Complex Numbers in N Dimensions

COMPLEX NUMBERS IN iVDIMENSIONS

NORTH-HOLLAND MATHEMATICS STUDIES 190 (Continuation of the Notas de Matematica)

Editor: Saul LUBKIN University of Rochester New Yorli, U.S.A.

2002 ELSEVIER Amsterdam - Boston - London - New York - Oxford Paris - San Diego - San Francisco - Singapore - Sydney - Tokyo

COMPLEX NUMBERS IN iV^DIMENSIONS

Silviu OLARIU Institute of Physics and Nuclear Engineering Magurele, Bucharest, Romania

2002 ELSEVIER Amsterdam - Boston - London - New York - Oxford Paris - San Diego - San Francisco - Singapore - Sydney - Tokyo

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Preface A regular, two-dimensional complex number x + iy can be represented geometrically by the modulus p = (a:^ + y^)^/^ and by the polar angle 6 = arctan(y/x). The modulus p is multiplicative and the polar angle 6 is additive upon the multiplication of ordinary complex numbers. The quaternions of Hamilton are a system of hypercomplex numbers defined in four dimensions, the multiplication being a noncommutative operation, [1] and many other hypercomplex systems are possible, [2]-[4] but these interesting hypercomplex systems do not have all the required properties of regular, two-dimensional complex numbers which rendered possible the development of the theory of functions of a complex variable. Two distinct systems of hypercomplex numbers in n dimensions will be described in this work, for which the multiplication is associative and commutative, and which are rich enough in properties such that exponential and trigonometric forms exist and the concepts of analytic n-complex function, contour integration and residue can be defined. [5] The n-complex numbers described in this work have the form u = XQ-i-hiXi H h/in-i^n-i? where hi,...,hn-\ are the hypercomplex bases and the variables XQ^...^Xn~i are real numbers, unless otherwise stated. If the n-complex number u is represented by the point A of coordinates XQ^X\^ ..,^Xn-i', the position of the point A can be described with the aid of the modulus d = {XQ -h a:f H h x^_iy^^ and of n — 1 angular variables. The first type of hypercomplex numbers described in this work is characterized by the presence, in an even number of dimensions n > 4 , of two polar axes, and by the presence, in an odd number of dimensions, of one polar axis. Therefore, these numbers will be called polar hypercomplex numbers in n dimensions. One polar axis is the normal through the origin O to the hyperplane v^ = 0, where v^ = XQ + xi A + Xn-i- In an even number n of dimensions, the second polar axis is the normal through the origin O to the hyperplane v^ = 0, where V- = XQ - xi H h Xn-2 ~ ^n-iThus, in addition to the distance d, the position of the point A can be specified, in an even number of dimensions, by 2 polar angles 6^^ 0_, by n/2 - 2 vu

Vlll

planar angles xj^k, and by n/2 - 1 aziniuthal angles (f)^. In an odd nnmber of dimensions, the position of the point A is specified by d, by 1 polar angle 6^, by (n - 3)/2 planar angles V^/t-i, and by (n - l)/2 azinuithal angles 0^. The multiplication rules for the polar hypercomplex bases h\,...,hn~\ are hjhk ~ hjj^k if 0 < j -f fc < n - 1, and hjli^ = hj^k-n if n < j + A: < 2n - 2, where /IQ = 1. The other type of hypercomplex numbers described in this work exists as a distinct entity only when the number of dimensions n of the space is even. The position of the point A is specified, in addition to the distance rf, by n/2 - 1 planar angles ipk and by n/2 aziniuthal angles 0/^. These numbers will be called planar hypercomplex numbers. The multiplication rules for the planar hypercomplex bases h\,...,hn-\ are hjhk = hj^k if ^^ j + k < n - \, and hjhk = -hj-^k-n if n < j + k < 2n - 2, where /lo = 1. For n = 2, the planar hypercomplex numbers become the usual 2-dimensional complex numbers x -f iy. The development of analytic functions of hypercomplex variables was rendered possible by the existence of an exponential form of the n-complex numbers. The azinuithal angles (pk^ which are cyclic variables, appear in these forms at the exponent, and lead to the concept of n-dimensional hypercomplex residue. Expressions are given for the elementary functions of n-complex variable. In particular, the exponential function of an ncomplex number is expanded in terms of functions called in this work ndimensional cosexponential functions of the polar and respectively planar type. The polar cosexponential functions are a generalization to n dimensions of the hyperbolic functions coshy,sinh?/, and tlie planar cosexponential functions are a generalization to n dimensions of the trigonometric functions cosy,sin?/. Addition theorems and other relations are obtained for the n-dimensional cosexponential functions. Many of the properties of 2-dimensional complex functions can be extended to hypercomi)lex numbers in n dimensions. Thus, the functions f{v) of an n-complex variable which are defined by power series have derivatives independent of the direction of approach to the point under consideration. If the n-complex function /(n) of the n-complex variable u is written in terms of the real functions Pk{x{),,..,Xn-\),k ~ 0, ...,n - 1, then relations of equality exist between the partial derivatives of the functions P^. The integral J^ f{u)du of an n-complex function betw(^en two points A,B is independent of the path connecting A,B. in regions where / is regular. If f{u) is an analytic n-complex function, then ^^ f{u)(lu/(ii-iiQ) is expressed in this work in terms of the n-dimensional hypercomplex residue /(no). In the case of polar complex numbers, a polynomial can be written

IX

as a product of linear or quadratic factors, although several factorizations are in general possible. In the case of planar hypercomplex numbers, a polynomial can always be written as a product of linear factors, although, again, several factorizations are in general possible. The work presents a detailed analysis of the hypercomplex numbers in 2, 3 and 4 dimensions, then presents the properties of hypercomplex numbers in 5 and 6 dimensions, and it continues with a detailed analysis of polar and planar hypercomplex numbers in n dimensions. The essence of this work is the interplay between the algebraic, the geometric and the analytic facets of the relations.

