47.47 46.19 45.39 44.84 44.43 44.13 43.88 43.69 43.39 43.08 42.78 42.62 42.47 42.31 42.15 41.99 41.83 4 31.33 26.28 24.26 23.15 22.46 21.97 21.62 21.35 21.14 20.97 20.70 20.44 20.17 20.03 19.89 19.75 19.61 19.47 19.32 5 22.78 18.31 16.53 15.56 14.94 14.51 14.20 13.96 13.77 13.62 13.38 13.15 12.90 12.78 12.66 12.53 12.40 12.27 12.14 6 18.63 14.54 12.92 12.03 11.46 11.07 10.79 10.57 10.39 10.25 10.03 9.81 9.59 9.47 9.36 9.24 9.12 9.00 8.88 7 16.24 12.40 10.88 10.05 9.52 9.16 8.89 8.68 8.51 8.38 8.18 7.97 7.75 7.65 7.53 7.42 7.31 7.19 7.08 8 14.69 11.04 9.60 8.81 8.30 7.95 7.69 7.50 7.34 7.21 7.01 6.81 6.61 6.50 6.40 6.29 6.18 6.06 5.95 9 13.61 10.11 8.72 7.96 7.47 7.13 6.88 6.69 6.54 6.42 6.23 6.03 5.83 5.73 5.62 5.52 5.41 5.30 5.19 10 12.83 9.43 8.08 7.34 6.87 6.54 6.30 6.12 5.97 5.85 5.66 5.47 5.27 5.17 5.07 4.97 4.86 4.75 4.64 11 12.23 8.91 7.60 6.88 6.42 6.10 5.86 5.68 5.54 5.42 5.24 5.05 4.86 4.76 4.65 4.55 4.44 4.34 4.23 12 11.75 8.51 7.23 6.52 6.07 5.76 5.52 5.35 5.20 5.09 4.91 4.72 4.53 4.43 4.33 4.23 4.12 4.01 3.90 13 11.37 8.19 6.93 6.23 5.79 5.48 5.25 5.08 4.94 4.82 4.64 4.46 4.27 4.17 4.07 3.97 3.87 3.76 3.65 14 11.06 7.92 6.68 6.00 5.56 5.26 5.03 4.86 4.72 4.60 4.43 4.25 4.06 3.96 3.86 3.76 3.66 3.55 3.44 15 10.80 7.70 6.48 5.80 5.37 5.07 4.85 4.67 4.54 4.42 4.25 4.07 3.88 3.79 3.69 3.58 3.48 3.37 3.26 16 10.58 7.51 6.30 5.64 5.21 4.91 4.69 4.52 4.38 4.27 4.10 3.92 3.73 3.64 3.54 3.44 3.33 3.22 3.11 17 10.38 7.35 6.16 5.50 5.07 4.78 4.56 4.39 4.25 4.14 3.97 3.79 3.61 3.51 3.41 3.31 3.21 3.10 2.98 18 10.22 7.21 6.03 5.37 4.96 4.66 4.44 4.28 4.14 4.03 3.86 3.68 3.50 3.40 3.30 3.20 3.10 2.99 2.87 19 10.07 7.09 5.92 5.27 4.85 4.56 4.34 4.18 4.04 3.93 3.76 3.59 3.40 3.31 3.21 3.11 3.00 2.89 2.78 20 9.94 6.99 5.82 5.17 4.76 4.47 4.26 4.09 3.96 3.85 3.68 3.50 3.32 3.22 3.12 3.02 2.92 2.81 2.69 21 9.83 6.89 5.73 5.09 4.68 4.39 4.18 4.01 3.88 3.77 3.60 3.43 3.24 3.15 3.05 2.95 2.84 2.73 2.61 22 9.73 6.81 5.65 5.02 4.61 4.32 4.11 3.94 3.81 3.70 3.54 3.36 3.18 3.08 2.98 2.88 2.77 2.66 2.55 23 9.63 6.73 5.58 4.95 4.54 4.26 4.05 3.88 3.75 3.64 3.47 3.30 3.12 3.02 2.92 2.82 2.71 2.60 2.48 24 9.55 6.66 5.52 4.89 4.49 4.20 3.99 3.83 3.69 3.59 3.42 3.25 3.06 2.97 2.87 2.77 2.66 2.55 2.43 25 9.48 6.60 5.46 4.84 4.43 4.15 3.94 3.78 3.64 3.54 3.37 3.20 3.01 2.92 2.82 2.72 2.61 2.50 2.38 26 9.41 6.54 5.41 4.79 4.38 4.10 3.89 3.73 3.60 3.49 3.33 3.15 2.97 2.87 2.77 2.67 2.56 2.45 2.33 27 9.34 6.49 5.36 4.74 4.34 4.06 3.85 3.69 3.56 3.45 3.28 3.11 2.93 2.83 2.73 2.63 2.52 2.41 2.25 28 9.28 6.44 5.32 4.70 4.30 4.02 3.81 3.65 3.52 3.41 3.25 3.07 2.89 2.79 2.69 2.59 2.48 2.37 2.29 29 9.23 6.40 5.28 4.66 4.26 3.98 3.77 3.61 3.48 3.38 3.21 3.04 2.86 2.76 2.66 2.56 2.45 2.33 2.24 30 9.18 6.35 5.24 4.62 4.23 3.95 3.74 3.58 3.45 3.34 3.18 3.01 2.82 2.73 2.63 2.52 2.42 2.30 2.18 40 8.83 6.07 4.98 4.37 3.99 3.71 3.51 3.35 3.22 3.12 2.95 2.78 2.60 2.50 2.40 2.30 2.18 2.06 1.93 60 8.49 5.79 4.73 4.14 3.76 3.49 3.29 3.13 3.01 2.90 2.74 2.57 2.39 2.29 2.19 2.08 1.96 1.83 1.69 120 8.18 5.54 4.50 3.92 3.55 3.28 3.09 2.93 2.81 2.71 2.54 2.37 2.19 2.09 1.98 1.87 1.75 1.61 1.43 1 7.88 5.30 4.28 3.72 3.35 3.09 2.90 2.74 2.62 2.52 2.36 2.19 2.00 1.90 1.79 1.67 1.53 1.36 1.00

