SPECIAL FUNCTJO;\,S OF MATHE:\IATICAL PHYSICS AND CII:EMISTRY
M:\'I'III~;MI\TICJ\L
UNIVElls!'ry
'I'I;:XTS
AT.l~X,\~Dlm
DANIEl.
C. AITKEN, D.Se" F.Il.S. Eo nU'I'JrEIlFOHD, D.Se" On. :'IATll.
D"TI!UMISANTII ANn i\IATlIICI!S STATllIT/CAI. :'IATlllmATICS
W,Wt;8 EU:CTIIICITY 1'/t().JI!C'1·/V'-: GI':O)lHT/lY
1:o."Tl'.G/lAT'ON 1'.\/11'110/. DII"'P./I/!STI.\TIOS
ISFlslTl.: SIWII!S,
Prof. A. C. Aitken, n,Se" 1'.11.5, I'rof. A. C. Aitken, D.Se.• F.rI.S, Prof. C, A, Coulson, D.Se., F.n.S, Prof. C. A, COllison, D.Se., F.II.S. 'I'. K Faulkner, Ph.D, Il. P. Gillcspie, Ph.D. n. P. Gillespie, Ph,D. ProF. ,1. :'1. IlyHloll, IJ.Se.
1:o."TEGlIolTIO:S 01' OllDINAII\' DlI'I'l!.H1!:o.'·IM. ~UAT/OSS ISTllOllUCTIOS
E. L. Ince, D.Se. '1"0 "IlI! 'I'llI!OILY 0/' I'IS/TI! Gnoups
\\'. Letlcr"mllll,
Ph.D .•
D.Se,
ASAL\'T/C'\L CI!OMlITRY 01' TURI!I! DUlr.SSIOSS
ProF. W. I-I. M'eren, Ph.D., F.Il.S. FUSCTIOSS 01' A emll'l.!!X VAR/AIII.I-:
D. E.
CUSS/CAl. :\II!ClIASICS • V"CTOll MCTIiOIll!
•
Vor.u~,,: AS/) h'TIWIIAI.
E. G.
Phillips,
I\I.A., M.Se.
nutherrortl, D.Se., Dr. :'lalh.
D. Eo Illllherrortl, D.Se., Dr. MUlh. .pror. W, W. Ilogosillski, Ph.D,
S/'/!CIM, FUSC'flOSS Ot· :.IATllE)IATICAI. I'U\'SICS .\SI) CllI!)IlSTlLY Prof. I. N. Sneddon, :'1.:\., n,Se,
ll"rry Spain, B.A., .\I.Sc., I'h.D. l'roF. II. W. Turnuull, F.ltS.
Tlmony 01' EQUATIOSS
hi
f~tqHlrlllion
TJIlwny 01" OUmNAUY DIl'I'ImENTIAI. EQU.... I'IO.'1S J. C. Butkill, Se.D.,
F.RS.
GEluI,\s·Escl.ISII :'IATIll!~IATICAI. VOCA .. UI .... IlY
S. Macintyre, M.A., I'h.D. TOl'OI.OO\·
•
K
,\1. PUllcrson, Ph.D.
SPECIAL FUNCTIONS OF MATHEMATICAL PHYSICS AND CHEMISTRY fly
IAN N. SNEDDON .'1.:\., D.Se. 1'f10FF.$son 01' MATUf:!ltATICS IN
TilE USl\'RHSITY COU.EG!: 0.' !'\ORTII STArrOROSlllRI:
OLIVER AND nOYD EDINllUUCIi AND LONDON NEW YOIlK: INTERSCIENCE PUlll.lSIIEIlS, INC.
1950
FIRST EOiTlOS
tu:;o
",UNTIP III 1l0LLA10.I
When /I is 11 ncgntivc fraction F(II) is defined by means of equation (ii); for example
By mellllS of the result (ix) we can derive all ill\.crcsting expression for I!:ulcr's constnnt, y, which is defined by the equation y = lim (1 n_oo
+ ~ + ... + -.:. -
log 'II) = 0.5772
(5.3)
/I
From (ix) we have
~{lo~r(=+ l)}= '"
liz
lim (100'11- - ' - _ - ' - - ... _ -'-) .::+1 z+2 :::+11
n ...., . , . .
so that letting z --'10- 0 we obtain the result (.iA)
lLnd from (5.1) we find
y= -
f:
e- l logfdl.
(5.5)
12
THE SPECIAL FUNCTIONS OF PHYSICS AND CHEMISTRY
§S
Integrating by parts we see that
f ,oc
f
log tllt = log::
+ for. -dl , t
so that - Y = lim
.~
(f• - t
""/:-I
dt
+ log z . )
(5.6)
Closely related to the glllumn fUllction are the cxpOllcntial-integrnl ei(x) defined by the equation ei(x) =
f
O'-" - d.. '"
(x> 0),
(5,7)
II·
and thc lognrillllllic-integrlli li(x) dcfined by li(X)=f'l
dU
o og
•
(5.8)
I~
Si(.r) Ci(x)
-1'-
-1
- 2 "--=~,,--,--~:-::c'-:-----:-~~----' Fig. t Variation of Ci(~1 nnd Si(z) with ~,
which are themselvcs connected by the relation ci(z) = - li(c"').
(5.9)
Other integrnls of importance arc the sine and cosine integ-nils Ci(x), Si(r), whieh arc uefincd by thc equations
INTRODUCTION
§ 5 ,a)
Ci{.v) = -
cos
Il
Si(x) =
J, --du, "
13
"sin I S,--,,, " U
(5.10)
and whose variation with J: is shown in Fig. 1. In heat. conduction problems solutions CUll often be expressed in terms of the error-function
" S'0 c-~Idrl,
crf(x) = .~
vn
whose vnrinl:ioll with
It
(5.11)
is exhibited grnphically in Fig. 2'·
'-0 ,--~-~--===-----, 0·8
+0-.
iI
(H
0·2
o
3·0
I·(l -T_
Fig. 2
Vurintion of erf(z) with x.
Sirnilndy in problems of wave Illotion the F,'csncl intc~rnls
• A. C. Aitken, SlalistiMI :'f(J/J,tntatiu, (Olivet.f1. Royd, Seventh Edition, In;,:!) p. G:! gi\'ClI n shott tallie of \'nluClI of erf (at).
14
THE SPECIAL FUNCTIONS OF PHYSICS AND CHEMISTRY
C(x)
=
s:
cos
(1 1lU2 )dll,
8(x)
=
J:
sin (!nIl 2)du
§5
(5.12)
occur. The \"llriatioll of thesc functions with x is shown in Fig. 3. 0·8 r----r--~-~--~-__,
--C(x) S(,T)
0·0
o·,~
U·2
II
Fig. II
:l _.1'_
.,
VariaUon of the Fresnel integmls, C(x) and S(x}, with x.
The importallcc of these fUllctions lies in the fuet thot it is OftCII possible to express solut.ions of physical problcms in terms of thcm. Thc corrcsponding Ilumerical values can thcn bc obtained from works such ns E. Jahnke !lnd F. Emdc, 'FlIlIkt;ollclj{a/dll' ('I'cubllCI', ].eipzig, 1H:J3) in which they IIfC tabulated. EX.~:UPLES
I.