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Contents Hyperbolic Complex Numbers in Two Dimensions 1.1 Operations with hyperbolic twocomplex numbers 1.2 Geometric representation of hyperbolictwocomplex numbers 1.3 Exponential and trigonometric forms of a twocomplex number 1.4 Elementary functions of a twocomplex variable 1.5 Twocomplex power series 1.6 Analytic functions of twocomplex variables 1.7 Integrals of twocomplex functions 1.8 Factorization of twocomplex polynomials 1.9 Representation of hyperbolic twocomplex numbers by irreducible matrices

1 2 4 6 8 10 12 14 15

Complex Numbers in Three Dimensions 2.1 Operations with tricomplex numbers 2.2 Geometric representation of tricomplex numbers 2.3 The tricomplex cosexponential functions 2.4 Exponential and trigonometric forms of tricomplex numbers 2.5 Elementary functions of a tricomplex variable 2.6 Tricomplex power series 2.7 Analytic functions of tricomplex variables 2.8 Integrals of tricomplex functions 2.9 Factorization of tricomplex polynomials 2.10 Representation of tricomplex numbers by irreducible matrices

17 19 20 27 31 35 38 42 44 47 50

Commutative Complex Numbers in Four Dimensions 3.1 Circular complex numbers in four dimensions 3.1.1 Operations with circular fourcomplex numbers . . . 3.1.2 Geometric representation of circular fourcomplex numbers

51 54 54

XI

16

56

Xll

3.1.3

The exponential and trigonometric forms of circular fourcomplex numbers 59 3.1.4 Elementary functions of a circular fourcomplex variable 64 3.1.5 Power series of circular fourcomplex variables . . . . 65 3.1.6 Analytic functions of circular fourcomplex variables 68 3.1.7 Integrals of functions of circular fourcomplex variables 70 3.1.8 Factorization of circular fourcomplex polynomials . . 73 3.1.9 Representation of circular fourcomplex numbers by irreducible matrices 76 3.2 Hyperbolic complex numbers in four dimensions 77 3.2.1 Operations with hyperbolic fourcomplex numbers . . 77 3.2.2 Geometric representation of hyperbolic fourcomplex numbers 80 3.2.3 Exponential form of a hyperbolic fourcomplex number 83 3.2.4 Elementary functions of a hyperbolic fourcomplex variable 85 3.2.5 Power series of hyperbolic fourcomplex variables . . 87 3.2.6 Analytic functions of hyperbolic fourcomplex variables 89 3.2.7 Integrals of functions of hyperbolic fourcomplex variables 90 3.2.8 Factorization of hyperbolic fourcomplex polynomials 91 3.2.9 Representation of hyperbolic fourcomplex numbers by irreducible matrices 92 3.3 Planar complex numbers in four dimensions 93 3.3.1 Operations with planar fourcomplex numbers . . . . 93 3.3.2 Geometric representation of planar fourcomplex numbers 95 3.3.3 The planar fourdimensional cosexponential functions 100 3.3.4 The exponential and trigonometric forms of planar fourcomplex numbers 105 3.3.5 Elementary functions of planar fourcomplex variables 108 3.3.6 Power series of planar fourcomplex variables 110 3.3.7 Analytic functions of planar fourcomplex variables . 113 3.3.8 Integrals of functions of planar fourcomplex variables 114 3.3.9 Factorization of planar fourcomplex polynomials . . 117 3.3.10 Representation of planar fourcomplex numbers by irreducible matrices 121 3.4 Polar complex numbers in four dimensions 121 3.4.1 Operations with polar fourcomplex numbers . . . . . 121 3.4.2 Geometric representation of polar fourcomplexnumbersl24

Xlll

3.4.3 3.4.4

The polar fourdimensional cosexponential functions The exponential and trigonometric forms of a polar fourcomplex number 3.4.5 Elementary functions of polar fourcomplex variables 3.4.6 Power series of polar fourcomplex variables 3.4.7 Analytic functions of polar fourcomplex variables . . 3.4.8 Integrals of functions of polar fourcomplex variables 3.4.9 Factorization of polar fourcomplex polynomials . . . 3.4.10 Representation of polar fourcomplex numbers by irreducible matrices Complex Numbers in 5 Dimensions 4.1 Operations with polar complex numbers in 5 dimensions . . 4.2 Geometric representation of polar complex numbers in 5 dimensions 4.3 The polar 5-dimensional cosexponential functions 4.4 Exponential and trigonometric forms of polar 5-complex numbers 4.5 Elementary functions of a polar 5-complex variable 4.6 Power series of 5-complex numbers 4.7 Analytic functions of a polar 5-complex variable 4.8 Integrals of polar 5-complex functions 4.9 Factorization of polar 5-complex polynomials 4.10 Representation of polar 5-complex numbers by irreducible matrices

128 133 136 139 141 142 145 147 149 150 151 154 159 161 161 163 163 164 165

Complex Numbers in 6 Dimensions 167 5.1 Polar complex numbers in 6 dimensions 168 5.1.1 Operations with polar complex numbers in 6 dimensions 168 5.1.2 Geometric representation of polar complex numbers in 6 dimensions 169 5.1.3 The polar 6-dimensional cosexponential functions . . 172 5.1.4 Exponential and trigonometric forms of polar 6-complex numbers 175 5.1.5 Elementary functions of a polar 6-complex variable . 176 5.1.6 Power series of polar 6-complex numbers 177 5.1.7 Analytic functions of a polar 6-complex variable . . 178 5.1.8 Integrals of polar 6-complex functions 179 5.1.9 Factorization of polar 6-complex polynomials . . . . 179

XIV

5.2

5.1.10 Representation of polar 6-complex numbers by irreducible matrices 181 Planar complex numbers in 6 dimensions 181 5.2.1 Operations with planar complex numbers in 6 dimensions 181 5.2.2 Geometric representation of planar complex numbers in 6 dimensions 183 5.2.3 The planar 6-diniensional cosexponential functions . 185 5.2.4 Exponential and trigonometric forms of planar 6-complex numbers 188 5.2.5 Elementary functions of a planar 6-complex variable 189 5.2.6 Power series of planar 6-complex numbers 190 5.2.7 Analytic functions of a planar 6-complex variable . . 191 5.2.8 Integrals of planar 6-complex functions 192 5.2.9 Factorization of planar 6-complex polynomials . . . 192 5.2.10 Representation of planar 6-complex numbers by irreducible matrices 193

Commutative Complex Numbers in n Dimensions 195 6.1 Polar complex numbers in n dimensions 198 6.1.1 Operations with polar n-complex numbers 198 6.1.2 Geometric representation of polar n-complex numbers 202 6.1.3 The polar n-dimensional cosexponential functions . . 208 6.1.4 Exponential and trigonometric forms of polar n-complex numbers 212 6.1.5 Elementary functions of a polar n-complex variable . 217 6.1.6 Power series of polar n-complex numbers 222 6.1.7 Analytic functions of polar n-complex variables . . . 224 6.1.8 Integrals of polar n-complex functions 225 6.1.9 Factorization of polar n-complex polynomials . . . . 229 6.1.10 Representation of polar n-complex numbers by irreducible matrices 231 6.2 Planar complex numbers in even n dimensions 232 6.2.1 Operations with planar n-complex numbers 232 6.2.2 Geometric representation of planar n-complex numbers236 6.2.3 The [)lanar n-dimensional cosexponential functions . 240 6.2.4 Exponential and trigonometric forms of planar ncomplex numbers 244 6.2.5 Elementary functions of a planar n-complex variable 247 6.2.6 Power series of planar n-complex numbers 251 6.2.7 Analytic functions of planar n-complex variables . . 253

XV

6.2.8 Integrals of planar n-complex functions 6.2.9 Factorization of planar n-complex polynomials . . . 6.2.10 Representation of planar n-complex numbers by irreducible matrices Bibliography Index