s21 S1 . S2 , where s21 ¼ S1 =m and s22 ¼ S2 =n are independent mean squares estimating a common variance 2 and based on m and n degrees of freedom, ¼ m n s22 respectively.

F¼

Z FðFÞ ¼ 0

n

m

1

2

3

4

5

6

F

m þ n m2 nmm=2 nn=2 xm=21 ðn þ mxÞðmþnÞ=2 dx ¼ :999 2 2 7

8

9

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12

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120

1

1 4053* 5000* 5404* 5625* 5764* 5859* 5929* 5981* 6023* 6056* 6107* 6158* 6209* 6235* 6261* 6287* 6313* 6340* 6366* 2 998.5 999.0 999.2 999.2 999.3 999.3 999.4 999.4 999.4 999.4 999.4 999.4 999.4 999.5 999.5 999.5 999.5 999.5 999.5 3 167.0 148.5 141.1 137.1 134.6 132.8 131.6 130.6 129.9 129.2 128.3 127.4 126.4 125.9 125.4 125.0 124.5 124.0 123.5 4 74.14 61.25 56.18 53.44 51.71 50.53 49.66 49.00 48.47 48.05 47.41 46.76 46.10 45.77 45.43 45.09 44.75 44.40 44.05 5 47.18 37.12 33.20 31.09 29.75 28.84 28.16 27.64 27.24 26.92 26.42 25.91 25.39 25.14 24.87 24.60 24.33 24.06 23.79 6 35.51 27.00 23.70 21.92 20.81 20.03 19.46 19.03 18.69 18.41 17.99 17.56 17.12 16.89 16.67 16.44 16.21 15.99 15.75 7 29.25 21.69 18.77 17.19 16.21 15.52 15.02 14.63 14.33 14.08 13.71 13.32 12.93 12.73 12.53 12.33 12.12 11.91 11.70 8 25.42 18.49 15.83 14.39 13.49 12.86 12.40 12.04 11.77 11.54 11.19 10.84 10.48 10.30 10.11 9.92 9.73 9.53 9.33 9 22.86 16.39 13.90 12.56 11.71 11.13 10.70 10.37 10.11 9.89 9.57 9.24 8.90 8.72 8.55 8.37 8.19 8.00 7.81 10 21.04 14.91 12.55 11.28 10.48 9.92 9.52 9.20 8.96 8.75 8.45 8.13 7.80 7.64 7.47 7.30 7.12 6.94 6.76 11 19.69 13.81 11.56 10.35 9.58 9.05 8.66 8.35 8.12 7.92 7.63 7.32 7.01 6.85 6.68 6.52 6.35 6.17 6.00 12 18.64 12.97 10.80 9.63 8.89 8.38 8.00 7.71 7.48 7.29 7.00 6.71 6.40 6.25 6.09 5.93 5.76 5.59 5.42 13 17.81 12.31 10.21 9.07 8.35 7.86 7.49 7.21 6.98 6.80 6.52 6.23 5.93 5.78 5.63 5.47 5.30 5.14 4.97 14 17.14 11.78 9.73 8.62 7.92 7.43 7.08 6.80 6.58 6.40 6.13 5.85 5.56 5.41 5.25 5.10 4.94 4.77 4.60 15 16.59 11.34 9.34 8.25 