Show that, in Iillhericnl polrLr coordinates r, 0, rp defined by x _ r ~ill 0 eo, rp, y _ r sin 0 Sill If" : _ r cos 0,
LJlllbce's equutiOll become'
INTRODUCTION
(r' 'lIrloP) + ~illO _'_ ~ (Sin 0 i'l~') -I- _1_ i'l''f' _ 0 i'lr i'lO i'lO sin'O op' ' ~
"
and pro\'e that. it possessCli soilltioll~ (If the form f·t''""B(cos 0), where el}l) !llIlisfil:s the ordinary ,Iiffcrtlliial equation (I - II') d'e __ - 2)1 -fie
1, whilc if I x I = 1 thc series con\'crgcs IIUsolutely if y > ex p.• ) It is convcrgcnt when ,'I: = - I, providcd that y > ex (J - 1. H wc introduce the nolnl ion r(ex r) (ex). = 0:(0: 1) _. _ (0: r - I) = -r~ (6.2)
+
+
+
+
+
we Illny wrile Ihe series (0.1) in thc fOl'm x) ~ " (·),(PI, x' (6.3) ... 1(.)" ,.-or.y. tlie suffixes 2 and I denoting thnt thcrc lIrc t\\"o parameters of the typc IX and onc of t.hc typc y. We shall gCllcrnlise this concept at n Intcr stnge (§ 12 below) but it is advisable at this stnge to dcnot.e t.he 'ordinary' hypergcomet.ric function by thc symbol 2"\ instend of simply F, if wc nrc ~
,.'I (., p." y'
.) See J. !II. 11)'slop. JlI/illilt Suit!t, FiUh Edition, (Oli,-cr k lloyd, IflM) p. 50.
"
i
HYPERGEOMETRIC FUNCTIONS
6
19
to nvoid confusion Inter. From the definition (0.3) it is obviolls that
A significant properly of the hypcrgcolllciric series follows imlllediat.ely from the definition ((UI). We have It
(Ix ~
P( 'I Cl':.
Now ( {J > O. The results hotd if z is com· plex pro\'idcc1 L1mt we choose the brnnch of (I - xl)-:O in such /I WilY that (1 - .r/)- --J> I us t -)- 0 and !l1(y) > !l1(P) > o. The flrslllpplicnlioll of (7.1) is the dcrivlllion of the vallie of the hypcrgcomctric series with unit argument. Pulling It = 1 in(i.1) we have I tFt(a,{Jii'i 1 )= H(P,y
~
f'
fJ) oP-t)l-2.-"-lt"-ldt
B(jJ, y-.-p) H(P, y P)
Cl P > 0, (J> O. If we express the bela function in terms of gamma fUllctions we have Cnuss' Theorem
if}' -
,"';(., p; !'ow if
IX
_ rly) I'ly - • - Pl
y; 1 ) -
Fiy
.j I'(y
Pf
= - ". n negative integer, we have
r(y - . - Pl r(y P) ~
('I-Pl.,
(1.2)
22
THE SPECIAL FUNCTIONS OF PHYSICS AND CHEMISTRY
§7
so that equation (7.2) reduees to t
P1 (-
R. •• ". fI'
) _
i"
I
-
(y - Pl. (y) ..
whieh is known, in clemen wry mathematics. as Vandermonde's theorem. Again. if we put x = - 1 Ilnd IX = 1 {1- 'Y we have, from equation (i.l)
+
R. R _ 2F I(IX. 1'1/' a.
+1,._1)_ T(I + /1- 0) J'I _ - F({1)l'(1 a.) 0 I
').' ,., / t d.
/
=
If we write ~ /2 in this int.egrnl we see that its vnlue is tB(~P, 1 - a.). Using this rcsult lind the relation !r(!p)fF(fJ) = F(I tfJ)jI'(1 Jl) we huve Kummer's theorem
+
+
T(I +p-o)r(l+ IPI
,F,(o,P,P-O+I;-l)~r(l+p)/'(I+!P
or
.
(7.31
Further we can deduce from the formula (i.l) relations between hypergcometric serics of nrt:l.Illlent x llnd those of argument :rJ(z - I). Pulling T t in C(luation (7.1). and noting that
= ]-
{l- x(1 - T)}-:-1l 2 /i\(y-iX, y-p; Yi .'c). (7.0)
Tf we put x =
t
2P l(a:,
in equation (TA) we obtain the relation
P; Yi t) =
2" 2P I(c:r., y-fJ;
)Ii
-1).
'fhe series all the right hand side of this cquntion cun be clcri\'cd frolll equation (7.3) provided either that
y = y - {J -
r.t
+ 1,
Le.
fJ =
1-
iX,
or that y
~
_ -
(y -
P) + 1,
i.o. y
~
i(- + P+ 1).
We then obtain the formulne (1.1) (1.8)
8. The Hypcrgcomclric Equation. Tn certain problems it is possible to reduce the solution to lhnl of solving the second order linear differential equation rr-y tt(l- x) d:r:~
in which
0:,
+ {y -
(1
dy + a. + ,8l·'!:} dx -
a.,8y = O.
(8.1)
,8 nnd y nrc constnnts. For instancc, thc Schro-
24
THE SPECIAL FUNCTIONS OF PHYSICS AND CHEMISTRY § a
dinger equation for n symmetricnl-top molecule, whieh is of importnnce in the theory of molecular spectra,l) CUll, by simple transformations, be reduced to this typc. An cquation of this typc also !Irises in the study of the flow of compressible fluids. In nddit.iOll certain other differential equations (such IlS that occurring in ex. 1 of Chapl.er 1) which nrisc in the solution of boundary "nlllc problems ill mnt.hemnieal physics can, by It simple change of "ariable, be transformed to an equal.ioll of type (8.1). Indeed it enn be shown that any ordinary linear differential equat.ion of the second order whose only singular points arc regular singular poinL", onc of which may be the point at infinity, ellll be transformed to the form (8.1). For that reasoll it is desirable to investigale the nature of t.he solutions of cquntion, which is called the hyperQ,eoll1ctric cquation. We may write the hypergeometric efjuation in the form it~lI"
+ x(l + + It~+ .. .){y -
+ +
(ex {J 1 )x}y' -«{J:c(1 +a:+x~+ ... )y= 0, so that, in the notation of § a, we see that ncar x = 0 it
Po = y, '10 = 0, nnd the indieial equation is
(22+(y-l)(?=0
with roots (! = 0 and (? = 1 - y. Similarly, the equation CUll be put ill the form (.T-l
)'y"-(x-l)(y-.-P-l-(y+.+P+l )(.T-l >+ .. .)y'
+.P(x-l)(l-(x-I)+ .. .}y~O, with indieial cquation
e'+ (.+ P- yl"
~
0,
of whieh the roots arc (2 = 0, Q = y - « - (J. Finally in the notnlion of § .~ wc hnvc for large "lillie:." of x I) See, for eXlllllple, I.. PlIuling lind E. n. \\"il!KIn. Illlr(J{/ucIiOli 10 QUllfltllm 1l1fChcllliCJ, with AIIlllka/iolls 10 Chemislry, plcGr:\w-Ui11, New York, lOa,,), JlJl. 275-2~lU, (Iud ex. 10 ucla\\'.
§8
HYPERGEOMETRIC FUNCTIONS
a(x)~ (.-I-~-I-
2S
P(x)~"!,
1>,
find so the indieinl equation llppropriate 10 the point at infinity is e' - (a -I- P)q -I- ap ~ 0, with roots I'J., p. Thus the regular singular poinL~ of the hypergeometrie equation are:(i) x = 0 with exponenL,> 0, 1 - y. (ii) x = co with expollelll.s (I., p. (iii) .'1: = I with exponent.s 0, y - (.( - p. These facts ure exhibited syrnholielllly by denoting the most general solution of the hypCI'gcollldrie eq\lntioll by u scheme of the form
y=r!g
1-y
~P
~
(8.2)
y-.-p
The symbol on the right is called the Ricmallll-P-fIlIlCtion of the equation. We shall now consider the form of the solutiolls in the neighbourhood of the regular singlllnr points. (a) x = 0: Corresponding to the root (} = 0 we have a solution of the form
• y=:i.:c,x'. ,..,
Substituting this series into eqllHtion (8.1) we obtnin the relation
•
(l-x)1:c,r(r-l)x H
,-0
• • -I- {y- (a-l-p-I- I ),t'} 1: r,I',I,r- 1 - rJ.fJ 1: crx' =
r_U which is readily seen to he equivlIlenl 1.0
•
,_u
:£ {cr+l[r(r+ 1 )+(1'+ 1 )yJ - c,(r+rJ.)(I'+fJJ },t" = ~O
0,
U
26
THE SPECIAL FUNCTIONS OF PHYSICS AND CHEMISTRY
so that (r Cr+l = (r
+ a)(r..L.. P) + 1)(r T 1') cr'
§8
(8.3)
from which it follows that -
(al,(p)"
',- (I' I, r.I
(8 .•')
o·
It follows that thc solution which reduces to unity when x= 0 is y
~
1
+ yI!x aP + a(. + J IP(P + J) , + 1'(;,+1)21 x ... y
i.e.