254 258 259 261 263

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Chapter 1

Hyperbolic Complex N u m b e r s in Two Dimensions A system of hypercomplex numbers in 2 dimensions is described in this chapter, for which the multipUcation is associative and commutative, and for which an exponential form and the concepts of analytic twocomplex function and contour integration can be defined. The twocomplex numbers introduced in this chapter have the form u = x -\- 5y, the variables x^y being real numbers. The multiplication rules for the complex units 1,6 are 1 • 5 = (^, (5"^ = 1. In a geometric representation, the twocomplex number u is represented by the point A of coordinates (x,y). The product of two twocomplex numbers is equal to zero if both numbers are equal to zero, or if one of the twocomplex numbers lies on the line x ~ y and the other on the line x = —y. The exponential form of a twocomplex number, defined for x-{-y > 0, x — y > 0, is w = pexp(5A/2), where the amplitude is p = {x? — y^Y^^ and the argument is A = Intan^, tan^ = (a: + y)/(x —y), 0 < 0 < 7r/2, The trigonometric form of a twocomplex number is w = d\/sin20exp{(l/2)(Jlntan0}, where (f = x^ + y^. The amplitude p is equal to zero on the lines x = ±y. The division l/(x+(5j/) is possible provided that p ^ 0. Ifwi = xi+6yi,U2 = ^2 + Sy2 are twocomplex numbers of amplitudes and arguments pi,Ai and respectively p2, A2, then the amplitude and the argument /9, A of the product twocomplex number uiti2 — X\X2 + y\y2 + 0, the quantity p=^v^l'\

//>0.

(1.14)

will be called amplitude of the twocomplex number x + 5y. The normals of the lines in Eq. (1.13) are orthogonal to each other. Because of conditions (1.13) these lines will be also called the nodal lines. It can be shown that if ?iw' = 0 then either u — 0, or u' = 0, or one of the twocomplex immbers u, u' is of the form x + Sx and the other is of the form x — 6x.

1.2

Geometric representation of hyperbolic twocomplex numbers

The twocomplex immber x -{• Sy can be represented by the point A of coordinates (x^y). If O is the origin of the two-dimensional space x^y, the distance from A to the origin O can be taken as d^ = x^ + y\

(1.15)

The distance d will l:)e called modulus of the twocomplex number x -f Sy. Since {x-hyf^{x-yf = 2d\

(1.16)

X + y and x — y can be written as x + y — \/2dsin9,

x - y — \/2dcos6,

(1-17)

so that X = dsin{e + 7r/4), y = -dcos{e + n/4).

(1.18)

If ti = X + Sy, ui = X] + Sy\, U2 = 0:2 + Sy2i and u ~ n\U2, and if Vjj^ = Xj + i/j, Vj- = Xj - yj, 2d] - vj_^ + Vj_,

Xj + yj = \pidjsin^j,

Xj - yj = dj\/2cos9j,

(1-19)

for j = 1,2, it can be shown that v^ = vi-^V2-{., v^ ~v\^V2-,

tan0 = tan^1 tan02-

(1.20)

Geometric representation of hyperbolic twocomplex numbers

5

The relations (1.20) are a consequence of the identities + y\y2) 4- {xiy2 4- y]X2) = {xi + yi){x2 + ^2),

(1.21)

(0:1x2 + yiy2) - {xiy2 + yiX2) = {xi - yi){x2 - ^2)-

(1.22)

(J:IX2

A consequence of Eqs. (1.20) is that if n = ^1^2, tlien ^ = ^1^2,

(1.23)

^j ="^j-f^j-.

(1-24)

where

for j = 1,2. If ^' > 0, vi > 0,1/2 > 0, then P = P1P2,

(1-25)

where Pi = i^f,

(1-26)

f o r i = 1,2. The twocomplex numbers

are orthogonal, 64.6-. = 0,

(1.28)

and have also the property that e\ = e-^, e l = e_.

(1.29)

The ensemble e+,e_ will be called the canonical hyperbolic twocomplex base. The twocomplex number u = x + Sy can be written as x + Sy^{x

+ y)e+ + {x - y)e-,

(1.30)

or, by using Eq. (1.12), u = v^e^ + t>_e_,

(1.31)

which will be called the canonical form of the hyperbolic twocomplex number. Thus, if Uj ~ Vj^e^ + Vj-C-, j — 1,2, and u — uiU2^ then the multiplication of the hyperbolic twocomplex numbers is expressed by the relations (1.20).

6

Hyperbolic Complex Numbers in Two Dimensions

The relation (1.23) for the product of twocomplex numbers can be demonstrated also by using a representation of the multipHcation of the twocomplex numbers by matrices, in which the twocomplex number ti = X -f dy is represented by the matrix

The product u = x + 6y oi the twocomplex numbers ui = Xi + 5yi,U2 = ^2 + ^JJ'ii can be represented by the matrix multiplication

\ y

X J

y yi

xi J y y2

X2 J

It can be chocked that det ( ^

^ ) = ..

(1.34)

The identity (1.23) is then a consequence of the fact the determinant of the product of matrices is equal to the product of the determinants of the factor matrices.

1,3

Exponential and trigonometric forms of a twocomplex number

The exponential function of the hypercomplex variable u can be defined by the series expu =:l-\-u-^ u^/2\ + w'V3! -f • • •.

(1.35)

It can be checked by direct multiplication of the series that exp(n + u') = expu • expu'.

(1.36)

The series for the exponential function and the addition theorem have the same form for all systems of conunutative hypercomplex munbers discussed in this work. If u = x + Sy, then expn can be calculated as expu = expx • exp(6y). According to Eq. (1.1), ^ 2 m ^ l ^ ^ 2 m + l ^ §,

(1.37)

where m is a natural number, so that exp{Sy) can be written as exp(^y) = coshy -f- (5sinh?/.

(1.38)

Exponential and trigonometric forms of a twocomplex number

7

From Eq. (1.38) it can be inferred that (cosht-f-5sinht)'^ = coshmt 4-(^sinhmt.

(1.39)

The twocomplex numbers u = x -^ Sy for which i?^. = a; + y > 0, i;_ = X — y > 0 can be written in the form X ^ Sy = e'^'^^y'.

(1.40)

The expressions of a:i, yi as functions of a:, y can be obtained by developing e^^i with the aid of Eq. (1.38) and separating the hypercomplex components, X = e^^ coshyi,

(1-41)

y = e^isinhyi.

(1.42)

It can be shown from Eqs. (1.41)-(1.42) that a;i = iln(T;+i;_), yi = i l n ^ .

(1.43)

The twocomplex number u can thus be written as u — pexp((5A),

(1-44)

where the amplitude is p = [x^-y^y/^ and the argument is A = (1/2) ln{(x+ y)/{x — y)}, for a: 4- y > 0, X — y > 0. The expression (1.44) can be written with the aid of the variables d^O^ Eq. (1.17), as u — p exp (-(5 In tan e ] ,

(1.45)

which is the exponential form of the twocomplex number M, where 0 < ^ < 7r/2.