7.57 7.09 6.74 6.47 6.26 6.08 5.81 5.54 5.25 5.10 4.95 4.80 4.64 4.47 4.31 16 16.12 10.97 9.00 7.94 7.27 6.81 6.46 6.19 5.98 5.81 5.55 5.27 4.99 4.85 4.70 4.54 4.39 4.23 4.06 17 15.72 10.66 8.73 7.68 7.02 6.56 6.22 5.96 5.75 5.58 5.32 5.05 4.78 4.63 4.48 4.33 4.18 4.02 3.85 18 15.38 10.39 8.49 7.46 6.81 6.35 6.02 5.76 5.56 5.39 5.13 4.87 4.59 4.45 4.30 4.15 4.00 3.84 3.67 19 15.08 10.16 8.28 7.26 6.62 6.18 5.85 5.59 5.39 5.22 4.97 4.70 4.43 4.29 4.14 3.99 3.84 3.68 3.51 20 14.82 9.95 8.10 7.10 6.46 6.02 5.69 5.44 5.24 5.08 4.82 4.56 4.29 4.15 4.00 3.86 3.70 3.54 3.38 21 14.59 9.77 7.94 6.95 6.32 5.88 5.56 5.31 5.11 4.95 4.70 4.44 4.17 4.03 3.88 3.74 3.58 3.42 3.26 22 14.38 9.61 7.80 6.81 6.19 5.76 5.44 5.19 4.99 4.83 4.58 4.33 4.06 3.92 3.78 3.63 3.48 3.32 3.15 23 14.19 9.47 7.67 6.69 6.08 5.65 5.33 5.09 4.89 4.73 4.48 4.23 3.96 3.82 3.68 3.53 3.38 3.22 3.05 24 14.03 9.34 7.55 6.59 5.98 5.55 5.23 4.99 4.80 4.64 4.39 4.14 3.87 3.74 3.59 3.45 3.29 3.14 2.97 25 13.88 9.22 7.45 6.49 5.88 5.46 5.15 4.91 4.71 4.56 4.31 4.06 3.79 3.66 3.52 3.37 3.22 3.06 2.89 26 13.74 9.12 7.36 6.41 5.80 5.38 5.07 4.83 4.64 4.48 4.24 3.99 3.72 3.59 3.44 3.30 3.15 2.99 2.82 27 13.61 9.02 7.27 6.33 5.73 5.31 5.00 4.76 4.57 4.41 4.17 3.92 3.66 3.52 3.38 3.23 3.08 2.92 2.75 28 13.50 8.93 7.19 6.25 5.66 5.24 4.93 4.69 4.50 4.35 4.11 3.86 3.60 3.46 3.32 3.18 3.02 2.86 2.69 29 13.39 8.85 7.12 6.19 5.59 5.18 4.87 4.64 4.45 4.29 4.05 3.80 3.54 3.41 3.27 3.12 2.97 2.81 2.64 30 13.29 8.77 7.05 6.12 5.53 5.12 4.82 4.58 4.39 4.24 4.00 3.75 3.49 3.36 3.22 3.07 2.92 2.76 2.59 40 12.61 8.25 6.60 5.70 5.13 4.73 4.44 4.21 4.02 3.87 3.64 3.40 3.15 3.01 2.87 2.73 2.57 2.41 2.23 60 11.97 7.76 6.17 5.31 4.76 4.37 4.09 3.87 3.69 3.54 3.31 3.08 2.83 2.69 2.55 2.41 2.25 2.08 1.89 120 11.38 7.32 5.79 4.95 4.42 4.04 3.77 3.55 3.38 3.24 3.02 2.78 2.53 2.40 2.26 2.11 1.95 1.76 1.54 1 10.83 6.91 5.42 4.62 4.10 3.74 3.47 3.27 3.10 2.96 2.74 2.51 2.27 2.13 1.99 1.84 1.66 1.45 1.00