2Jo'I(IX,
=
Pi
Yi x).
(8.5)
Similarly, if 1 - I' is not zero nor:l. positive nor negativc integcl', thc solution corresponuing to tllc !'Oot IJ = I - Y is y =
•
~
c,m1-1'+'
~"
where
•
(I - .'1:) L c,(r+ 1- y)(r - y)x'-Y
,-0
• + {y- (et.+,B + 1 ).c} ,..0 ~ c,(r+ 1 -
+ •
y):tr-y -et.{J L c,xl-y+' = 0, ,-0
which is equivalent. to
•
E,,{('+ 1 - YH'-Y)+Y('+ I-yllx'''''-
.-0
•
_ L c,{(r+l-y)(r-y)+{et.+P+ 1 )(r+ l-y)+IX{J}x'-Y+!=O, ~O
implying that Cr+l=
(, + a - Y + 1)(' + (r+l)(r+2
Py)
y + 1)
c,.
Comparing this relation with (8.3) and and taking Co = 1
27
HYPERGEOMETRIC FUNCTIONS
§8
we sec that this solution is xl-YzF1{a.-y+l, {J-i'+'!j 2-y;
xl·
Combining equations (8.5) and (8.6) we sec that the general solution valid in the neighbourhood of the origin is
y=A ZF\(a.,
Pi y; x)+B:rI-Y zlr't(a.-,+l. j1-y+ 1; 2~y; xl, (8.71
provided that 1 - Y is not zero or n positive integer. If y = I, the solutions (8.5) and (8.6) arc identical. If we write
und put YA:c) = Yl(X) log x
+,-,1:• crx.
we find on substituting in (8.1), with y = I, tlillt
1,+1)',
.-+1
-'I
a.
+P+ 1)'r + I·I,IPI,I·P-·-P-'I rl{r+ 1)1
0
from which the cocfficicnL1..... (!" nre constants. Furthermore it is rendily shown that this eljlllltion is satisfied by the series ~ (ot')n(Cl,,ln'" (Cl,,+-1)n. x .. .._0 (Ql )n(!.'2)n .. ,(Q,,)n /I!'
(12.3)
which is. itself. n genernlisfllioll of tile series «(j,]). Such ~enerallsed hypergeomelrlc series nnd is denoted by the symbol Hoi P.(Clt...., Cl...... ; QI' .. '. Q.), It is left as an cxereise to the render 10 show that, if no two of the numbers 1. QI' (!~ •. , '. (!. dirrer by nn integer (or zero) that the other lJ linearly independent solutions of equation (12,~) nrc a series is culled a
xl-f, rilF.(1
+ Cl
i -
Q, .. '. 1
+ IXril -
l+t?I-Q~."', I+(!.-(!/;:t).
!?/; 2- (l/.
(i=1,2, .... II).
As it stands (12.3) is a gCllcmlisllljoll or thc scries (6,1)
§12
HYPERGEOMETRIC FUNCTIONS
37
but it is not sufficiently wide to cover a simple series of the type (11.03). To cover such cases we generalise, not the differential equation, but the series defining the function. The gelleralisation of (6.1) \\'hich indmlcs (12.:J) is the series
I: n..(l
(1(1 +. fi y)r(! + .)F(' + j. + p)
_
T(·)F(fi)f'(y)
P
- r(l+o::)rP+0::
y)~
l' [
fi.
y;
1 l+!«.
'J '
This ~Ji'l series with unit argument eun be summed by Gauss' formula (7.2) nlld the expression for S fOllnel. It the follows that 3
'J
F' [.. fi. y; ~ 1+o::-fJ, 1+O::-Yi _ r( IH.)r( l+jx-fi-y)r(! +.-fi)r(l+.-y) - r(l+.)r(l+. fi y)r(l+j. fi)r(l+j. y)
a result which is known as Dixon's theorem.
EXAMPLES
1.
Sholl' that (I) (ii)
.""(a, {J; (J; ::) ,{'''(la,
~a
+ !;
(I - ::)-"; j; ::) - H(I
I
:)-0:
(iii)
tF,(j',{:J.,
P+ I: y; z):
01,1-"(11+ I: jl': y;:r) - (i'- Ih"',(OI. (I: y-l: z) - (01+ t -i'),P,(.:r., (I; i': ,f);
(I -Z),f',{II.
p: i': z) -,"',(ot-I. (I-I: i'; 2') -
(\'1)
p: y: z);
(I-P"'"
,
,
a.+I1-y-1
z,F,(ot. (I: y+ I: z):
---,F,(II,p:y~l;z)
- ,F,{ot-I. P-I: y: 2') -,f',(:z, /1-1: y; z);
fJ: y: z) (II-y);r • - ,P,(II,p-l:y:;r) + ._- ,",(11.
(I-Z),1-',(II.
,
(i)
,P,(II., 11+ I: y+ I::) -
p: i'+I: ...);
,,,',(11.. p: y: :) II(Y-P)
•
- i'-(--I y+ 1 :,",(a. + 1, P+ I:
y+~; :),
"
HYPERGEOMETft.IC FUNCTIONS
,"'1(11. P: y;:)
(ii)
• 11-/1-1- I - ,".(II-1-I,p-I;i';:) -I- ---:,F.(lI+l;fJ;y+l;:). y Deduce n simple el:jlrCSliion for lhe hypc:rgoomelric Jeries ,~\(lI. (J; fJ - I: :). 5.
H n ill n JlO!lilh'c integer.
,P,( -II, lI-I-'I; y; z) _
pro~oe
Oml
~I-)O'(I_~)r- .1". rCliI."" IX.: o LPI' ' ,.. P.,
birJ
(,-":1',,-1
pl')(1)- '1,F(r+~)
"
- -
r(j)
1'(1 + n+ r)
r(l
+ 2,)("
,)1
From the duplication formula for lhe gamma function
2-r(, rU)r(l
+ !) = I 1 + :!r) rl2' =(1-),2"
and from example 9(iii) of Chapter I, r(1+11+r) 1'(011
r)l
=(-lnll+ 1 ),(-n),.
so that /-.frI(I) = (_ly(Il+ 1M-II),. .. (1),2r
(15.1)
Now, b)' Taylor's theorem P ,,(It) =
t
(J-l -I J Y p~I(1) .
.-0
'
Substituting the exprcssion (15.1) in this expansion wc obtain thc relation
P .. (p.)=~(-1I),(ll-:--1),(1 ~
(l).r.
... ~' _ -J
whieh gh'cs Murphy's formula (15.2)
for lhe Lcgcndre polynomilll P,,{.II). If 1I0W we put ox = I ill c."nrnple 5 of Clmpter I r wc scc that equation (15.2) is equivalcnt to I
tI"
P (p.)= I','_l)", .. 2"111 lip" v
(15.S)
LEGENDRE FUNCTIONS
§15
55
which is Rod&i~ues' formula for the Legendre polynomial. Rodrigues' formula is of great lise in the evaluation of definite integrals involving Legendre Consider, for instance, the integral
polynomials.
(15.4) Dy Rodrigues' formuln we lllay write this integral ns I ""I _ II
f1
dn j(x) /,,(£1_ 1 )"lIx, -I (X
and nn integration by parL I and 11/. is It positive integer then l
fl"'Qn(fl>-tf
r"P"~)d~=!fl fl,m_~'''Pn{~)d~
It -1 It - ~ unci the integral on the right is equivalent to the finite sum _I
"I-I
tE
....0
I
ftm-I-rf ;rp,Mld;
(18.2)
_I
Tfll~:S;;:'/I
it follows from equation (15.0) that eaeh term of this series vfinishes so that we have 118.3)
provi
(C:-I)"
c (1;-1' )"+"+1
2=ti
d~
(22.2)
0 we may lake the contour C to be the circle
IC-" 1- I vIi"~ - 1II· Integrating round this contour we obtl\in from equnlion (22.2) the equlltioll 1 f2.~
"
_7t
0
cos
VI+·V(/I~-l)eos(tp-'P)}n.
