Exponential forms and path integrals for complex numbers in n dimensions

PII. S016117120102004X http://ijmms.hindawi.com © Hindawi Publishing Corp.

EXPONENTIAL FORMS AND PATH INTEGRALS

FOR COMPLEX NUMBERS IN

n

DIMENSIONS

SILVIU OLARIU

(Received 25 August 2000)

Abstract.Two distinct systems of commutative complex numbers inndimensions are described, of polar and planar types. Exponential forms ofn-complex numbers are given in each case, which depend on geometric variables. Azimuthal angles, which are cyclic variables, appear in these forms at the exponent, and this leads to the concept of residue for path integrals ofn-complex functions. The exponential function of ann-complex number is expanded in terms of functions called in this paper cosexponential functions, which are generalizations tondimensions of the circular and hyperbolic sine and cosine functions. The factorization ofn-complex polynomials is discussed.

2000 Mathematics Subject Classification. Primary 32A45, 33E20, 46F15, 58J15.

1. Introduction. Hypercomplex numbers are a generalization to several dimen-sions of the regular complex numbers in 2 dimendimen-sions. A well-known example of hypercomplex numbers are the quaternions of Hamilton, which are a system of hyper-complex numbers in 4 dimensions, the multiplication being a non-commutative opera-tion [1]. Many other hypercomplex systems are possible [2,3,4,5,6,7,9,10,11,12,13], but these interesting systems do not have all the required properties of regular, 2-dimensional complex numbers which rendered possible the development of the the-ory of functions of a 2-dimensional complex variable.

Two distinct systems of complex numbers inndimensions are described in this paper, for which the multiplication is associative and commutative, and which are rich enough in properties such that exponential forms exist and the concepts of analytic n-complex function, contour integration and residue can be defined. The first type ofn-complex numbers described in this article is characterized by the presence, in an odd number of dimensions, of one polar axis, and by the presence, in an even number of dimensions, of two polar axes. Therefore, these numbers will be called polar n-complex numbers. The other type ofn-complex numbers described in this paper exists as a distinct entity only when the number of dimensionsnof the space is even. These numbers will be called planarn-complex numbers. The planar hypercomplex numbers become forn=2 the usual complex numbersx+iy.

given for the elementary functions ofn-complex variables. The exponential function of ann-complex number is expanded in terms of functions called in this papern -dimensional cosexponential functions of the polar and planar type, respectively. The polar cosexponential functions are a generalization tondimensions of the hyperbolic functions coshy,sinhy, and the planar cosexponential functions are a generalization tondimensions of the trigonometric functions cosy, siny. Addition theorems and other relations are obtained for then-dimensional cosexponential functions.

In the case of polarn-complex numbers, a polynomial can be written as a product of linear or quadratic factors, although several factorizations are in general possible. In the case of planarn-complex numbers, a polynomial can always be written as a prod-uct of linear factors, although, again, several factorizations are in general possible.

A study of commutative complex numbers in 2,3,4,5, and 6 dimensions and further properties of polar and planar complex numbers inndimensions can be found in [8].

2. Polarn-complex numbers

2.1. Operations with polar n-complex numbers. A hypercomplex number inn dimensions is determined by itsncomponents(x0,x1,...,xn−1). The polarn-complex numbers and their operations discussed in this paper can be represented by writing then-complex number (x0,x1,...,xn−1)asu=x0+h1x1+h2x2+ ··· +hn−1xn−1, whereh1,h2,...,hn−1are bases for which the multiplication rules are

hjhk=hl, l=j+k−n(j+k)/n, (2.1)

forj,k,l=0,1,...,n−1, whereh0=1. In this relation,[(j+k)/n]denotes the integer part of(j+k)/n, defined as[a]≤a < [a]+1, so that 0≤j+k−n[(j+k)/n]≤n−1. In this paper, brackets larger than the regular brackets,[], do not have the meaning of integer part. The significance of the composition laws in (2.1) can be understood by representing the baseshj,hkby points on a circle at the anglesαj=2πj/n,αk= 2πk/n, as shown inFigure 2.1, and the producthjhkby the point of the circle at the angle 2π(j+k)/n. If 2π≤2π(j+k)/n <4π, the point represents the basishlof the angleαl=2π(j+k)/n−2π.