The relation between the amplitude p and the distance d is p^dW^2e.

(1.46)

Substituting this form of p in Eq. (1.45) yields u = dsini/2 20exp(i(51ntan6l), which is the trigonometric form of the twocomplex number u.

(1.47)

8

1.4

Hyperbolic Complex Numbers in Two Dimensions

Elementary functions of a twocomplex variable

The logarithm ni of the twocomplex mimber u^ ui = hin, can be defined for v^ > 0, v- > 0 as the sohition of the equation tx-e^S

(1.48)

for wi as a function oi u. From Eq. (1.45) it results that Inn = l n p + -SlntanO,

(1.49)

It can be inferred from Eqs. (1.49) and (1.20) that \\\{u\U2) ~ Inwi + \\\ui.

(1.50)

The explicit form of Eq. (1.49) is ln(a: + Jy) = ^(1 + b) \\\{x + y) + ^(1 - ^) ln(x - y),

(1.51)

SO that the relation (1.49) can be written with the aid of Eq. (1.27) as In II = e_^ In 7;-^ + e_ lnt;_.

(1.52)

The power function u^ can be defined for v^ > 0,7;_ > 0 and real values of 71 as n^=e^'"".

(1.53)

It can be inferred from Eqs. (1.53) and (1.50) that {uiU2r ==u'^u^.

(1.54)

Using the expression (1.52) for \nu and the relations (1.28) and (1.29) it can be shown that (X + Syf =^-{l + S)ix + yr + ^(1 - S){x - ?/)".

(1.55)

For integer n, the relation (1.55) is valid for any x,j/. The relation (1.55) for n = —1 is 1 = 1 fi ± i + i ^ y (1.56) X + oy 2 \x + y x - yj The trigonometric functions cos u and sin u of the hypercomplex variable u are defined by the series cosu = 1 - wV2! + u^/A] -f •..,

(1.57)

Elementary functions of a twocomplex variable s'mu^u-

u^/3\ + u^/5\ + • • •.

9 (1.58)

It can be checked by series multiplication that the usual addition theorems hold for the hypercomplex numbers ^1,^2, cos(wi +1x2) = COSU1COSW2 — sinwi sinn2,

(1.59)

sin(ui + U2) = sinui cos 1x2 + costxi sinw2-

(1.60)

The series for the trigonometric functions and the addition theorems have the same form for all systems of commutative hypercomplex numbers discussed in this work. The cosine and sine functions of the hypercomplex variables 6y can be expressed as cos6y = cosy, sinSy = 6s\ny.

(1-61)

The cosine and sine functions of a twocomplex number x + Sy can then be expressed in terms of elementary functions with the aid of the addition theorems Eqs. (1.59), (1.60) and of the expressions in Eq. (1.61). The hyperbolic functions cosh u and sinh u of the hypercomplex variable u are defined by the series cosh u = l+ u^l2\ 4- wV4! + • • •, sinhn = ?/ -f ^^^/3! -I- vC"lh\ + • • •.

(1.62) (1.63)

It can be checked by series multiplication that the usual addition theorems hold for the hypercomplex numbers xi\,uy V v/2;

2/3'

l +h+k

+

:;

tanO (2.88)

( ^ * )

Substituting in Eq. (2.88) the expression of the amplitude p, Eq. (2.39), yields

^\/I(H

fe . ^ 1-f/l + fc -V2cos(9Jexpf-yr- 0^ from Eq. (2.86) it results that Inn = ln/9 + -(/i + k) In ( - ^ )

+ -7={h - k)(t).

(2.95)

It can be checked with the aid of Eqs. (2.25) and (2.32) that l n ( W ) = lntA + lnu',

(2.96)

which is valid up to integer multiples of 27t{h - k)/\/3. The trigonometric functions of the hypercomplex variable u and the addition theorems for these functions have been written in Eqs. (1.57)(1.60). The trigonometric functions of the hypercomplex variables hy.ky

36

Complex Numbers in Three Dimensions

can be expressed in terms of the cosexponential functions as 1 cos{hy) = -[ex (iy) + ex (-iy)] + ~h[mx {iy) + mx (-jy)] + -A;[px(?:y) + px(-M/)],

(2.97)

1 1 cos(%) = -[ex (iy) + ex {-iy)] + -h[px (iy) + px {-iy)] + -k[mx {itj) + rnx {-iy)],

(2.98)

sin(/iy) = —[ex {iy) - ex {-iy)] + —.h[mx {iy) - mx {-iy)] 1 + ^A:[px(«y)-px(-i?/)],

(2.99)

sin(A;y) = —[ex {iy) - ex {-iy)] + —h[px {iy) - px {-iy)] Zl

2,1

1 (2.100) + —•A;[mx {iy) — mx {—iy)], 2i where i is the imaginary unit. Using the expressions of the cosexponential functions in Eqs. (2.68)-(2.70) gives expressions of the trigonometric fiuictions of hy, hz as /, V 1 2 , / ^ \ y cos{hy) = - cos y + - cosh I -^y I cos - +

+h 1

1 ./N/3 \ y ^ '^ • u ( ^ \ • y - cos y - - c o s h l^-^-yj c o s - + - ^ smh l^—yj s m -

\ . y +k -1c o s y - -1c o s, h/ Nl ~/ 3y l\ c o sy- - - ^1s m• h. l/"N/S —ylsm(2.101)

cos(fcy) = - cos y+-

+h

1

1

cosh ( — y J cos - -t, /\/3 \

y

^

• u(^

\

• y

3 '""'y ~ 3 '"'^ VT^j '-"' 2 - 7! '' [^V ""' 2

+k 1

1

,(y/^\

y

1

• v./^ ^ • y (2.102)

Elementary functions of a tricomplex variable

sm{hy) = - siny - - cosh I —y - siny + - cosh \^—yj

37

I sin sm - + - ^ smh (^—yj cos -

- s m y + -cosh ( — y 1 s m - - - ^ s m h \—y

cos(2.103)

sin(A;y) = ^ siny — - cosh

. y -y I sm -

1 . 1 , +/i - s m y + -cosh

sinh

1 . 1 ,. - s m y + -cosh

sinh

4-fc

y\cosy

-y I cos 2 ^/ 2 (2.104)

The trigonometric functions of a tricomplex number x + hy -h kz can then be expressed in terms of elementary functions with the aid of the addition theorems Eqs. (1.59), (1.60) and of the expressions in Eqs. (2.101)-(2.104). The hyperbolic functions of the hypercomplex variable u and the addition theorems for these functions have been written in Eqs. (1.62)-(1.65). The hyperbolic functions of the hypercomplex variables /ly, ky can be expressed in terms of the elementary functions as 1 ^ 2 ~ cosh y -f- - cos - ^ ? / ) c o s h | +

cosh.{hy)

*5

o

+h l cosh y - i COS l^-^yj

cosh | - i = sin ^ - ^ y j sinh |

+k I cosh y - i cos f - ^ y j cosh | + - L sin ( - i y j sinh | (2.105) u/, X 1 I 2 /N/3 \ , y cosh(A;y) = - coshy + - cos I — y I cosh - +

+h

1

..