*Multiply these entries by 100.

A-109

SOURCES OF PHYSICAL AND CHEMICAL DATA In addition to the primary research journals, there are many useful sources of property data of the type contained in the CRC Handbook of Chemistry and Physics. A selected list of these is presented here, with emphasis on print and electronic sources whose contents have been subject to a reasonable level of quality control.

A. Data Journals 1. Journal of Physical and Chemical Reference Data – Published jointly by the National Institute of Standards and Technology and the American Institute of Physics, this quarterly journal contains compilations of evaluated data in chemistry, physics, and materials science. It is available in print and on the Internet. [ojps.aip.org/jpcrd/] 2. Journal of Chemical and Engineering Data – This bimonthly journal of the American Chemical Society publishes articles reporting original experimental measurements carried out under carefully controlled conditions. The main emphasis is on thermochemical and thermophysical properties. Review articles with evaluated data from the literature are also published. [pubs.acs.org/journals/jceaax/index.html] 3. Journal of Chemical Thermodynamics – This journal publishes original research papers that include highly accurate measurements of thermodynamic and thermophysical properties. [http://www.sciencedirect.com] 4. Atomic Data and Nuclear Data Tables – This is a bimonthly journal containing compilations of data in atomic physics, nuclear physics, and related fields. [www.sciencedirect.com] 5. Journal of Phase Equilibria – This journal presents critically evaluated phase diagrams and related data on alloy systems. It is published by ASM International and is the successor to the previous ASM periodical Bulletin Of Alloy Phase Diagrams. [www.asm-intl.org.] 6. Journal of Chemical Information and Computer Sciences – Although not a true data journal, it contains many papers on the prediction of physical property data from molecular structure. It is published by the American Chemical Society. [pubs.acs.org/journals/jcisd8/index.html]