Two polarn-complex numbersu=x0+h1x1+h2x2+ ··· +hn−1xn−1, u=x0+ h1x

1+h2x2+ ··· +hn−1xn−1are equal if and only ifxj=xj,j=0,1,...,n−1. The sum of the polarn-complex numbersuandu_is

u+u_=x0+x

0+h1x1+x1+···+hn−1xn−1+xn−1. (2.2)

The product of the polarn-complex numbersu,u_is

uu_=x0x

0+x1xn−1+x2xn−2+x3xn−3+···+xn−1x1 +h1x0x

1+x1x0+x2xn−1+x3xn−2+···+xn−1x2

+h2x0x

2+x1x1+x2x0+x3xn−1+···+xn−1x3

.. .

+hn−1x0xn−1+x1xn−2+x2xn−3+x3xn−4+···+xn−1x0

.

hn−2

hn−1 1

α1

h1

h2

Figure2.1. Representation of the polarn-complex bases 1,h1,...,hn−1by points on a circle at the anglesαk=2πk/n.

The productuu_{can be written as}

uu₌n−1 k=0

hk n−1

l=0

xlxk−l+n[(n−k−1+l)/n]. (2.4)

Ifu,u_{,uaren-complex numbers, the multiplication is associative(uu)u=u(uu)} and commutativeuu_{=uubecause the product of the bases, defined in (2.1), is} as-sociative and commutative.

The inverse of the polarn-complex numberuis then-complex numberu_having the property thatuu_{=1. This equation has a solution provided that the} correspond-ing determinantνis not equal to zero,ν=0. Ifnis an even number, it can be shown that

ν=v+v− n/2−1

k=1 ρ2

k, (2.5)

and ifnis an odd number,

ν=v+ (n−1)/2

k=1 ρ2

k, (2.6)

where

ρ2

k=vk2+v˜k2, vk= n−1

p=0 xpcos

_2πkp

n

, v˜k= n−1

p=0 xpsin

_2πkp

n

. (2.7)

Thus, in an even number of dimensionsn, ann-complex number has an inverse unless it lies on one of the nodal hypersurfacesv+=0, orv−=0, orρ1=0, or...orρn/2−1=0. In an odd number of dimensionsn, ann-complex number has an inverse unless it lies on one of the nodal hypersurfacesv+ =0, orρ1=0, or...orρ(n−1)/2=0.

For evenn,

d2₌ 1 nv+2+

1 nv−2+

2 n

n/2−1

k=1 ρ2

and for oddn,

d2₌ 1 nv+2+

2 n

(n−1)/2

k=1 ρ2

k. (2.9)

From these relations it results that if the product of twon-complex numbers is zero, uu_{=0, thenρ}

+ρ+=0,ρ−ρ−=0,ρkρk=0,k=1,...,n/2, which means that either u=0, oru_{=0, oru,ubelong to orthogonal hypersurfaces in such a way that the} afore-mentioned products of components should be equal to zero.

2.2. Geometric representation of polarn-complex numbers. The polarn-complex numberx0+h1x1+h2x2+···+hn−1xn−1can be represented by the pointAof coor-dinates(x0,x1,...,xn−1). IfOis the origin of then-dimensional space, the distance from the originOto the pointAof coordinates(x0,x1,...,xn−1)has the expression

d2_=x2

0+x12+···+x2n−1. (2.10)

The quantitydwill be called modulus of the polarn-complex numberu=x0+h1x1+ h2x2+···+hn−1xn−1. The modulus of ann-complex numberuwill be designated by d= |u|. Ifν >0, the quantityρ=ν1/n_{will be called amplitude of the polarn-complex} numberu.

The exponential and trigonometricforms of the polarn-complex numberucan be obtained conveniently in a rotated system of axes defined by the transformation

v+=√nξ+, v−=√nξ−, vk= n₂ξk, v˜k= n₂ηk, (2.11)

fork=1,...,[(n−1)/2]. This transformation from the coordinatesx0,...,xn−1to the variablesξ+,ξ−,ξk,ηkis unitary.

The position of the pointAof coordinates(x0,x1,...,xn−1)can also be described with the aid of the distanced, equation (2.10), and ofn−1 angles defined further. Thus, in the plane of the axesvk,v˜k, the azimuthal anglesφkcan be introduced by the relations

cosφk=v_ρk

k, sinφk= ˜ vk

ρk, (2.12)

where 0≤φk<2π, so that there are[(n−1)/2]azimuthal angles. If the projection of the pointAon the plane of the axesvk,v˜kisAk, and the projection of the point Aon the 4-dimensional space defined by the axesv1,v1,v˜ k,v˜kisA1k, the angleψk−1 between the lineOA1kand the 2-dimensional plane defined by the axesvk,v˜kis

tanψk−1=ρ1_ρ

k, (2.13)

for 0≤ψk≤π/2,k=2,...,[(n−1)/2], so that there are[(n−3)/2]planar angles. Moreover, there is a polar angleθ+, which can be defined as the angle between the line