1

/\/3 \

, y

1

. /\/3 \

. , y

3 '°'^ y - r ° n T^) ""'^ 2 + ;^''" 1 ^^J'' 2

38

Complex Numbers in Three Dimensions 1

+k - cosh y — - cos

1 2 smh{hy) = - sinhy - - cos

. sin

sinh 2 (2.106)

^ y ) sinh I

+h 1 • ,

1 (^ \ . , y 1 . - s m h y + - c o s l - y l s m h - + -^sm

-^y] cosh I

• t y + -1 cos ^ , j s i n h | - - L s i n V3 +k -1 sinh

2 ^) ^^^'^ 2J 2 (2.107)

1 2 I sinh i: sinh(A;t/) = - sinhy — - cos -—V 2 '^/ 2

1 . ,

1

(\f^

\ . , y

1

-iv """"^ 2

- smhw + - cos ——y snih 7= sni 3 - ' 3 1 2 ^ ; 2 v/3 1 • . 1 ^ y j s i n h | + - L s i n ----?/ I cosh --sinhy+-cos 2 ^ 2

(2.108) The hyperboHc functions of a tricomplex number x -f hy + kz can then be expressed in terms of the elementary functions with the aid of the addition theorems Eqs. (1.64), (1.65) and of the expressions in Eqs. (2.105)~(2.108).

2.6

Tricomplex power series

A tricomplex series is an infinite sum of the form (2.109) where the coefficients a^ are tricomplex numbers. The convergence of the series (2.109) can be defined in terms of the convergence of its 3 real components. The convergence of a tricomplex series can however be studied using tricomplex variables. The main criterion for absolute convergence remains the comparison theorem, but this requires a number of inequalities which will be discussed further. The modulus of a tricomplex number u = x + hy + kz can be defined as \u\=.{x'+y^

+ Z^)^'\

(2.110)

Tricomplex power series

39

Since \x\ < \u\^\y\ < \u\^\z\ < |n|, a property of absolute convergence established via a comparison theorem based on the modulus of the series (2.109) will ensure the absolute convergence of each real component of that series. The modulus of the sum U1+U2 of the tricomplex numbers Ui^U2 fulfils the inequality IKI - \U2\\ < |U1 + U2\ < K l + \U21

(2.111)

For the product the relation is \uiU2\ < v/3|t/i||i/2|,

(2.112)

which replaces the relation of equality extant for regular complex numbers. The equality in Eq. (2.112) takes place for a^i = yi = zi and 0:2 = 2/2 = ^2^ i.e when both tricomplex numbers lie on the trisector line (i). Using Eq. (2.91), the relation (2.112) can be written equivalently as lsi5l

+ \a\al

< 3 (^J? + ^ a ? ) ( ? 5 i + ^ a l ) ,

(2.113)

where S'j = xj + y'j A-z'j - Xjpj -XjZj -yjZj.aj = Xj + yj + ZjJ = 1,2, the equality taking place for c^i = 0, J2 = 0- A particular form of Eq. (2.112) is \u^\ < V3\u\^,

(2.114)

and it can be shown that |^/| f{u)du=--={h-k)J2

\nt{ujuJu)aj,

(2.155)

where the functional int(M, C), defined for a point M and a closed curve C in a two-dimensional plane, is given by ' j^^TiT ^\ J X xL j M is an Ulterior point of (7, ,_ ^^^. int(M,C) = ^ . • . ^ (2.156) \ n 0 if-r «^ • V ' ^ » ^ ^^ M is exterior to C, ^ ^ and Ujii^ Fn are the projections of the point Uj and of the curve F on the nodal plane 11, as shown in Fig. 2.9.

2.9

Factorization of tricomplex polynomials

A polynomial of degree m of the tricomplex variable u =- x + hy + kz has the form Pm[u) = u'^ + aiu"^"^ + • • • + am-\u + a^,

(2.157)

where the constants are in general tricomplex numbers, a/ = p/ + hqi + fcr/, / = l , - " , m . In order to write the polynomial Pm(^) as a product of

48

Complex Numbers in Three Dimensions

Figure 2.9: Integration path F, pole Uj and their projections Tn.Ujji on the nodal plane H. factors, the variable u and the constants a/ will be written in the form which explicits the transverse and longitudinal components, 7n

m

Prn{u) = ^ ( a n e i +anei)(7j,ei -f ^;iei)^^-^ + e4 5 3 a / + < - ^ (2.158) /=0

/=:0

where the constants have been defined previously in Eq. (2.132). Due to the properties in Eq. (2.128), the transverse part of the polynomial Pm(^) can be written as a product of linear factors of the form m

m

^(aaei+a/iei)(?;iei+i)iei)'"~' = 1=0

]]^[{v[-vn)eii-(vi~vn)ei],{2Abd) 1=1

where the quantities vn^vn are real numbers. The longitudinal part of Pyn(?/), Eq. (2.158), can be written as a product of linear or quadratic factors with real coefficients, or as a product of linear factors which, if imaginary, appear always in complex conjugate pairs. Using the latter form for the simplicity of notations, the longitudinal part can be written as m

rn

5 ^ a H ^ r ' = n(^^+-^^/+), 1=0

i=\

(2.160)

Factorization of tricomplex polynomials

49

where the quantities vi^ appear always in complex conjugate pairs. Due to the orthogonality of the transverse and longitudinal components, Eq. (2.128), the polynomial Pmi'^) can be written as a product of factors of the form m

Pm{u) = Ylii'^i - vii)ei + {vi - vii)ei + {v^ - ^/+)e+].

(2.161)

1=1

These relations can be written with the aid of Eqs. (2.125) as m

Pm{u) = l[{u-ui),

(2.162)

where 111 = viiei + vnei + vi^e.^.

(2.163)

The roots vi^ and the roots vnei + vnei defined in Eq. (2.159) may be ordered arbitrarily. This means that Eq. (2.163) gives sets of m roots u i , . . . , u ^ of the polynomial Pm{u)^ corresponding to the various ways in which the roots vi^.vnei + vnei are ordered according to / in each group. Thus, while the tricomplex components in Eq. (2.158) taken separately have unique factorizations, the polynomial PmiQ) can be written in many diflFerent ways as a product of linear factors. If P{u) = u^ — 1, the degree is m = 2, the coefficients of the polynomial are a\ = 0,a2 = ~ 1 , the coefficients defined in Eq. (2.132) are a2i = ~l,a22 = 0,a2u = —1- The expression of P(ii), Eq. (2.158), is (eit;i + ^i^i)^ — ei -i-e+(t;^ — 1). The factorizations in Eqs. (2.159) and (2.160) are (ei?;i-f ei?)i)^-ei = [ei(t?i+ l) + ei?)i][ei(i;i - l ) + ei?}i] and v\-l = {v+~{l){v^'-l). The factorization of P(n),Eq. (2.162), isP(n) = {u-ui){n-U2), where according to Eq. (2.163) the roots are ui = ±ei ±e^,U2 = —ui. If ei and e^ are expressed with the aid of Eq. (2,127) in terms of h and fc, the factorizations of P{u) are obtained as u^ -1 = {u + l){u-l),

(2.164)

or as u'-l

( \-2h-2k\( («+ 3

)(„

l-2h-2k\ 3

It can be checked that (±ei ± e-f )^ = ei + e+ = 1.