B. Data Centers This section lists selected organizations that perform a continuing function of compiling and critically evaluating data in specific fields of science. 1. National Institute of Standards and Technology – Under its Standard Reference Data program, NIST supports a number of data centers in chemistry, physics, and materials science. Topics covered include thermodynamics, fluid properties, chemical kinetics, mass spectroscopy, atomic spectroscopy, fundamental physical constants, ceramics, and crystallography. Address: Office of Standard Reference Data, National Institute of Standards and Technology, Gaithersburg, MD 20899 [www.nist.gov/srd/]. 2. Thermodynamics Research Center – Now located at the National Institute of Standards and Technology, TRC maintains an extensive archive of data covering thermodynamic, thermochemical, and transport properties of organic compounds and mixtures. Data are distributed in both print and electronic form. Address: Mailcode 838.00, 325 Broadway, Boulder, CO 80305-3328 [www.trc.nist.gov] . 3. Design Institute for Physical Property Data – Under the auspices of the American Institute of Chemical Engineers [www.aiche.org/dippr/], DIPPR offers evaluated data on industrially-important chemical compounds. The largest project deals with physical, thermodynamic, and transport properties of pure compounds. Address: Brigham Young University, Provo, UT 84602 [dippr.byu.edu] . 4. Dortmund Data Bank – Maintains extensive databases on thermodynamic and transport properties of pure compounds and mixtures of industrial interest. The data are distributed through DECHEMA, FIZ CHEMIE, and other outlets. An abbreviated database system is also available for educational use. Address: DDBST GmbH, Industriestr. 1, 26121 Oldenburg, Germany [www.ddbst.de]. 5. Cambridge Crystallographic Data Centre – Maintains the Cambridge Structural Database of over 250,000 organic compounds. The data files and manipulation software are distributed in several ways. Address: 12 Union Rd., Cambridge CB2 1EZ, UK [www.ccdc.cam.ac.uk]. 6. FIZ Karlsruhe – In addition to many bibliographic databases, FIZ Karlsruhe maintains the Inorganic Crystal Structure Database in collaboration with the National Institute of Standards and Technology. The ICSD contains the atomic coordinates and related data on over 50,000 inorganic crystals. Address: Fachinformationszentrum (FIZ) Karlsruhe, Hermann-von-Helmholtz-Platz 1, D-76344 EggensteinLeopoldshafen, Germany [crystal.fiz-karlsruhe.de]. 7. International Centre for Diffraction Data – Maintains and distributes the Powder Diffraction File (PDF), a file of x-ray powder diffraction patterns used for identification of crystalline materials. The ICDD also distributes the NIST Crystal Data file, which contains lattice parameters for over 235,000 inorganic and organic crystalline materials. Address: 12 Campus Blvd., Newton Square, PA 19073-3273 [icdd.com]. 8. Research Collaboratory for Structural Bioinformatics – Maintains the Protein Data Bank (PDB), a file of 3-dimensional structures of proteins and other biological macromolecules. Address: Department of Chemistry and Chemical Biology, Rutgers University, 610 Taylor Road, Piscataway, NJ 08854-8087 [www.rcsb.org]. 9. Toth Information Systems – Maintains the Metals Crystallographic Data File (CRYSTMET). Address: 2045 Quincy Ave., Gloucester, ON, Canada K1J 6B2 [www.tothcanada.com]. 10. Atomic Mass Data Center – Collects and evaluates high-precision data on masses of individual isotopes and maintains a comprehensive database. Address: C.S.N.S.M (IN2P3-CNRS), Batiment 108, F-91405 Orsay Campus, France [csnwww.in2p3.fr/amdc/]. 11. Particle Data Group – International center for data of high-energy physics; maintains database of properties of fundamental particles, which is published in both print and electronic form. Address: MS 50-308, Lawrence Berkeley National Laboratory, Berkeley, CA 94720 [pdg.lbl.gov]. 12. National Nuclear Data Center – Maintains databases on nuclear structure and reactions, including neutron cross sections. The NNDC is the U. S. node in an international network of nuclear data centers. Address: Brookhaven National Laboratory, Upton, NY 11973-5000 [www.nndc.bnl.gov].

B-1

SOURCES OF PHYSICAL AND CHEMICAL DATA (continued) 13. International Union of Pure and Applied Chemistry – Address: PO Box 13757, Research Triangle Park, NC 27709-3757 [www.iupac.org]. IUPAC supports a number of long-term data projects, including these examples: a. Solubility Data Project – Carries out evaluation of all types of solubility data. The results are published in the Solubility Data Series, whose current outlet is the Journal of Physical and Chemical Reference Data. [www.unileoben.ac.at/~eschedor/] b. Kinetic Data for Atmospheric Chemistry – Maintains a comprehensive database on the kinetics of reactions important in the chemistry of the atmosphere. [www.iupac-kinetic.ch.cam.ac.uk/] c. International Thermodynamic Tables for the Fluid State – Prepares definitive tables of the thermodynamic properties of industrially important fluids. Thirteen volumes have been published by IUPAC. [http://www.iupac.org/publications/books/seriestitles/]