OA1+and the axisv+, whereA1+is the projection of the pointAon the 3-dimensional space generated by the axesv1,v1,v˜ +,

tanθ+= √

2ρ1

ν1

O ν+

θ+ A1+

˜

ν1

ρ1

A1 ν1

O ν−

θ− A1−

˜

ν1

ρ1

A1

O

˜

νk Ak

νk φk ρk

O Ak

ρk ψk−1

A1

A1k

ρ1

Figure2.2. Angular variables for the description ofn-complex numbers.

where 0≤θ+≤π, and in an even number of dimensionsnthere is also a polar angle

θ−, which can be defined as the angle between the lineOA1−and the axisv−, where

A1−is the projection of the pointAon the 3-dimensional space generated by the axes

v1,v1,v˜ −,

tanθ−= √

2ρ1

v− , (2.15)

where 0≤θ−≤π. Thus, the position of the pointAis described, in an even number of dimensions, by the distanced, byn/2−1 azimuthal angles, byn/2−2 planar angles, and by 2 polar angles. In an odd number of dimensions, the position of the pointAis described by(n−1)/2 azimuthal angles, by(n−3)/2 planar angles, and by 1 polar an-gle. These angles are shown inFigure 2.2. The variablesν,ρ,ρk,tanθ+/√2,tanθ−/√2, tanψkare multiplicative and the azimuthal anglesφkare additive upon the multipli-cation of polarn-complex numbers.

2.3. Then-dimensional polar cosexponential functions. The exponential function of the polarn-complex variableucan be defined by the series expu=1+u+u2_/2!+ u3_{/3!+···. It can be checked by direct multiplication of the series that exp(u+u)=} expu·expu_{, so that expu=expx0·exp(h1x1)···exp(h}

n−1xn−1). It can be seen with the aid of the representation inFigure 2.1that

hn_k+p=hp_k (2.16)

forpinteger,k=1,...,n−1. Thenehky_{can be written as}

ehky₌ n−1

p=0

where the functions gnl, which will be called polar cosexponential functions in n dimensions, are

gnl(y)= ∞

p=0 yl+pn

(l+pn)! (2.18)

forl=0,1,...,n−1. Ifnis even, the polar cosexponential functions of even indexk are even functions,gn,2l(−y)=gn,2l(y), and the polar cosexponential functions of odd index are odd functions,gn,2l+1(−y)= −gn,2l+1(y),l=0,1,...,n/2−1. For odd values ofn, the polar cosexponential functions do not have a definite parity. It can be checked that

n−1

l=0

gnl(y)=ey (2.19)

and, for evenn,

n−1

l=0 (−1)k_g

nl(y)=e−y. (2.20)

The expression of the polarn-dimensional cosexponential functions is

gnk(y)=_n1 n−1

l=0 exp

ycos

_2πl

n

cos

ysin

_2πl

n

−2πkl_n

(2.21)

fork=0,1,...,n−1. It can be shown from (2.21) that n−1

k=0 g2

nk(y)=_n1 n−1

l=0 exp

2ycos

_2πl

n

. (2.22)

It can be seen that the right-hand side of (2.22) does not contain oscillatory terms. If nis a multiple of 4, it can be shown by replacingybyiyin (2.22) that

n−1

k=0

(−1)k_g2

nk(y)=_n2

 

1+cos2y+

n/4−1

l=1 cos

2ycos

_2πl

n

_

 (2.23)

which does not contain exponential terms.

Addition theorems for the polarn-dimensional cosexponential functions can be obtained from the relation exph1(y+z)=exph1y·exph1z, by substituting the ex-pression of the exponentials as given byeh1y₌n−1

p=0hpgnp(y), gnk(y+z)=gn0(y)gnk(z)+gn1(y)gn,k−1(z)+···+gnk(y)gn0(z)

+gn,k+1(y)gn,n−1(z)+gn,k+2(y)gn,n−2(z)+···+gn,n−1(y)gn,k+1(z) (2.24)

fork=0,1,...,n−1. It can also be shown that

gn0(y)+h1gn1(y)+···+hn−1gn,n−1(y)l

=gn0(ly)+h1gn1(ly)+···+hn−1gn,n−1(ly).