).

,„ _ , , (2.165)

50

Complex Numbers in Three Dimensions

2.10

Representation of tricomplex numbers by irreducible matrices

If the matrix in Eq. (2.34) representing the tricomplex number u is called f/, and

/l" —L (2.166)

i I v/3

v^

which is the matrix appearing in Eq. (2.17), it can be checked that 2

TUT-'

=

V

•^{y-z) 2 \y 0

2

^j

X —

2

0 0

(2.167)

0

The relations for the variables x — {y + z)l2, {\/3/2){y — z) and x + y-\-ziov the multiplication of tricomplex numbers have been written in Eqs. (2.26), (2.28) and (2.29). The matrices TUT~^ provide an irreducible representation [7] of the tricomplex numbers ii = x + hy + kz, in terms of matrices with real coefficients.

Chapter 3

Commutative Complex N u m b e r s in Four Dimensions Systems of hypercomplex numbers in 4 dimensions of the form u = x + ay + I3z + jt are described in this chapter, where the variables x, y, z and t are real numbers, for which the multiplication is both associative and commutative. The product of two fourcomplex numbers is equal to zero if both numbers are equal to zero, or if the numbers belong to certain four-dimensional hyperplanes as discussed further in this chapter. The fourcomplex numbers have exponential and trigonometric forms, and the concepts of analytic fourcomplex function, contour integration and residue can be defined. Expressions are given for the elementary functions of fourcomplex variable. The functions f(ii) of fourcomplex variable defined by power series, have derivatives limw-^uo[/(^) ~~ /(^o)]/(?^ — ^o) independent of the direction of approach of u to UQ. If the fourcomplex function f{u) of the fourcomplex variable u is written in terms of the real functions P{x^y,z,t),Q{x,y^z,t),R{x^y,z,t),S{x,y,z^t), then relations of equality exist between partial derivatives of the functions P, Q, i?, S. The integral JA f{'^)du of a fourcomplex function between two points A,B is independent of the path connecting A, B. Four distinct types of hypercomplex numbers are studied, as discussed further. In Sec. 3.1, the multiplication rules for the complex units a,/3 and 7 are a^ = - 1 , /3^ = —1, ^'^ = l,a/3 = /3a = —7, 07 = 7 a = /5, ^7 = 7/3 = a. The exponential form of a fourcomplex number is ii = pexp [7 In tan ip + {l/2)a{(p -h x) + (l/2)/3(0 - x)] 1 where the amphtude is p^ = [{x + t)'^ + (y + z)'^] [{x - t)^ + {y - zY\ , 0, X are azimuthal angles, 51

52

CouitnutHtive Complex Numbers in Fom' Dimensions

0 < 0 < 27r, 0 < X < 27r, and i/? is a planar angle, 0 < V^ < 7r/2. The trigonometric form of a fourcomplex number is ?/ = ri[cos(V^ —7r/4) + 7sin(t/^~7r/4)] exp [{l/2)a{(t) + x) + (l/2)/3(0 - x% where (f = .r^ + y^ + ^'^ -f f. The amplitude /r> and tani/^ are multiplicative and the angles 0, x are additive upon the multiplication of fourcomplex numbers. Since there are two cyclic variables, (f) and x? these fourcomplex numbers are called circular fourcomplex numbers. If f{u) is an analytic fourcomplex function, then §^^f{u)du/{u-uo) = 7T[{a + P) ini{uQ^yS^y)-¥{a-^) intKrC^Tr^)] /(wo), where the functional int takes the values 0 or 1 depending on the relation between the projections of the point UQ and of the curve F on certain planes. A polynomial can be written as a product of linear or quadratic factors, although the factorization may not be unique. In Sec. 3.2, the nniltiplication rules for the complex units a, (5 and 7 are a^ = 1, f3^ = 1, 7'^ == l.c'^ft = l^oc = 7, rv7 = 7 a = ft, /J7 = 7 ^ = a. The exponential form of a fourcomplex number, which can be defined for sz=. x-i-y-\- z-^tX). s' = X - y -^ z - t > 0, s" = x •]-y - z - t > 0, s^" = X — y ~ z -{• t > 0, is n = //exp(ai/i 4- ftz\ + 7^1), where the amplitude is /i = [ss's^s'^yi'^ and the arguments are jyi = (1/4) ln(.s,s'7.s'5'"), ^1 = {llA)\n{ss"'/s's''), U = (l/4)ln(.s,s7.9".s'"). Since there is no cyclic variable, these fourcomplex numbers are called hyperbolic fourcomplex numbers. The amplitude // is multiplicative and the arguments y\,z\,t\ are additive upon the multiplication of fourcomplex numbers. A polynomial can be written as a product of linear or (quadratic factors, although the factorization may not be uniciue. In Sec. 3.3, the multiplication rules for the complex units o,ft and 7 are OL^ = /?, fi'^ = - 1 , 7^ — -ft^otft — i^Oi — 7, 0^7 = 70' = - 1 , /?7 = r^^ = —a . The exponential function of a fourcomplex number can be expanded in terms of the four-dimensional cosexponential functions fu){x) — 1 - rz;V4! + .x^S! - • • •, /HI-X) - x - ^ 1^ + ;i;V9! - • • •, jn{x) = X-V2! - a;V6! + x^VlO! , f^'^{x) = x^I'M - x^ll\ + x^^j\\\ . Expressions are obtained for the four-dimensional cosexponential functions in terms of elementary functions. The exponential form of a fourcomplex number is u — pexp {(l/2)(a - 7) Intani/; +(l/2)[/i -f (a -f 7)/\/2] -(l/2)[/i - (« + 7)/\/2]x}, where the amplitude is p^ = | [j; + (jy -

i)l\[^

+ [^ + (?/ + f ) / v / 2 ] ' | | [ : r - ( ? / - f ) / v / 2 ] ' + [ ^ - ( y + ^ ) / v / 2 ] ' | , (/).x are azimuthal angles, 0 < 0 < 27r,0 < x < 27r, and t/? is a planar angle, 0 < ^ < 7r/2. The trigonometric form of a fourcomplex number is u = d [cos (V; - 7r/4) +.(l/v^)(a - 7) sin (t/^ - 7r/4)] exp {(l/2)[/:i + [a + 7)/v^]