C. Major Multi-Volume Handbook Series 1. Chapman & Hall/CRC Chemical Dictionaries – These originally appeared in print form as the Dictionary of Organic Compounds, Dictionary of Natural Products, etc. They are now published in electronic form and are available in CDROM format [www.crcpress.com] and on the Internet [www.chemnetbase.com]. The consolidated version, called the Combined Chemical Dictionary, has data on more than 450,000 compounds spanning all branches of chemistry. The coverage includes physical properties, biological sources, hazard information, uses, and literature references. 2. Properties of Organic Compounds – Originally published in three editions as the Handbook of Data on Organic Compounds, it is now in electronic form as Properties of Organic Compounds. The database includes about 30,000 compounds; physical properties and spectral data (mass, infrared, Raman, ultraviolet, and NMR) are covered. It is offered as CDROM [www.crcpress.com] and web access [www.chemnetbase.com]. 3. Beilstein Handbook of Organic Chemistry – The classic source of data on organic compounds, dating from the 18th century, Beilstein was converted to electronic form in the last decade of the 20th century. Over 8 million compounds and 5 million chemical reactions are now covered, with a broad range of physical properties as well as synthetic methods and ecological data. The database is accessed by the CrossFire software [www.mdli.com]. 4. Gmelin Handbook of Inorganic and Organometallic Chemistry – A subset of the information in the print series has been converted to electronic form and is now distributed in the same manner as Beilstein. In addition to the standard physical properties, the coverage includes a wide range of optical, magnetic, spectroscopic, thermal, and transport properties for about 1.4 million compounds [www.mdli.com]. 5. DECHEMA Chemical Data Series – DECHEMA distributes the DTHERM database, which emphasizes data used in process design in the chemical industry, including thermodynamic and transport properties of about 20,000 pure compounds and 90,000 mixtures. Access is available through in-house databases and via the Internet. [www.dechema.de]. 6. Landolt-Börnstein Numerical Data and Functional Relationships in Science and Technology - Landolt-Börnstein covers a very broad range of data in physics, chemistry, crystallography, materials science, biophysics, astronomy, and geophysics. Hard-copy volumes in the New Series (started in 1961) are still being published, and the entire New Series is now accessible on the Internet [www.landolt-boernstein.com].

D. Selected Single-Volume Handbooks The following handbooks offer broad coverage of high-quality data in a single volume. This list is only representative; an extensive listing of handbooks in all fields of science may be found in Handbooks and Tables in Science and Technology, Third Edition (Russell H. Powell, ed., Oryx Press, Westport, CT, 1994). 1. American Institute of Physics Handbook – Although an old book, it contains much data that is still useful, especially in acoustics, mechanics, optics, and solid state physics. (Dwight E. Gray, ed., McGraw-Hill, New York, 1972) 2. Constants of Inorganic Substances - This book presents physical constants, thermodynamic data, solubility, reactivity, and other information on over 3000 inorganic compounds. Since it draws heavily on Russian literature, it contains a great deal of data that does not make its way into most U. S. handbooks. (R. A. Lidin, L. L. Andreeva, and V. A. Molochko, Begell House, New York, 1995) 3. Handbook of Chemistry and Physics – Now in the 84th Edition, the CRC Handbook covers data from most branches of chemistry and physics. The annual revisions permit regular updating of the information. Also available on CDROM [www.crcpress.com] and the web [hbcpnetbase.com]. (David R. Lide, ed., CRC Press, Boca Raton, FL, 2002) 4. Handbook of Inorganic Compounds – This book covers physical constants and solubility for about 3300 inorganic compounds. Also available on CDROM [www.crcpress.com]. (Dale L. Perry and Sidney L. Phillips, eds., CRC Press, Boca Raton, FL, 1995) 5. Handbook of Physical Properties of Liquids and Gases – This is a valuable source of data on all types of fluids, ranging from liquid and gaseous hydrocarbons to molten metals and ionized gases. Detailed tables of physical, thermodynamic, and transport properties are given for temperatures from the cryogenic region to 6000 K. Both Western and Russian literature is covered. (N. B. Vargaftik, Y. K. Vinogradov, and V. S. Yargin, Begell House, New York, 1996) 6. Handbook of Physical Quantities – The range of coverage is somewhat similar to the CRC Handbook of Chemistry and Physics, but with a stronger emphasis on physics than on chemistry. Solid state physics, lasers, nuclear physics, geophysics, and astronomy receive considerable attention. (Igor S. Grigoriev and Evgenii Z. Meilikhov, eds., CRC Press, Boca Raton, FL, 1997) 7. Kaye & Laby Tables of Physical and Chemical Constants – Kaye & Laby dates from 1911, and the 16th Edition was prepared in 1995 by a committee of experts. The coverage extends to almost every field of physics and chemistry; data on a limited number of representative substances or materials are given for each topic. (Longman Group Limited, Harlow, Essex, UK, 1995)