(2.25)

The polarn-dimensional cosexponential functions are solutions of thenth-order differential equation

dn_ζ

whose solutions are of the formζ(u)=A0gn0(u)+A1gn1(u)+···+An−1gn,n−1(u). It can be checked that the derivatives of the polar cosexponential functions are re-lated by

dgn0

du =gn,n−1, dgn1

du =gn0, ...,

dgn,n−2

du =gn,n−3,

dgn,n−1

du =gn,n−2. (2.27) Forn=1 andn=2 the polar cosexponential functions areg10(y)=ey_,g20(y)= coshy, and g21(y)=sinhy. For n=3 the cosexponential functions areg30(y)= 1+y3_/3!+y6_{/6!+ ···,g31(y)=y+y}4_/4!+y7_{/7!+ ···,g32(y)=y}2_/2!+y5_/5!+ y8_{/8!+···, and they fulfill the identityg}3

30+g331+g303 −3g30g31g32=1.

2.4. Exponential and trigonometric forms of polarn-complex numbers. In order to obtain the exponential and trigonometricforms of polar n-complex numbers, a new set of hypercomplex bases will be introduced for evennby the relations

e+=_n1 n−1

p=0

hp, ek=_n2 n−1

p=0 hpcos

_2πkp

n

, ˜ek=_n2 n−1

p=0 hpsin

_2πkp

n

, (2.28)

wherek=1,...,[(n−1)/2]and, ifnis even,

e−=_n1 n−1

p=0 (−1)p_h

p. (2.29)

The multiplication relations for the new hypercomplex bases are

e2

+=e+, e−2=e−, e+e−=0, e+ek=0, e+˜ek=0, e−ek=0, e−e˜k=0, e2

k=ek, e˜2k= −ek, ek˜ek=e˜k, ekel=0, ek˜el=0, e˜k˜el=0, k=l, (2.30) wherek,l=1,...,[(n−1)/2]. It can be shown that, for evenn,

u=e+v++e−v−+ n/2−1

k=1

ekvk+e˜kv˜k, (2.31)

and for oddn,

u=e+v++ (n−1)/2

k=1

ekvk+e˜kv˜k. (2.32)

The exponential form of the polarn-complex numberuis

u=ρexp

    

n−1

p=1 hp

 1

nln √

2

tanθ++F(n)

(−1)p n ln

√ 2 tanθ−

−2 n

[(n−1)/2]

k=2

cos2πkp n

lntanψk−1



₊[(n−1)/2]

k=1 ˜ ekφk

    ,

(2.33)

whereF(n)=1 for evennandF(n)=0 for oddn, and

for evenn, and

ρ=v+ρ21···ρ(n2−1)/2

1/n _(2.35)

for oddn.

The trigonometricform of the polarn-complex numberuis

u=dn 2

1/2 ₁ tan2_ψ

++

F(n) tan2_ψ

−+1+ 1

tan2_ψ1+_tan12_ψ2+···+_tan2_ψ1 [(n−3)/2]

−1/2

×

 e+√2

tanθ++F(n)

e−√2 tanθ−+e1+

[(n−1)/2]

k=2 ek tanψk−1

 _exp

 [(n−1)/2]

k=1 ˜ ekφk

 _.

(2.36) 2.5. Elementary functions of a polarn-complex variable. The logarithmu1of the polarn-complex numberu,u1=lnu, can be defined as the solution of the equation u=eu1_{. For evenn, lnuexists as an n-complex function with real components if} v+>0 andv−>0. For oddnlnuexists as ann-complex function with real compo-nents ifv+>0. The expression of the logarithm is

lnu=lnρ+ n−1

p=1 hp  1 nln √ 2

tanθ++F(n)

(−1)p n ln

√ 2 tanθ−

−_n2 [(n−1)/2]

k=2 cos

_2πkp

n

lntanψk−1



₊[(n−1)/2]

k=1 ˜ ekφk.

(2.37)

The function lnuis multivalued because of the presence of the terms ˜ekφk.

The power functionum_{of the polarn-complex variableucan be defined for real} values ofmasum_=emlnu_{. It can be shown that}

um_=e

+v+m+F(n)e−v−m+ [(n−1)/2]

k=1 ρm

k

ekcosmφk+˜eksinmφk. (2.38)

For integer values ofm, this expression is valid for anyx0,...,xn−1. The power func-tion is multivalued unlessmis an integer.

2.6. Power series of polar n-complex numbers. A power series of the polarn -complex variableuis a series of the form

a0+a1u+a2u2_+···+a

lul+···. (2.39)

Using the inequality

_{uu ≤}√_{n u u (2.40)} which replaces the relation of equality extant for 2-dimensional complex numbers, it can be shown that the series (2.39) is absolutely convergent for |u|< c, where c=liml→∞|al|/√n|al+1|.