Sill
=
(II
(IIltp)chp
( )pm() +III 1/1)1 COS sin IIlIp II 1'1
from whieh follows immediately the Fourier expansion
(p
+ V(P' -1) 00' (9'- 9'))" 111 )ll~:"{JI)cosm('P-9')' ",_1 '1+'" .
= P,,{JI)+2i (
(22.3)
I) Eo G. I'hillil)S, "'wlt/ioM of a Compute Variable, (Oliver,," lloyd, HHO,) p. 0:1.
80
THE SPECIAL FUNCTIONS OF PHYSICS AND CHEMISTRY § 22
Chunging 11 to - (II VI'
+ 1) we
obtnin the expansion
+ ·V(/I''!. -
1) cos .,,}-"-I " (11- /Il)1 = P .(p') 2 1: (-I)'" I ' P':(p') cos m~. (22.4)
+
_I
II.
Appl)1ing Purse,·nl's theorem for Fourier scries to thc series (22.3) and ('!:!A) wc find that the scries ~ (II-m)!... '" P.{fl) P .
0"
"
Jo
2,.. .
sinOclOJ X" ... Y"..... drp=O
(23.5)
0"
for nil integral values of II, n', til Ilnd Ill' with '" s;: II, m' :s;; II'. DeclIlIse of these orthogonality rclnliollships we ciln cslnblish nn expansion theorem which is II slraightrorwnrd gcncrali7.ation of the Legendre series (16.3). It is rcndilr shown lhat for II huge class of fUllctions /. Ull~ fUllction 1(0, 'P) cun be represented by the series
•
• •
,,-0
,,_I ... _1
}".:c"P,,(cosO)+ E
}".:{x"",X", ..(O.rp)
+ V"",Y",,,,(O,rp)} (23.6)
where the coecricicnL.. c", x"",. V" .. lire gi"clI by the expressions
.>//+ I J~'lip e" = ---.I:t
0
=
V" .. =
(23.i)
0
f,sin 0 dO J, X".",(O, p)f(O, rp)drp,
(23.8)
f"osin
(23.9)
.~
X" ...
f' /(0, rpjP ,,(cos 0) sin 0 dO, 2.~
OliO
J2.\-•• (0. rp)J(O, rp)tlrr· o·
THE SPECIAL FUNCTIONS OF PHYSICS AND CHEMISTRY § 23
82
For finy gi"en fUllclion/(O. p) the series (23.G) CAll therefore in principle be computed by a series of simple intcgmtions. 'I'he functions X .... and Y ..... which nre known ns surface spherical harmonics can be constnlcled ensil)' from the known e:'<prcssions for the nssocintcd functiOllS T':.
We find, for instnnce, that , ..\ 1,1(0, (p) = -
:l I Car,)
sin 0 cos p.
15)1 sin 0 cos 0 cos p, .\':.1(0, If)= - ( 47z:
-"2,!(0,9') =
15)1 . ( -167t SlIl'OCOS'1.rp,
-"3.1(0,9') = -
G!2~)i sin 0 (5 cos! 0 -
X 3 ,,.(O, rp) =
(105)1 sin! 0 COS 0 cos 2tp. 16n:
X 3 3(0,9') •
= -
1) cos !P,
(~)I sin' 0 cos 3p. 32:r
lind the corresponding expressions for the Y II II' nre ob· tained by rcplncing cos (1119') by sin (IIUp) in Lhe c'xprcssions for the X" ",' The functions X" m nnd Y" '" have the important property that they arc solutions of t.he ]lartial differential equation 1
a (. 0 ax) 1 0' x ,~ an +~o-a--+II("+I).\'=O (:!:J.1O)
~oao sm
Sin
Slll~
p-
so that thc function
(A,"
+ Br"-
l
)X....(0.9')
+ (C,,, + IJr.. -1j Y ....(O, 9').
where A. Il, C and JJ are constants, is a solution of Laplaec's equation. It follows immediately from equations (23.1)(23.9) takcn with the cxpnnsioll (2:J.6) that thc fUllction
LEGENDRE FUNCTIONS
I"
. (')" +:E. . (')"
Ip(r, 0, rp) = E e,. ..-0
P,.(cos 0)
(I
:E -
tI
II_I ",_I
+
{.:r... .Y .,,,,(0. If) + Y..... Y .....(D.Ip)}
satisfies Lnplnces efIulllion ill the region 0 ~ T ~ a, is finite at r = 0, and takes the value /(0,9') on the sphere r = fl.
For example, suppose we wish to find the solution of Laplacc's equation which lnkc.'l 011 the value :c2 on the surface of the spltelc r = a. Here we ha\'c a~ ll~ /(0,11')= 112sin20cos 2tp ="3 -"3
=
(a COS'O-l) 2 + ~1I2sin20cos ':!.rp
,,2 (12 ('\.it\ I 3" -"3 /J2 (COS 0) + 15J ...'(2,2(0, lp)l/2.
Thus the required solution is "'(T, 0, m) = T 'r
2 2.-3 11 2 _ 2.. m) 3 r P,(cos 0) + (2.)1 1:; r2X....(0, 'r ~.M
Substituting the YlllllCS of P .. nnd X .. .. And Ir:lnsforming back to cartesian coordinates we sc·c· that the required solution is
24.
Usc of Associated
Functions in Wave of Ilssocintcd Legcndre functions in wnvc mechanics, wc shnll consider one of the simplest problems in thnt subject - that of soh-iug SchrOdinger's equntion ~ 8.,,2 m J7-IjI+~(IV- 1')'1'=0 (2·U) Lc~cndrc
l\.lcchnnlcs. To iIIustratc t.he
IlSC
for the rotntor with free nxis, thnt is for n pnrticle moving 011 t.he surfncc of u sphere_ Tn equution (2·U), \I' represents the totnl cncrgy of the syslem, V thc potential cucrg)'. In thc easc 'Indcr collsidcrnl iOIl V is Il l:ollslnnt, Vo
84
THE SPECIAL fUNCTIONS Of PHYSICS AND CHEMISTRY § 24
say, and tJle wavc function 'P will be II function of 0, 11 only. rr the radius of the sphere is dcnoted by a thcn equation (2'k 1) is of the fOl'llL 1
ClIp
cotO
a" ao"+ 7
alj1
1
I12p
ao + 112 sin 0 a,,: +
8:z 2m( 11'- Vo) II" '1'=0. (2.L2)
If we consider solutions of the form
'1'=
et'*''''''
thcn
e' 0 + 81t2ml1~( /"II' e" +'cot ,Substituting (:N.3)
I' = cos 0,
wc find lhnt this cqunlion reduces to J~cgclldrc's associntcd equation (21.1) and hence has solution
e=
II P:(cos 0)
+ BQ:'(cos 0).
Howevcr for the same reason as in the case of potential theory (§ 20 abo,'c) we must take n = O. The solulions of equation (2.L2) will thcrefore be made up of com· binations of solutions of thc form 'Pm ..(0, tp) = A",,, 1::Hm 9' 1'::' (cos 0). where A .... is a constant. Thc physical conditions imposed 011 the wave function 'P nrc that it should be single-valucd and continuous. Obviously thcn thc 'physical' solutions will havc III lIll inl.cgcr, sincc 'P... "(O.!p 2:t) must equnl !jJ... "(0, p). Further in order Uillt the series for P'::(/l) should COil verge for the vnlllC5/1 = ± I it is necessary that it should havc only a finitc number of terms. This is possiblc only if II is a positive integer. If therefore the solution (2.~..q is to be ,'alid for 0 = 0 nncl 0 =:z we must hn,'e " n positin'
+
"
LEGENDRE FUNCTIONS
integer. The phy!>ieul conditions on the wave function arc therefore not satisfi(.'(l by !>)'stellls with IIIl arbitrary value for lhe energy 11' hill only by systems for whieh
h' + -.-,11(/1 + 1). 8...-11I(1
11' = 1'0
(2.1.,5)
where 71 is a positive inleger. Tn other words, the energy of such II mechnnical system does not vary continuously, but is enpnble of nssuming valucs takcn from thc discrete set (2.1.5), 1':XA~II'LI':S
Show tlmt, if" u odd
1.