B-2

SOURCES OF PHYSICAL AND CHEMICAL DATA (continued) 8. Lange’s Handbook of Chemistry – Provides broad coverage of chemical data; last updated in 1998. Also available on the web [www.knovel.com]. (John A. Dean, ed., McGraw-Hill, New York, 1998) 9. Recommended Reference Materials for the Realization of Physicochemical Properties – This IUPAC book emphasizes highly accurate data on substances and materials that can be used as calibration standards. It covers physical, thermal, optical, and electrical properties. (K. N. Marsh, ed., Blackwell Scientific Publications, Oxford, 1987) 9. The Merck Index – Now in its 13th Edition (published in 2001), The Merck Index is a widely used source of data on over 10,000 compounds, chosen particularly for their importance in biology, medicine, and ecology. A short monograph on each compound gives information on the synthesis and uses as well as physical and toxicological properties. Also available on CDROM [www.camsoft.com]. (Maryadele J. O’Neil, ed., Merck & Co., Whitehouse Station, NJ, 2001)

E. Summary of Useful Web Sites for Physical and Chemical Properties Most of the web sites in the following list provide direct access to factual data on physical and chemical properties. However, the list also includes portals that link to different property databases or describe the procedure for gaining access to electronic sources of property data. There are also a few chemical directory sites, which are useful for obtaining formulas, synonyms, and registry numbers for substances of interest. Web Site

Address

Comments

Acronyms and Symbols Advanced Chemistry Development

www3.interscience.wiley.com/stasa/ www.acdlabs.com

Alloys Online Atomic Mass Data Center Beilstein Cambridge Structural Database Chapman & Hall/CRC Combined Chemical Dictionary Chemfinder Chemical Acronyms Database

alloys.asminternational.org csnwww.in2p3.fr/amdc/ www.mdli.com www.ccdc.cam.ac.uk www.chemnetbase.com/scripts/ ccdweb.exe www.chemfinder.com www.oscar.chem.indiana.edu/cfdocs/ libchem/acronyms/ acronymsearch.html chem.sis.nlm.nih.gov/chemidplus/ www.chemindustry.com/chemicals/ www.chemnetbase.com

ChemIDplus ChemIndustry CHEMnetBASE ChemWeb Databases Coblentz Infrared Spectra CODATA Home Page DECHEMA (DTHERM) DIPPR Pure Compound Database Dortmund Data Bank Enzyme Nomenclature Database FDM Reference Spectra Databases FIZ Chemie Berlin FIZ Karlsruhe - ICSD Fundamental Physical Constants Gmelin Handbook of Chemistry and Physics Hazardous Substances Data Bank IUPAC Home Page IUPAC Kinetics Data IUPAC Nomenclature Rules IUPAC Solubility Data Project Knovel.com

www.chemweb.com/databases/ www.galactic.com/coblentz/ www.codata.org www.dechema.de dippr.byu.edu www.ddbst.de www.expasy.ch/enzyme/ www.fdmspectra.com/ www.fiz-chemie.de crystal.fiz-karlsruhe.de physics.nist.gov/cuu/ www.mdli.com hbcpnetbase.com toxnet.nlm.nih.gov/cgi-bin/sis/ htmlgen?HSDB www.iupac.org www.iupac-kinetic.ch.cam.ac.uk/ www.chem.qmw.ac.uk/iupac/ www.unileoben.ac.at/~eschedor/ www.knovel.com