The convergence of the series (2.39) can also be studied with the aid of the formulas (2.38), which for integer values ofmare valid for any values ofx0,...,xn−1. Ifal=

_n−1

p=0hpalp, and

Al+= n−1

p=0

alp, Alk= n−1

p=0 alpcos

_2πkp

n

, A˜lk= n−1

p=0 alpsin

_2πkp

n

fork=1,...,[(n−1)/2], and for evenn

Al−= n−1

p=0 (−1)p_a

lp, (2.42)

the series (2.39) can be written as ∞

l=0

 _e+Al+vl

++F(n)e−Al−v−l+ [(n−1)/2]

k=1

ekAlk+e˜kA˜lkekvk+e˜kv˜kl



_{. (2.43)}

The series in (2.39) is absolutely convergent for

|v+|< c+, |v−|< c−, ρk< ck, (2.44) fork=1,...,[(n−1)/2], where

c+=lim l→∞

|Al+|

|Al+1,+|, c−=liml→∞ |Al−|

|Al+1,−|, ck=liml→∞

A2 lk+A˜2lk

1/2

A2

l+1,k+A˜2l+1,k

1/2. (2.45)

These relations show that the region of convergence of the series (2.39) is an n -dimensional cylinder.

2.7. Analytic functions of polarn-complex variables. The derivative of a function f (u)of then-complex variableuis defined as a functionf_{(u)having the property} that

_{f (u)−f}

u0−f_{u0u−u0 →0 as u−u0 →0. (2.46)} If the differenceu−u0is not parallel to one of the nodal hypersurfaces, the definition in (2.46) can also be written as

f_{u0= lim} u→u0

f (u)−fu0

u−u0 . (2.47)

The derivative of the functionf (u)=um_{, withman integer, isf(u)=mu}m−1_{, as} can be seen by developingum_{=[u0+(u−u0)]}m_as

um₌m p=0

m! p!(m−p)!u

m−p 0

u−u0p, (2.48)

and using the definition (2.46).

If the functionf_{(u)defined in (2.46) is independent of the direction in space along} whichuis approachingu0, the functionf (u)is said to be analytic, analogously to the case of functions of regular complex variables [14]. The function um_{, withm} an integer, of then-complex variableuis analytic, because the differenceum_−um 0 is always proportional tou−u0, as can be seen from (2.48). Then series of integer powers ofuwill also be analyticfunctions of then-complex variableu, and this result holds in fact for any commutative algebra.

If the n-complex function f (u)of the polar n-complex variable u is written in terms of the real functionsPk(x0,...,xn−1), k=0,1,...,n−1 of the real variables

x0,x1,...,xn−1as

f (u)= n−1

k=0

then relations of equality exist between the partial derivatives of the functionsPk. The derivative of the functionfcan be written as

lim ∆u→0

1 ∆u

n−1

k=0

 _hkn−1

l=0 ∂Pk ∂xl∆xl



_{, (2.50)}

where∆u=nk=−01hl∆xl.

The relations between the partial derivatives of the functionsPkare obtained by set-ting successively in (2.50)∆u=hl∆xl, forl=0,1,...,n−1, and equating the resulting expressions. The relations are

∂Pk ∂x0=

∂Pk+1

∂x1 = ··· = ∂Pn−1 ∂xn−k−1=

∂P0

∂xn−k= ··· = ∂Pk−1

∂xn−1 (2.51)

fork=0,1,...,n−1. The relations (2.51) are analogous to the Riemann relations for the real and imaginary components of a complex function. It can be shown from (2.51) that the componentsPkfulfill the second-order equations

∂2_Pk ∂x0∂xl=

∂2_Pk

∂x1∂xl−1= ··· =

∂2_Pk ∂x[l/2]∂xl−[l/2] =

∂2_Pk ∂xl+1∂xn−1

=_∂x∂2Pk

l+2∂xn−2= ··· =

∂2_Pk

∂xl+1+[(n−l−2)/2]∂xn−1−[(n−l−2)/2]

(2.52)

fork,l=0,1,...,n−1.

2.8. Integrals of polarn-complex functions. The singularities of polarn-complex functions arise from terms of the form 1/(u−u0)m_{, withm >0. Functions containing} such terms are singular not only atu=u0, but also at all points of the hypersurfaces passing throughu0and which are parallel to the nodal hypersurfaces.

The integral of a polarn-complex function between two pointsA,Balong a path situated in a region free of singularities is independent of path, which means that the integral of an analyticfunction along a loop situated in a region free of singularities

is zero, _!