".to) _
O. ami that, if II (-I )i·
n'
i~
.:....'11.
2·{UIl)!l· 2,
I'ro\'c thut
I:
JI··'
.._en +1 I f ' _ I'
3.
(I)
(ii)
+ "lV"
(1 - "O-~(I
-
P_VI)_iIOg{I+JI} I-p
I) show thut
•
-lltcH - 1: I,·P.V')
.-,
r(II"'U
".VI) - IIlru)C·.F,ll.
-II;
i-no C-')
lkduco: tlmt (lU)
-4.
Ir"
...... U. -II;
i~
f hen~.
l
P .VI)( I
-
mnking \I$e or
I_,l I
-1'--
I) -
(II
+
il
" l).(.r)d..r
I
(ii)
2JU'
-I (I
I -
h'
Ifn'_1_21u:+',', Ilto\'cthllt iflhJ I)
~')
pro,"" thnt
2tl.f"
I
J
-I If".. f1)'>.(z)dl" -
19.
If
= it
real nnd
.• J
1=1
111.
Is n positive inlelter lind C_ J'
"I'·{cosO)liinOrIO 12.(;,) - COl 2CCOliO + t'
I
+ v{,,' -
I),
90
THE SPECIAL FUNCTIONS OF PHYSICS AND CHEMISTRY
Deduce thllt, if
It I >
Q.(P)-
I,
1: t-.. f" J>.(oosO) ain mOdO.
",_I
0
Uy cvahutiug thi, inteJ,:ral sho..' thnt
\1;01
Q,·,)_---C-·-I P(I .\1 r(lI+t) "" 26.
Pro\'c tlml, ir
III
is n po5lll\'C (nleltCr,
..
1: 1,·-" 7'''(/l) -
r1_'" 27.
"+I"'+t·C-') , .
•
Show IILut. if III I > I,
(_I)"'(2".)I(I_/l")i"'
2-ml(I_~/l"+lllj"'ti
f1> - I then
.. l'(n+m+ll (Pl-Ili-fl (I-P)·d; Q.(P)- 1'("+1) ~,- -I (P (')..... , Deduce thnt. if III
.. Q.(P)-
+ I I>
2 then
V:lno+m+11 vl"-I)i· ( , ) 2.·'r(Il+0 lfl+IJ•••• 11".. (1I+I,F1+m+I;211+2;JI+l
Find n llimille exprusion for Q:4'(p).
28.
Pro\'e the fol1owlnJ: recurrence relations for Ferrer'a t\nocinlcd Legcndre Functlons:-
(iiI (iii)
(0-11I+1)7':.. 01) -
(211+lj/l'l':01) + (11+11I)1':_1(,11) _ 0;
r,:"'(ll) - 1:::(p) _ (ll_m)(t_,,'ji T =(p).
De.ri'·e thc exprnsiol1ll for 1';"(0,1') nnd .\';"(0, '1') for m _I. 2, 3, ,~, ,ill 0 cos" 0 em 'I' in teTIllS of lurfllce Illherical hmnolilcs. 29.
Express the l'lInctions sin"O Iln" IF. lin' 0 01)51 fI"
30. Find the runction which interior or the III here z' y" IIlld tnk~ the \ouluc oz.rl fly'
5Rthrl~
l.nlll:lce·, equnlion In Ihe
+ + ;1 _ f1" remnirl!l finite lit thc origin + + y:" 011 the lurfnee of the .pheft'.
CIIAI"'1o:1I IV
BESSEL FUNCTIONS 25. The Origin of Bessel FUllc(ions. Bessel functions were first introduced by Bessel, ill 18::?l, in the discussion of n problem ill dynnmienl nstronomy. which mny be described ns follows. If P is n planet moving in nn ellipse whose foclls S in the sun and whose centre nnd mnjor axis nrc C nnd A'A respectively (cr. Fig. 8), then the angle
Q
c
S
A
Fig. 8
ASP is cnUcd thc trite ""omtl1y of the plnnet. It is found thnt. in nstronomical cnlculntions, the tme nnomnly is not n vcry convenient angle with which to denl. Inslend we use the IIICOIl ol/ollloly, C. whieh L~ defined to be 2n: times the mHo of the nrca of the elliptic seclor ASP to the IIren of t.he ellipse. Another nngle of significance is the eccentric llIlOlI/tlly, Il, of the pin net defined to be the nngle ACQ where Q is the point ill which the ordinate through p met:L~ the nuxilinry circle of I he ellipse. It is rendily shown l ) by n simple gcometricnl argument I) Cf: D. K HUlherfonl, C/lutiool.ltuhania, (Olin!r« Uo~'d, 1051 )f.1::l.
"
92
THE SPECIAL FUNCTIONS OF PHYSICS AND CHEMISTRY § 25
that, if e is the ~eenb'ieity of the ellipse, the relation between the mco.n anomaly and the eccentric anomaly is C=fI-e.sinu. (25.1) The problem seL by Dcssel was that of expressing the difference bctwecn the meflll nnd eccentric tlllomnlics, u· ns a series of sines of lmlltiplcs of tIle Olelln nllolllnly, i.e. thnt of determining the coefficicnts ",(r = I, 2, :J, ... ) stich thnt
e,
•
I; = Z c, sin (rC).
(25.2)
-,
1/ -
To obtain the \"alues of the cocrficienls c, we multiply both sides of equation (25.2) by sin (.e> and integrnte with respect to C from 0 to =to We then obtain f"(t£ o
C) sin (8C)(11; =
I: c, f"sin (rC) sin (se)(lC 0
,_I
Now
!:;rJ,.•
J:5ill (rC) sin (sC)dC =
fmc! nn intCf:,rrntioll by pnrls shows that
I:(U - C) sin (.rC)dC =
~ [,,- II) cos (.rC)]: + ~
r: (~; - I)
cos (.reldC.
From (25.2), C- II is zero when t; = 0 and when t; =:'1:, so that the 5qunre bracket vllllishcs: the integral enn be written in the form
, f",cos al; dl£
-I
.f
ami hence, using equntion (25.1) we obtain the rcsult c.
=...:.... "cos {'(II :r.r
0
e sin lI)}dlt.
(2ij.3)
The intcgrnl on the right-hand side of equation (25.3) is
12S
BESSEL FUNCTIONS
93
n function of , and of the eccentricity e of the plnnet's orbit. I r we write
J .. (x) =
..!.. roos{z sin 0 :t J 0
110),10
(:!5.,~)
it follows from cqullliollS (25.3) and (25.2) that , ~ ( )sin (rC) u-r,=2,L,J.cr . r
(25.!:i)
._1
The funclion .1,,(3:) so defined is called Bessel's cocfrldent or order n. We shall now show that J .(x) is equal 10 the codficicnl of t" in the expansion of cxp Ox(t - ,.-1 in other words we may define J ,,(x) by means of the c"xpnnsion
no
cx p {!z(t_2-)}= I: '/,,(:z:W. I ,, __ - i. If we develop exp (ixt) in asccnding powers of ixt we sec that the vnllle of t.his integral is ... (ix)' L -
I' (1 -
f 2 )"-lt·(lt.
$1 _I If 8 is an odd integer then the corresponding integral occurring in this series is zero, and if 8 is an even integer, 2r, say, then the integral has the vnlue • -0
(I _ I o so tbat I
+
+
11)"-1,,'-1 dll = r(1I !)r(r !) r(r1 r 1)
+ +
l~ E(-I)'x'l1"+!('+.1
+ r + 1) (_ l)'x 2' P(! )r(ll + !) !o rlT(1I + r + 1 )2 .-0
(2r)!
(" ...