Introduction to Chapter 3

53

-{l/2)[/3 - (a + 7 ) / \ / 2 ] x } , where (f = x'^ -\-y'^ + z'^ + i\ The ampUtude p and tan i/? are multipUcative and the angles (/>, x ^r^ additive upon the multipHcation of fourcomplex numbers. There are two cychc variables, cf) and X? so that these fourcomplex numbers are also of a circular type. In order to distinguish them from the circular hypercomplex numbers, these are called planar fourcomplex numbers. If/(w) is an analytic fourcomplex function, then §p f{u)du/{u - UQ) — TT m + (a + 7 ) / \ / 2 j mt(uo^v^^^v) + (p — [a + 7 ) / \ / 2 j int(worC?r7^^) f{uo) , where the functional int takes the values 0 or 1 depending on the relation between the projections of the point 1^0 and of the curve F on certain planes. A polynomial can be written as a product of linear or quadratic factors, although the factorization may not be unique. The fourcomplex numbers described in this chapter are a particular case for n = 4 of the planar hypercomplex numbers in n dimensions discussed in Sec. 6.2. In Sec. 3.4, the multiplication rules for the complex units a, P and 7 are Q;2 = ^^ ^2 ^ I ^2 ^ p^^^f^ = /Ja = 7, Of7 = 7Qf = 1, ^7 = 7/3 = a. The product of two fourcomplex numbers is equal to zero if both numbers are equal to zero, or if the numbers belong to certain four-dimensional hyperplanes described further in this section. The exponential function of a fourcomplex number can be expanded in terms of the four-dimensional cosexponential functions ^40G'^) = l+.T^/4!-f x^/8! + -• •, (/4i(^) = x-\-x^/b\+x^/9\-{• • •, g42{x) = x^/2\-\-xy6\+x^yiO\^'. •, g^^ix) = x^'M^x^/7\^x^^/n\ + •••. Addition theorems and other relations are obtained for these fourdimensional cosexponential functions. The exponential form of a fourcomplex number, which can be defined for x+y+z+t > 0, x—y+z—t > 0, is u = pexp [(l/4)(a + /3 + 7) l n ( v ^ / tan(9+) - (l/4)(a ~ /3 + 7) ln(v/2/ tan 6>_) +{a - 7)0/2], where p = {p^P-^^'^ pi = {x - zf + (y - if, pi = {x + zf -{y + f)^ e^ = {l+a~\-P-\- 7)/4, e^ = (I - a + 0 - 7)/4, ex = (1 —/3)/2,ei = (a —7)/2, the polar angles are tanS^. = \/2p^/v^^ta.n6= \/2//+/?;_, 0 < ^+ < 7r,() < ^_ < TT, and the azimuthal angle / is defined by the relations x ~ y — p^coscj), z - t — pj^siwcf), 0 < (f) < 27r. The trigonometric form of the fourcomplex number u is u = (1 + 1/ tan^ e^ + 1/ tan^ 9-)'^^^ {ej + e+\/2/ tan/9+ + e_\/2/ tan/9_} exp[ei, and there are two axes v^^V- which play an important role in the description of these numbers, so that these hypercomplex numbers are called polar fourcomplex numbers. If f{n) is an analytic fourcomplex function, then §yf{u)dul{u — UQ) =

54

Coimnutntive Complox Numbers in Four Dimensions

^ ( ^ "• 7) int(wo^i;, r^^;)/(no) , where the functional int takes the values 0 or 1 depending on the relation between the projections of the point UQ and of the curve F on certain planes. A polynomial can be written as a product of linear or quadratic factors, although the factorization may not be unique. The fourcomplex numbers described in this section are a j)articular case for n = 4 of the polar hypercomplex numbers in n dimensions discussed in Sec. 6.1.

3.1

Circular complex numbers in four dimensions

3.1.1

Operations with circular fourcomplex numbers

A circular fourcomplex number is determined by its four components (x, y, z, t). The sum of the circular fourcomplex numbers (x, y, z, t) and (x', y', z', t') is the circular fourcomplex number {x -h x',y + y',z -]- z',t -\- t/). The product of the circular fourcomplex numbers (x^y^z^t) and {x\y\z\f/) is defined in this section to be the circular fourcomplex number (.xx' — yy' — zz' + tt\ xy^ 4- yx' + zt' + tz', xz' -f zx' -h yt' -f ty\ xt' -f tx' - yz' - zy'). Circular fourcomplex numbers and their operations can be represented by writing the circular fourcomplex number [x^y^^^t) as u — x-\-ay + (iz^ 7^, where a, ^ and 7 are bases for which the multiplication rules are a ^ - - l , /3^^ = - l , 7^ = 1, ap = 0a =^ ~ 7 , aj •= ^a = (3, 13^ — ^ft = a.

(3.1)

Two circular fourcomplex numbers u ~ x •{• ay -{• fiz -{- ^yt, v! = x' + ay' + ^z' + 7^' are equal, u — u'^ if and only if x = x'^ y = ?/, z = z'^t = t'. If u — x+ay+pz+'yt, u' = .//-hay'+^^'-f 7^' are circular fourcomplex numbers, the sum u -j- u' and the product \LU' defined above can be obtained l)y applying the usual algebraic rules to the sum {x + ay + liz + ^t) + {x'-{-ay'-]-fiz' + ^t') and to the product {x -f ay + (3z + 'yt){x' -f ay' + /3z' -f -yf/), and grouping of the resulting terms, u + u' = x + x' + a(y + y') + fi(z + z') + ^(t + //),

(3.2)

nu = xx/ - yy - zz + it' + a{xy 4- yx' + zt' + iz) +(i(xz' + zx' + yt' + fy) + j{xt' + tx' - yz' - zy'). Uu.u'.u" tive

(3.3)

are circular fourcomplex niunbers, the multiplication is associa-

{uu')u" = u{u'u")

(3.4)

Circular complex numbers in four dimensions

55

and commutative uu^ = u'u^

(3.5)

as can be checked through direct calculation. The circular fourcomplex zero isO + a O - f / ? - 0 + 7 0 , denoted simply 0, and the circular fourcomplex unity i s l - f a - 0 + ;9-0 + 7 ' 0 , denoted simply 1. The inverse of the circular fourcomplex number u = x + ay + /3z + ^t is a circular fourcomplex number tz' = x' + ay'+ /3z' + jt' having the property that uu' = 1.

(3.6)

Written on components, the condition, Eq. (3.6), is xx' yx' zx' tx'

— yy' — zz' + tt' = 1, + xy' + tz' + zt' = 0, + ty' + xz' + yt' = 0, - zy' - yz' + xt' = 0.