Γf (u)du=0, (2.53) where it is supposed that a surfacespanning the closed loopΓ is not intersected by any of the hypersurfaces associated with the singularities of the function f (u). Using the expression, equation (2.49), forf (u)and the fact thatdu=kn=−01hkdxk, the explicit form of the integral in (2.53) is

!

Γf (u)du=

!

Γ n−1

k=0 hk

n−1

l=0

Pldxk−l+n[(n−k−1+l)/n]. (2.54)

ξk O

u0 Γ

u0ξkηk Γξkηk

ηk

Figure2.3. Integration pathΓand poleu0, and their projectionsΓξkηkand

u0ξkηkon the planeξkηk.

The quantitydu/(u−u0)is

du u−u0=

dρ ρ +

n−1

p=1 hp

 1

ndln √

2

tanθ++F(n)

(−1)p n dln

√ 2 tanθ−

−_n2

[(n−1)/2]

k=2 cos

_2πkp

n

dlntanψk−1



₊[(n−1)/2]

k=1 ˜ ekdφk.

(2.55)

Since ρ,ln(√2/tanθ+),ln(√2/tanθ−), and ln(tanψk−1) are single-valued variables, it follows that "Γdρ/ρ = 0,

"

Γd(ln √

2/tanθ+) = 0, "Γd(ln √

2/tanθ−) = 0, and

"

Γd(lntanψk−1)=0. On the other hand, sinceφkare cyclic variables, they may give contributions to the integral around the closed loopΓ.

The expression of"Γdu/(u−u0)can be written with the aid of a functional which will be called int(M,C), defined for a pointMand a closed curveCin a 2-dimensional plane, such that

int(M,C)=

 

1 if_{0 if}M_Mis an interior point of_{is exterior toC.} C, (2.56)

Iff (u)is an analyticfunction of a polarn-complex variable which can be expanded in a series in the region of the curveΓ and on a surface spanningΓ, then

!

Γ

f (u)du u−u0 =2πf

u0 [(n−1)/2]

k=1 ˜

ekintu0ξkηk,Γξkηk

, (2.57)

whereu0ξkηkandΓξkηkare respectively the projections of the pointu0and of the loop Γ on the plane defined by the axesξkandηk, as shown inFigure 2.3.

2.9. Factorization of polarn-complex polynomials. A polynomial of degreemof the polarn-complex variableuhas the form

whereal,l=1,...,m, are in general polarn-complex constants. Ifal=np−=10hpalp, and with the notations of (2.41) and (2.42) applied forl=1,...,m, the polynomial Pm(u)can be written as

Pm=e+

 _vm

+ + m

l=1

Al+v+m−l



_+F(n)e−  _vm

− + m

l=1

Al−v−m−l

 

+ [(n−1)/2]

k=1



_{ekvk+˜ekv˜k}m₊m l=1

ekAlk+e˜kA˜lkekvk+e˜kv˜km−l

 _,

(2.59)

where the constantsAl+,Al−,Alk,A˜lkare real numbers.

These relations can be written with the aid of (2.28) and (2.29) as

Pm(u)= m

p=1

u−up, (2.60)

where

up=e+vp++F(n)e−vp−+ [(n−1)/2]

k=1

ekvkp+˜ekv˜kp, (2.61)

forp=1,...,m. The rootsvp+, the rootsvp−and, for a givenk, the rootsekvkp+˜ekv˜kp defined in (2.59) may be ordered arbitrarily. This means that (2.61) gives sets ofm rootsu1,...,umof the polynomialPm(u), corresponding to the various ways in which the rootsvp+,vp−,ekvkp+e˜kv˜kpare ordered according topin each group. Thus, while the polar hypercomplex components in (2.59) taken separately have unique factoriza-tions, the polynomialPm(u)can be written in many different ways as a product of linear factors.

For example,u2_{−1=(u−u1)(u−u2), where for even n, u1= ±e+±e−±e1±} e2± ··· ±en/2−1, u2= −u1, so that there are 2n/2independent sets of rootsu1,u2 ofu2_{−1. It can be checked that(±e}

+±e−±e1±e2± ··· ±en/2−1)2=e++e−+e1+

e2+ ··· +en/2−1=1. For oddn,u1= ±e+±e1±e2± ··· ±e(n−1)/2,u2= −u1, so that there are 2(n−1)/2_{independent sets of rootsu1,u2 ofu}2_{−1. It can be checked that} (±e+±e1±e2±···±e(n−1)/2)2=e++e1+e2+···+e(n−1)/2=1.

2.10. Representation of polarn-complex numbers by irreducible matrices. The polarn-complex numberucan be represented by the matrix

U=

         

x0 x1 x2 ··· xn−1 xn−1 x0 x1 ··· xn−2 xn−2 xn−1 x0 ··· xn−3

..