=
2'
sincc, by the duplication formula for the gamma function, r(!)(2r)1 = 2~'r!T(r+ !). It follows immediately from the series expansion (2;.2) for J ,,(x) that (Iz)" J "z ( ) -- T(i)F(lI+!)
I' (1 -1
: ",,(.11-1-
•
y)I," =:E
•1:
".(.'1:)".(y)l,r+.
Equating eoefficicnts of '" wc obtain thc addition formula
•
J,,(x-l-Y)= :E ".(.11).J,,_.(y)
(29.1 )
~-.
'1'0 put this in a form which invoh'cs only Bessel coefficients of positive order wc write the right hand sidc in thc form _1
"
m
1: J.(X)./K_'(Y) -I-:E J.(x)J .. _.(y) -I.... 0
r_a:>
>:
.Ir(x).I,,_.(y)
._,,+1
and notc that bceause of thc relation (25.7) thc first tcrm can bc writtcn as
.
-,
l: (- I )'J _,(.)J ._,(Y) = l: (-
I)'J ,(.)J o+«Y).
r_1
...._:J]J.1bt- (.I + ,I) +(l-·I + ,I )(.1 +.I)} 3: •
lUl[1
aq lsnUl
1i0llS
J;)
SIU;)P!JP0;'
;)In lUlU ;);)S ;).\\ ((;'OS) uonunb" U! S:l]J;)S SPI1 ::lll!1nmsqnS ,.,
'.+.x J ;) 3:
(c'm:)
•
= fi
\UJOJ ;nll JO S! \lonnlOS lS-l!J ;)111 U;H!.f, 'J;>::lalU! uu lUlIl asoddns nuqs :).\\ lIu JO lSJ!~[ ',1 1= = a 5100.1 sull S!111 puu
JOli OJ;)Z J;)lll!;)U S! "
0=;; 0, x".!,,(x) --)- 0 ns x --)- 0 so that the [ower limit is zero and we obtnin the integrlll
,
fOiV".!,,-l(Xjd,'1:
= «".1,,(00 F I(I' + I',
refl+V+
_jn2x2)X"+pc-P:t.(/x
I )n' ( 2. r(v+ I )p"'+rit !F1!!I+!V+!,
=
!.,t+!I+ I; v+ 1;
-
a\
p:)"
(32.6) If we make use of equation (:l0.8) on the left-hand side of this CfJlllltiOIl and of equation (7.'Q 011 tile right-hund side.
we see that this result is equivalent to the formula
F(p+l+ t )(1"
a:>
fo J • (ax)XI'c-P"'I!X= 2" F(I'+ 1 p2)il'+Ir+i ~ .' ) , f', (!/l+b+l; h-l/lj .. + ._-- v+lj a-p)((J~+
where p
>
0,
,II
+ > o. V
X
(32.7)
112 THE SPECIAL FUNCTIONS OF PHYSICS AND CHEMISTRY § 32
The hypcrgcolllclric series occurring 011 the righl-hnnd of this equation nssumes tl pnrticularly simple form if either I' = )' or I' = .' 1, find we obillin the formulae
+
'"" Z' r(v+!) a· So J.(az):.r!"e-uth:= rei) - . (ll~+p~)*r
So oD
+ *l
va'
r(') - ' . . .. I ' (32.9) ! «(I.+~)~ of the forlllulu (:J2.8) which occur
J.(ax)X-+1r.-PZrfz=
Two special CIHICS frequently lIrc
2o+lr(••
(32.8)
" So Jo(a:t)r'z dx =
,\/(a'
1
+ p')'
(112.10)
a " • ."". So ;r.!.(oz)rJ>Zt[.z:= (a-+P-r
(32.11 )
IntcgTtlling both sides of Cpdx=~a2[ {J ~()'(llF+( 1- J.~(:2)] {J ,,().a)}2. (ll5.S) Suppose now that J. nnd II arc positi\rc roots of the transeendcnLnl e(}tllltion hJ,,().a)
+ 1i).{jJ~(J.a) =
0
(U5.fj)
where h nnd k are constants. 1 t follows than that
J:.T.J"P..t:)J
lI
(/ltC)d.1J = CAOA.i"
(:J5.7)
where CA = {J ;~~;'~}2
{J.-~).2{j2 + 112 _
If wc !lOW suppose that we can expand functiou I(x) in the form I(~)
~
=
, a.,J
lI
(35.8)
k2112}. all
nrbitrary
()./x)
(35.!l)
where the sum is taken over the positive !"Oots of the equntion (35.G) thcn wc can determine the eocffieienL.11 )
then the coefficients of the sum (35.n) flrc given by fl J
2J·7
= {J (' n
"Jo
»)2'
2 ~1
(}.J
~
/1")
0- -
f";1:/(:r:)J,,(J.p:)clx 0
(35.12)
Similarly if t.he sum is Luken over the positive roots of the equation J,,(i.n) = 0 (85.13) we fiud lhat the coefficients
(I J = 2{1.t. n . "
{lJ
lIrc given by the [ol'ollila
Wfa:r/(X)Jn().sX)dX
I'lll
(35.H)
0
In this section lIO tlttcmpt has UCCIl made to discllss the VCIT difficult problem of the cOll\'crgcncc of FourierBessel series. For l\ vcry full discussion of this topic the render is referred to Chapter XVI I r of G. N. Watson's A 'l'reatise on the Theory 01 I1cssel Functions, 2nd. edit., (Cambridge University jJ ress, I!H.q. 36. Thc Use of Bessel Functions in Potcntial Theory. As nn example of the use of Bessel functions in potential theory we shall consider the problem of determining a fUlletion tp(/.>, z) for t.he hnlf-spuce fI :2 e :2 0, z:2 0 sutisfying the differential equation
a2lj! I alp aZlj! oe' + e oe + 0" ~ 0 and the boundnry conditions:
(q (ii)
•
,~lle),
on
,~o;
'1'-> 0 as
:J -i'"
co;
(30.1 )
122 THE SPECIAL FUNCTIONS Of PHYSICS AND CHEMISTRY
(,.,..,) a, ae + xrp = (iv)
0
i
36
on f! = ll;
e-+ O.
'P remains finite as
We saw in § 1 that a fUllction of the form 'P = R(e)Z(::) is n solutioll of IXjuntioll (36.1) provided that
cPZ dz 2
2~
).,Z = 0
-
(36.2)
find that (/21l
1 (Ill
( Q-
Q (!?
-1.+--1
.2]"' +1.j~=O
(00.3)
wherc).( is n constant of separation. To sntisfy the boundary condition (ii) we lUllst take solutions of equation (30.2) of the form
and to satisfy the condition (iv) we must lake as the solutions of equat.ion (36.3) functions of the form R = JO().&I) since the second solutions YO(),ie) would become infinite in the region of the nxis e = O. The differential equation (36.1) and the boundary conditions (ii) llnd (iv) urc satisfied by nny SlIlllS of the form 'f'(r, =)= :Ea1c-Aj'Jo()"!?)
,
(BOA)
wherc the a/ and )./ nrc constants. But if we arc to $illtisfy the boundary condition (iii) we must luke thc sum over tbc posili\"c roots of thc equulion I)
+
)'/Jo().,.a) xJo()".a) = O. (36.5) The solution is dctermined therefore if we elUl find constants a/ such that condition (i) is snlisfied, i.c. such thnt
,
!(fJ) = :EatJ(}.;!?)·
(30.6)
1) For propcrliC!l of lhe rools oUhi! c{llmlion!l« exnmple 16 below.
BESSEL FUNCTIONS
§J6
From equations (35.10) nnd (35.8) we sec thnl we must take t).2
il =
_J.,
+ ;.,:2)
ll'!().~
j
f n'J(n')J ()..n'}tle' (Jo(J.,a)}2 It
0II'LES
MlIking use of Example 2 of Chapter II and of the expallsioll expaud cos (:I: /jin 0) RS n power serie.'! ill .'lin 0 in two WHy.'!. lienee hy equaling powef:5 of lIin"' 0 IIhow Ihat if Il is a positive integer
1.
(~5.8)
:1:" _ :!"+1 ; (n + Il - 1 P Ju(x). n_. (n Il)!