Landolt-Börnstein MatWeb

www.landolt-boernstein.com www.matweb.com

Metals Crystallographic Data File NASA Chemical Kinetics Data

www.tothcanada.com jpldataeval.jpl.nasa.gov

B-3

Free servcie; useful for indentifying acronyms for chemicals Chemical directory, with programs for estimating physical and spectral properties Physical, electrical, thermal, and mechanical properties of alloys See B.10 See C.3 See B.5 See C.1 Chemical directory, with links to several property databases Useful for associating chemical names and acronyms

Chemical directory Chemical directory Portal to C&H/CRC Chemical Dictionaries, Handbook of Chemistry and Physics, Properties of Organic Compounds, etc. Portal to many databases IR spectra on CDROM Thermodynamic key values and fundamental constants See C.5 See B.3 See B.4 IUBMB nomenclature for enzymes Infrared spectra Portal to DETHERM (C.5) and Dortmund Data Bank (B.4) See B.6 CODATA fundamental constants See C.4 Web version of CRC Handbook Physical and toxicological properties of chemicals of health or environmental importance See B.13 See B.13.b Useful site for organic and biochemical nomenclature See B.13.a Portal to Lange’s Handbook, Perry’s Chemical Engineers’ Handbook, etc. See C.6 Thermal, electrical, and mechanical properties of engineering materials See B.9 Kinetic and photochemical data for stratospheric modeling

SOURCES OF PHYSICAL AND CHEMICAL DATA (continued) Web Site National Center for Biotechnology Information National Nuclear Data Center National Toxicology Program NIST Atomic Spectra Database NIST Ceramics Webbook NIST Chemistry Webbook NIST Data Gateway NIST Physical Reference Data NLM Gateway Particle Data Group Polymers — A Property Database Powder Diffraction File Properties of Organic Compounds Protein Data Bank SpecInfo Spectra Online STN Easy STN Easy-Europe STN Easy-Japan Syracuse Research Corporation Table of Isotopes Thermodynamics Research Center TOXNET Wiley Interscience

Address www.ncbi.nlm.nih.gov

Comments Portal to GenBank and other sequence databases

www.nndc.bnl.gov ntp-server.niehs.nih.gov physics.nist.gov/cgi-bin/AtData/ main_asd www.ceramics.nist.gov/webbook/ webbook.htm webbook.nist.gov srdata.nist.gov/gateway/ physics.nist.gov/PhysRefData/

See B.12 Chemical health and safety data Energy levels, wavelengths, and transition probabilities of atoms and atomic ions See B.1

gateway.nlm.nih.gov/gw/Cmd pdg.lbl.gov www.polymersdatabase.com/ polymers/ icdd.com www.chemnetbase.com/scripts/ pocweb.exe www.rcsb.org www.chemicalconcepts.com spectra.galactic.com/SpectraOnline/ stneasy.cas.org stneasy.fiz-karlsruhe.de stneasy-japan.cas.org esc.syrres.com/interkow/database.htm ie.lbl.gov/education/isotopes.htm www.trc.nist.gov toxnet.nlm.nih.gov www3.interscience.wiley.com/ reference.html

B-4

Broad range of physical, thermal, and spectral properties Portal to all NIST data systems; see B.1 Atomic and molecular spectra, cross sections, x-ray attenuation, and dosimetry data Portal to all National Library of Medicine databases See B.11 Properties of commercial polymers See B.7 See C.2 See B.8 IR, NMR, and mass spectra IR, UV, NMR, Raman, and mass spectra (unreviewed) Chemical directory (and access to Chemical Abstracts)

Properties of environmental interest Nuclear energy levels, moments, and other properties See B.2 Portal to HSDB and other databases on hazardous chemicals Portal to Kirk-Othmer Encyclopedia of Chemical Technology, Ullmann’s Encyclopedia of Industrial Chemistry, Encyclopedia of Reagents for Organic Synthesis, etc.

CRC Handbook of chemistry and physics

CRC Handbook of Chemistry and Physics

CRC Handbook Chemistry and Physics, 85th Edition