. ... ... ··· ... x1 x2 x3 ··· x0

         

. (2.62)

coefficients is, for evenn,

         

v+ 0 0 ··· 0

0 v− 0 ··· 0

0 0 V1 ··· 0

..

. ... ... ··· ... 0 0 0 ··· Vn/2−1

         

(2.63)

and, for oddn, _

        

v+ 0 0 ··· 0

0 V1 0 ··· 0

0 0 V2 ··· 0

..

. ... ... ··· ... 0 0 0 ··· V(n−1)/2

         

, (2.64)

where

Vk=

vk v˜k −v˜k vk

, (2.65)

k=1,...,[(n−1)/2]. The relations between the variablesv+,v−,vk,v˜kfor the multi-plication of polarn-complex numbers arev+=v+v+,v−=v−v−,vk=vkvk−v˜kv˜k, ˜

vk=vkv˜k+v˜kvk.

3. Planar hypercomplex numbers in evenndimensions

3.1. Operations with planarn-complex numbers. A planar hypercomplex number inndimensions is determined by itsncomponents(x0,x1,...,xn−1). The planarn -complex numbers and their operations discussed in this paper can be represented by writing then-complex number(x0,x1,...,xn−1)asu=x0+h1x1+h2x2+ ··· +

hn−1xn−1, whereh1,h2,...,hn−1are bases for which the multiplication rules are

hjhk=(−1)[(j+k)/n]hl, l=j+k−n(j+k)/n, (3.1)

forj,k,l=0,1,...,n−1, whereh0=1. The rules for the planar bases differ from the rules for the polar bases by the minus sign which appears whenn≤j+k≤2n−2. The significance of the composition laws in (3.1) can be understood by representing the baseshj,hkby points on a circle at the anglesαj=πj/n,αk=πk/n, as shown in Figure 3.1, and the producthjhkby the point of the circle at the angleπ(j+k)/n. If π≤π(j+k)/n <2π, the point is opposite to the basishlof angleαl=π(j+k)/n−π. In an odd number of dimensions n, a transformation of coordinates according tox2l=xl, x2m−1= −x(n−1)/2+m, and of the bases according to 2l=hl, h2m−1= −h

(n−1)/2+m,l=0,...,(n−1)/2,m=1,...,(n−1)/2, leaves the expression of ann -complex number unchanged,nk=−01hkxk=nk=−01hkxk, and the products of the bases h

−hn−2

−hn−1 1

h1

h2

hn−2

hn−1

−1

−h1

−h2

α1

Figure3.1. Representation of the planarn-complex bases 1,h1,...,hn−1by points on a circle at the anglesαk=πk/n.

Two planarn-complex numbersu=x0+h1x1+h2x2+ ··· +hn−1xn−1,u=x0+ h1x

1+h2x2+ ··· +hn−1xn−1are equal if and only ifxj=xj,j=0,1,...,n−1. The sum of the planarn-complex numbersuandu_is

u+u_=x0+x

0+h1x1+x1+···+hn−1xn−1+xn−1. (3.2)

The product of the planar numbersu,u_is

uu_=x0x

0−x1xn−1−x2xn−2−x3xn−3−···−xn−1x1 +h1x0x

1+x1x0−x2xn−1−x3xn−2−···−xn−1x2 +h2x0x

2+x1x1+x2x0−x3xn−1−···−xn−1x3

.. .

+hn−1x0xn−1+x1xn−2+x2xn−3+x3xn−4+···+xn−1x0

.

(3.3)

The productuu_{can be written as}

uu₌n−1 k=0

hk n−1

l=0

(−1)[(n−k−1+l)/n]_x

lxk−l+n[(n−k−1+l)/n]. (3.4)

If u,u_{,u are n-complex numbers, the multiplication is associative, (uu)u =}

u(u_{u), and is commutative, uu = uu, because the product of the bases,} de-fined in (3.1), is both associative and commutative. Forn=2 the product is uu₌

x0x

0−x1x1+h1(x0x1+x1x0). In 2 dimensions, the notation forh1ish1=i,ibeing the conventional imaginary unit.