Derive the corresponding result for x"·t! rmd show thnt the two resull!l IlIny be comhined into lhe single formula ... ;~: ( r + I l - I ) ! ( r x' __'
n-o
+ O)J _11 .,u () or
"I
(r_l,:!,ll, ... )
2. It" lind show Umt
C dellOte
.
t.he eccentric and mean nnOllllllie.'l of n plnnet
cos ('IlI) _
/I
sin ('Ill) _"
,
1:: _.I.. _.(me) cos (me) m__ t
3. :'>lllking \l$C of the eXllression for p. (cosO) given in EXIIIIlplc 19 of Chnpter III show th"t
. !..... ~
n..()
4.
Show lhat 8.1;'(:) _ .I._ I {:)
(i)
.U~"(~)
(ii) 5.
P .(cos 0) _ trc:os8J.(r sin 0).
"I
-
3J._,(:)
+ a./~(~) + .1.(::)
+ 3J.,,(:)
- J.,,(::).
_ O.
Prove thllt
_+ ' .(r.1: ) , ~_ { - l)J :!
_ S"'COS(l\,+i)lI -=~"'~'--c;" (_1)"" 1I 0 cos lu ,{(X'II')
r-l
nnd dcdu(:e tllat
_• + •~ :!
r-l
(-l)'.I.{rx) _ O.
128 THE SPECIAL FUNCTIONS OF PHYSICS AND CHEMISTRY 6.
Shol'l" lh.Dl lhe 21 _
with freedom equatiollll t - sin f, y _ costl(l - cos I)
C'l1I'.... e
may he rCl'fell(:I11ecl ill lile inlerval 0
0
show Uml
f: eo ... •
cos(z sill 0)diJ - J.( ,/(21' - a'))
P"we lhat, it - 1 I
I lllld
til
+" +
1 > 0, prove thnt
f:C-·'J .. i(I)'.--1flt-l/~ where Q=(z) kind.
denol~
the fl.SIOCillted
c
l.(.c +y) -
~ l ..{.cl/ .._..{y) ..-41
(z'_q-i·Q;(z)
I~gemlre
tunetioll ot the 5eCOnd
_
+
~ (J ..(z)/ •••{y) ... _1
+ l .....(.cll.. (yl)
130 THE SPECIAL FUNCTIONS OF PHYSCIS AND CHEMISTRY 19.
Show that
10.
Prove that
..
-(t-
~
" _ _ Ill
./.{la)l· _ e-I'
I )
..
..
~
.. __ 00
1.··t·J.{~)
lind deduce thnt J.{re'8) _
(i)
..
~
J •••(r)e/'·+"l8
(-ir sin 0)· ,
.. -0 l.(~)
(ii)
_
....
e.-
:E - , Ju.(z).
"'-0'" Using Lhe expllnsioll of the la~L (llll:stion prove 1I1l1L J,(ae''''+be'j!) Is the coefficient of t' ill the expnn~ioll of
21.
eXj' { _
+
i(a !lin a. t b lin
(I)} __.. i: _ ;: .. e'.':H'"",J.(a}J..(b)I.'.
+
Df putting If _ a cos a. boos{l. 0 _ alina. prove XeUJlIIIIIl\', additiOil theorem J.(ll) where Il' _ Il' 22.
+ b'
'
~
+ bain{J.{J-a.+(J-:;
_ /.Ie-'& in ...
be
'8)
__'" E .I•••(Il)J.(b)e- lod
- :lab cos O.
Prove that ".(=}J.(Iu:)
where /I' _
(I'
+ /)' -
_...!.. I J.(U.r)dO, :J Jo
:lu/) cos 0 ami deduce thllt
where 1:
Pro"C, ill " similar way, thnl
1
-
4a'
(Il
+ 1t)1 + c'
131
BESSEL FUNCTIONS where I: find N(kl are
f
l"
0 ,/(1 - !;l s ill''1')tlrp
/:;(1:) -
13.
defined lIhove and
liS
Show that
J ..(z)J ~(z) _ .:.. ell~-.l!T/ fn. 1~ __ (:!.e sin O)eIR+.. IOI dO
,
"
nnd hel1t'e tlmt (JR(Z))" _ -
Deduce lhnt
24.
, f",
J u (2xsinO)dO
"
Prove lhal
25. We ddine lhe Bessel-lnteJ.1rlll runctlon or order II by lhe e(lliation
"".1 R(Il) --1111
f,
./iR(:r) _ Prove t hul ir " is evell
_.2.-
"
rem
(1I0)ci(zsin 0) 110, :r 0 where ci(.r) denotes the cosille illlegr:!!, Pill! derive the correSl>onding expression wilen " is odd. Show that:,/iR(Z)
(i) (ii) (iii)
(iv)
(v) (vi)
Ii (e!·"-l/") .. Ji~(z)
•
~
J
I"Ji.(x):
_ J .(z)/or;
+
(II - 1).I;._,(z) ~ (II I )./i~.,(Z) _ 2ItJj~(z): ci(:e) _ Ji.(ie) - ~./i,(z) + 2./i.(z) :
ai(z) _ 2./i.(z) - 2./i.(x) Ji.(z) _
r + logHz) -
+ ~./i.(z)
-
;
"8,1".(1, 1,2,2,2, -z"/4).
CIIA1''rEll V
THE FUNCTIONS OF HERMITE AND LAGUERRE 38. Thc HCI"mltc Polynoml:lls. The Hcrmite polynomial lJ"(x) is defincd for intcgrnl valucs of Il and all rcal vnlllcs of oX by the idcntity _,21~_1'
I,
__
~ lI,,(IV) t"
..... _ _ ,_.
,,-0
]f
(38.1)
"/I.
we write
thcn it follows from Taylor's theorcm thnt.
1I ,,(x) ~ (O"~) at ~ ,., [0: at ,-I-"'J (.. 0 1-0
Now it is obvious from the form of thc function c:.:p {- (It - t)2} that [
0"
_
at"
J
C-IZ-I)"
= (_ 1)" _d" (c-",)
dx"
1-0
nnd so wc havc thc form
d"
//"(x) = (- 1)"C"I--' (r z ') IX" 1
(as.2)
for the cnlculntion of the polynominl //,,(."1:). It follows from this formula thnt the first eight I rcrmile polynominls nre://o(x) //l(X) //2(X) //3(X)
1, = 2;1:, = ·k/:2 - 2, = 8.v3 - 12x.
=
§38
133
,THE FUNCTI ONS OF HERMITE AND LAGUERRE
lf 4 (x} = 1I~(x) = 1/ 8 (x) = 111(x) =
+ +
12, lOr' - ·ISr 120x, :J2z$ - IOOr 04x' - .ISOr; - i20x' - 120, :J:JOOr - lGSOx. 128z - 13Hz" '
+
In general we have
II ,,(x)=(2 .:r)" -
"(11-1 ) (2X)"-2 11
+ 11(1l -
1)(1l - 3)(11 - ,~) (" )04
_x
2!
+ ...
or, ill the notatio n of Section 12,
1I.(x) = ('lx)"
~Po (- !", ! - 1"; - ~).
(38.3)
Recurr ence formul ae for the Hermit e polyno mials follow dirccLly from the definin g I'cllllion (a8.1). If we differen tiate both sides of that equatio n with rcspee llo.t: we obtain the relation
• /I' ( ) 2tc~ZI-I' = I: ~ t n ..-0
II!
(rom which it follows directly thal 2/1/1 n_dx) = Jl~(x).
(88"1)
On lhe other hund if we differe ntiate hoth sides of the identit y (38.1) with respect to t we oblnin the relation 2{x -
I)c~'_11
IJ (x) ~ 1"-1 .. = 1: 1)1 ,,_I (11
which cnn be wrillen in the form 21: ~ 11,,(:1:) t" _ 2 ~ 11,,(:1:) 1"+1 = .. -0
II!
.. -0
III
to yield the identit y 2xll,,(.%) = 21111 11 _ 1(x)
1': 11,,(:1:) t'l-I t)l .._1(11
+ 11 +1(z) I1
by the idenlifi ciltion of ".'ocfficicnl.!i of I".