(3.7)

The system (3.7) has the solution ,

X =

x{x^ + y'^ + z^-4

' ;/(-a:^ -y^ + z^y = -4 .' z{-x' z = ^, _ tj-x^

+ y^-z'-t^) -5

f) - 2yzt

(3.8)

,

f) + 2xzt ,

(3.9)

,

(3.10)

+ 2xyt

+ y^ + z'' + t^) - 2xyz

provided that p 7^ 0, where p'i=a:'+y'

+ z' + t' + 2{x'y'' + x'^z' - x^t" - y^z^ + yH^ + zH^)

-Sxyzt.

(3.12)

The quantity p will be called amplitude of the circular fourcomplex number x + ay + 0z + jt. Since p' = pip-,

(3.13)

where

PI = {X + if + (y + z)\ pi = {x- tf + {y- zf,

(3.14)

56

Commutative Complex Numbers in Four Dimensions

a circular foiircomplex number u = x^ay

+ pz + jt has an inverse, unless

x + t = 0,y + z = 0,

(3.15)

x-t

(3.16)

or = 0, y-z

= 0.

Because of conditions (3.15)-(3.16) these 2-dimensional surfaces will be called nodal hyperplanes. It can be shown that if nu' = 0 then either ti = 0, or ix' = 0, or one of the circular fourcomplex numbers is of the form a: -f ay + /3y -f 7X and the other of the form .T' -f ay' — fiy' — ^x'.

3.1.2

Geometric representation of circular fourcomplex numbers

The circular fourcomplex number x-fay 4-/3^+7^ can be represented by the point A of coordinates {x, y, z, t). If O is the origin of the four-dimensional space X, y, z, t^ the distance from A to the origin O can be taken as d^ = x^ -^y^ + z^ ^-f.

(3.17)

The distance d will be called modulus of the circular fourcomplex number x + ay + I3z + 7^, d = \u\. The orientation in the four-dimensional space of the line OA can be specified with the aid of three angles 0, x, V' defined with respect to the rotated system of axes ^4-^

x-t

y-hz

y- z

The variables ^ , 7 ; , T , ( will be called canonical circular fourcomplex variables. The use of the rotated axes ^, v. r, C for the definition of the angles , X? ^ is convenient for the expression of the circular fourcomplex numbers in exponential and trigonometric forms, as it will be discussed further. The angle (f) is the angle between the projection of A in the plane ^, v and the O^ axis, 0 < (/) < 27r, X is ^he angle between the projection of A in the plane r, C, and the Or axis, 0 < x < 27r, and V^ is the angle betwe^en the line OA and the plane rOC, 0 < t/^ < 7r/2, as shown in Fig. 3.1. The angles (f) and X will be called azimuthal angles, the angle V^ will be called planar angle . The fact that 0 < V^ < n/2 means that ip has the same sign on both faces of the two-dimensional hyperplanc DOC,. The components of the point A in terms of the distance d and the angles (/>, x, i^ are thus —-p:- = d cos (f) sin ijj, v2

(3.19)

Circular complex numbers in four dimensions

57

Figure 3.1: Azimuthal angles , x ^^^ planar angle ip of the fourcomplex number x + ay + pz + ^t^ represented by the point A, situated at a distance d from the origin O. X — t

y+2 y-

z

V2

= dcosxcos^,

(3.20)

= rising sin V',

(3.21)

= dsinxcosV^.

(3.22)

It can be checked that p^ = \/2dsmip,p= \/2dcos'0. The coordinates X, y^ z^ t in terms of the variables d, 0, Xi ^ ^re X — --^(cosV^cosx + sini/;coS(/)), v2

(3.23)

y— -—(cos t/j sin X + sin i/? sine/)), v2

(3.24)

z = - - p ( ~ c o s ' 0 s i n x + sin'0sin0), V2

(3.25)

f = ~?=(~ cosi/'cosx + sin^cos 1, so that Eq. (3.57) has always a real solution, and ti = 0 for xt + yz ^ 0. It can be shown similarly that ^2 ,

cos2yi =

2 p^

cos 2^1 =

2{xy ^ Zt)

2 ^ ^2 ^ ^2

^—^ P

' sm2?/i =

^ p^

, sin 2^1 = -^^

'

(3-58)

~-^,

(3.59)

P

It can be shown that {x'^ — y^ A- z^ ~ \?)^ < p^^ the equality taking place for xy — zt, and {x'^ + y^ — z^ — t^)'^ < p^, the equality taking place for xz = yt, so that Eqs. (3.58) and Eqs. (3.59) have always real solutions. The variables

1

. 2{xy-zt)

yi = - arcsm 2

^ p^

1

.

2{xz-yt)

' ^i == o ^^^^^^ 2

2 p^

'

t, = 1 rgsinh^i^^lM

(3.60)

z p^ are additive upon the multiplication of circular fourcomplex numbers, as can be seen from the identities {xJ - yy^ - zz' 4- tt'){xy' 4- yx' + zt' + tz') ~{xz' + zx' + yt' -f V)(^*' - y^' - ^y' + ^^') - (a;y - ;^t)(a;'2 - y'^ + ^'^ - t'^) + (a;^ - y^ ^ ;,2 ^ t2)(a;'y' - z't% (3.61) ( r a ' - yy' - 2:2:' + ttO(a;2:' + zx' + yt' -f ty') -{xy' + yx' 4- zt' + ]t^')(^^^ - y^' - V + *^0 = {xz - yt){x'^ 4- y^2 -^ ^'2 - ^^2) -f {x^ Ay^-z^--

t^){x'z' -

y't'), (3.62)

{xx' ~ yy' - zz' 4- tt'){xt' - yz' - zy' + tx') +{xy' + yx' 4- ^^' + tz'){:xz' + 2:0;' 4- yi' 4- ty') = (^i + yz){x'^ 4- y'2 4- ^'' 4-1!^) 4- (a:^ 4- y ' + ^^ + e){x'^

+ y'^'). (3.63)

62

Commutative Complex Numbers in Four Dimensions

The expressions appearing in Eqs. (3.57)-(3.59) can be calculated in terms of the angles 0, x, i' with the aid of Eqs. (3.23)-(3.26) as o P ^ _ J _ 2{xt + yz) p^ sin 2V'' p2 ^ ' ' ^ y ' y ' - ' '

1_ tan2V'

^

= cos( + x), - ^ S

= «'" ,T, ( are related, according to Eqs. (3.34)-(3.37) by ^ = V2{ee - v'v"), V = V2{^'v" + v'n. C = v^(r'C" + CV'),

r = x/2(rV" -

('("). (3.111)

which show that, upon multiplication, the components ^,t? and r, ( obey, up to a normalization constant, the same rules as the real and imaginary components of usual, two-dimensional complex numbers. If the coefficients in Eq. (3.101) are ai = aio + aan + (3ai2 + iciis,

(3.112)

and An = a/o+a/3, i n = aa+a/2, A12 = aio-ais,

i/2 = a a - a / 2 , (3.113)

68

Commutative Complex Numbers in Four Dimensions

the series (3.101) can be written as oc

^ 2 ' / ^ [{eiAn + e,i/,)(ei^ + civ)' + (e2^