(38.5)
134 THE SPECIAL fUNCTIONS Of PHYSICS AND CHEMISTRY
i
39
Eliminating 2/111.._t(x) from equntions (8S.·Q and (38.5) we obt.nin the rclnlioll 1I~(x) = 'lor-lI,,(x) -
1I"+I(x).
(:.18.0)
Differentialing both sides of this identity we find that Jl::(x) =
2xll~(x)
and, by equation (:lS.4). 11::(z) -
+ 211..(x) -
1I~+I{x)
lJ~+I(Z) = 2(11
2xJl~(x)
In other words y = JI ,,(x) is ferentinl C differe ntial equatio n is thcrefo re (30.8)
For genern l values of thc pnrame ler II the two series for YI(Z) and Y~(x) arc infinitc . Yrom equlllio ns (3!l.·1) and (39.6) it follows that for both series ar+z""
rr
we write c.xp (x2)
then
=
bo
•
"7"(/"
as r
~
00.
(30.0)
+ b~ + ... + b,xr + br+2zrH + ... •
br+!! "" ; b"
as r ~ CO
(30.10)
Suppos e thnt a,V IbN is equal to a consta nt y, which may be (3D.O) and ~mlfdl or lnrgc. then it follows form equatio ns
136 THE SPECIAL FUNCTIONS OF PHYSICS AND CHEMISTRY §"'O
(3!1.1 0) thnl. for large cnough vnlues of N. Q,v+'! ...lb.v+t............ y. In other words the higher temlS of the series for YI(X), y=(z) differ from those of exp (z2) only by multiplicnti\·c constnuts YI' Y:t. so thnt for large "alues of 1z I, y.(:c) ........ YleZ"' y~(.J:) ........ YseZ·
sincc for such values the lower terllls nrc llllimportnnl. We shull sce laler (Section 41 below), that in qUlIlltUnl meehllnics we require solutions of Hermite's differential equation which do not become infinite morc rapidly t.Il1ln exp(!x 2) liS I x 1- 00. IL follows from the above considerations that sueh solutions llrc possible only if either y.
1 tp 1211,'/;' = 1.
If we lct
then the eq\ll1lion
(.~1.1)
hecomes
,Fiji (211' ~) Ijl=O _+ de:: --~~ Ill'
(.u .2)
lind the conditions (i) and (ii) become I) N. F. MoU IIlld I. N. Sneddon, "'m.... ,Hull/wit.' till/I pljtalil)lu. (OxforU, \fIlS), p, 50.
1/11 .'1/),
141
THE FUNCTIONS OF HERMITE AND LAGUERRE
(;')
lp--+ OIlS
IE 1-+ 00;
f-.• I'P[:di =
(ii')
~7l
,fmo. f '+-,-,
The nrgulllcnt ghocli at the beginning of Section '10 shows that equation (-n.:!) possesses solutions 'P which satisfy the condition (ii') if and only if the parall1eter (211'11/1') wldeh OCCllrs in the equation lllkcs one of the \"3111C5 1 ~II where 11 is n positive integer. In olher words solutions of this type, which IIrc known by the probability interpretation of the w:wc function l;" to correspond to sial iOllllry stales of the :')"SIClll enn exist if nnd only if
+
II' = IIr(lI where
II
+!)
(41.3)
is a positi\"c integer. '''hen this is the case the form
of the wave [unction 'P is known from Section ·W to be 'P = ClJ1 ,,(;)
(.HAl
where C is It constant.. Applying condition (ii') unci equalion (.IO.I:!) we see thnt
C _ (.t-rml)I__'_ -
21"{1d)i
Ii
Thus the wlIve function corresponding to nn ndmissable energy {It ! )hl' is
+ ,
If.
~
(1:1:11/1)1 h
YJ,,{~)
::I"(I/!)I'
~=
'2,-r,r;;;; z VT
In quantum theory the matrix clements (1/ defined by t.he equntion (II
J:r 11)
=
-.
(.H.O)
I it: 11')
fa'> XV,,{.r)v'..(z)tl.T
fire of consiclernble importance in the elise of the hnrmonic oscillator. In terms of tile vnrinble ; wc hn\'e
(" 1'1,»
~
h f·
0,
(-1)"(111)'(11-1)·-" C-";t" I,. (;t) 1/.1: - '-=,!::-"'C';f:~-'-ftH
lind tieduce that if n
m)!fl
(II
~
I
+
I {(Il+l)I)I (_1)•• 1:;:11 .. (II I I)!
9.
Sho\\' thaI (1l!)~1
I..
J
lI''"(l-:r)·[•• (fl.J')IU _ (-i) •• '
o
iO.
{(11+1!)l)'
"
·'(11). '
I'rove Ihnt the function
xiI' c-j~ 1.:+,,(;t) iJ " solution or the differential equlIlion li"y ,,-)'+1 u}I ( lixl+--.--d.l:+
I
-"'4+
2p+,,+1 _ }'(2"-,) ) y _ O. ~ .1,1:'
THE FUNCTIONS OF HERMiTe AND LAGUERRE
157
II. The potential cnergy ror lhe nuclear molion or 0 dintomie molC'Cule ill c1Ol1e-I)' "I'llroxillllltetl h)' the )IOMle runction I'(r) _ Dc'" - '.!Ve-, Show ahut the spherically symlllctrical solutiOll.'! or the Schrlltlillger c'luntion with Illi~ I'0tcillini lire
C.
I
\'(r) ~ ~ {exp( - be--J}(~r)·-·- I.:,::~:~' (2be--)
(0.s;1/ :iA' -
H
where b _ :!.:l(~mV)! '(all), C. iJ II nonn:l.liZ3tion L'Onstant, and the \'Iduoo or the ene~' ore
oorn::spolltlin~
1I' __ lJ
1
[
(II
+ b
j)
+ (II + _HI) .u.'
12. Prove tl'ut the normLllimtiou constant C. in the "revi/l!ls eXnlnl'le Il:I~ the ,·:tlue
I'Illcre
13. Show lImt in pllrn\)olie L'Oonlin:ltlS e'llllitions Sdnvdinccr'~ Clll.llltion
~.
'I, Ip dcfim,
'1'01.11:5 of function!! 13, H, 51,
fllllCliulI~
5ll.
Ordillltr)' point G. OrUr0l,>OlIal &et"1"CIlL'e 59. Orlhonormal IICt GO. Parnbolic co-onlillfttes 15, 157, l'uulillg, L., 2.1. P-funclion :!5. !'hiJliI)!!. K G. 70. Point III infinity O. Potcntial UlCOr)' '10, 57, ,0, 83, 121.
1~-1.
Tnylor 5eriC!! 5.
F. K 115.
:"eum:ulII's adLlition lhcorcm 130 Bessel funeliOIl 105: formuln
"", N'orrrllllisctl
.~3,
Serie.. of Legendre function!! 57: or Bessel flll1eljolL~ II U. !::iillc·jntcgral I:.!, 17. !::iinJluhtr IKllnt II. Snedtlon, I. :-:. t·IO, 1110. !::il'cci:tl funelions 3. Sl'herie:tl Hessel functions lOS: oo-ordinate!l I·S, ':"0. !::itieltjel inlegml 162. ~urf:lI:e .r;pheric~ll h:muonics 80. S)'metrical-toll molecule 2'~, ·12. 110,
~eurmUIll,
,I:.!,
Ka. 140, 1[,0, Hi5, 157.
VllndcrlllOlltle'~ tl,t'(lrem ~2. 'VHI.~(lIl, G. l';. 1:!1. "",llIOIl'S ULL'(lrcr1\ 45. WIIVC mechnnics ,I:!, 43, 8:1, 1.10,
t,m.
Wclier';j Bessel function. tOn. Whil1llle's theorem 45. \\'llillllker'!1 fUllction, as. "'ilion, f;. B. 21. ...~fO!II
of
of U1.'endrc function. S,' frmcllons I:.!O.
I~l