It can be shown that

ν= n/2

k=1 ρ2

k, (3.5)

where

ρ2

k=vk2+˜vk2, vk= n−1

p=0 xpcos

_π(2k−1)p

n

, v˜k= n−1

p=0 xpsin

_π(2k−1)p

n

. (3.6)

Thus, a planarn-complex number has an inverse unless it lies on one of the nodal hypersurfacesρ1=0, orρ2=0, or...orρn/2=0. It can also be shown that

d2₌ 2 n

n/2

k=1 ρ2

k. (3.7)

From this relation it results that if the product of twon-complex numbers is zero, uu_{=0, then ρ}

kρk=0, k=1,...,n/2, which means that either u=0, or u=0, oru,u_{belong to orthogonal hypersurfaces in such a way that the afore-mentioned} products of components should be equal to zero. Forn=2,v1=x0, ˜v1=x1, that is, v1and ˜v1are the real and imaginary parts of a 2-dimensional complex number.

3.2. Geometric representation of planarn-complex numbers. The planarn -com-plex numberx0+h1x1+h2x2+ ··· +hn−1xn−1 can be represented by the pointA of coordinates (x0,x1,...,xn−1). If O is the origin of the n-dimensional space, the distance from the originO to the point Aof coordinates (x0,x1,...,xn−1)has the expression written in (2.10). The quantitydwill be now called modulus of the planar n-complex number u=x0+h1x1+h2x2+ ··· +hn−1xn−1. The modulus of an n -complex numberuwill be designated byd= |u|. The quantityρ=ν1/n_{will be called} amplitude of then-complex numberu.

The exponential and trigonometricforms of the planarn-complex numberucan be obtained conveniently in a rotated system of axes defined by the transformation

vk= n₂ξk, v˜k= n₂ηk, (3.8)

fork=1,...,n/2. This transformation from the coordinatesx0,...,xn−1to the vari-ablesξk,ηkis unitary.

The position of the pointAof coordinates(x0,x1,...,xn−1)can also be described with the aid of the distanced, equation (2.10), and ofn−1 angles defined further. Thus, in the plane of the axesvk,v˜k, the radiusρkand the azimuthal angleφkcan be introduced by the relations

cosφk=v_ρk

k, sinφk= ˜ vk

ρk, (3.9)

ψk−1between the lineOA1kand the 2-dimensional plane defined by the axesvk,v˜kis

tanψk−1=ρ1_ρ

k, (3.10)

where 0≤ψk≤π/2, k=2,...,n/2, so that there aren/2−1 planar angles. Thus, the position of the pointAis described by the distanced, byn/2 azimuthal angles and byn/2−1 planar angles. The variablesν,ρ,ρk,tanψkare multiplicative and the azimuthal anglesφkare additive upon the multiplication of polarn-complex numbers. 3.3. The planarn-dimensional cosexponential functions. The exponential func-tion of the planar n-complex variableu can be defined by the series expu= 1+ u+u2_/2!+u3_{/3!+ ···. It can be checked by direct multiplication of the series that} exp(u+u_{)=expu·expu, so that expu=expx0·exp(h1x1)···exp(h}

n−1xn−1). It can be seen with the aid of the representation inFigure 3.1that

hnk+p= −hpk, (3.11)

for integerp,k=1,...,n−1. Forkeven,ehky_{can be written as}

ehky₌ n−1

p=0

(−1)[kp/n]_h

kp−n[kp/n]gnp(y), (3.12)

wheregnpare the polarn-dimensional cosexponential functions. For oddk,ehkyis

ehky₌ n−1

p=0

(−1)[kp/n]_h

kp−n[kp/n]fnp(y), (3.13)

where the functionsfnk, which will be called planar cosexponential functions inn dimensions, are

fnk(y)= ∞

p=0

(−1)p yk+pn

(k+pn)!, (3.14)

fork=0,1,...,n−1.

The planar cosexponential functions of even indexkare even functions,fn,2l(−y)= fn,2l(y), and the planar cosexponential functions of odd index are odd functions, fn,2l+1(−y)= −fn,2l+1(y),l=0,...,n/2−1.

The planarn-dimensional cosexponential functionfnk(y)is related to the polar n-dimensional cosexponential functiongnk(y)by the relation

fnk(y)=e−iπk/ngnkeiπ/ny, (3.15)

fork=0,...,n−1. The expression of the planarn-dimensional cosexponential func-tions is then

fnk(y)=_n1 n

l=1 exp

ycosπ(2l−1) n

cos

ysinπ(2l−1) n

−π(2l−1)k n

, (3.16)

It can be shown from (3.16) that

n−1

k=0 f2

nk(y)=_n1 n

l=1 exp

2ycosπ(2l−1) n

. (3.17)

It can be seen that the right-hand side of (3.17) does not contain oscillatory terms. If nis a multiple of 4, it can be shown by replacingybyiyin (3.17) that

n−1

k=0

(−1)k_f2

nk(y)=_n4 n/4

l=1 cos

2ycos

_π(2l−1)

n

, (3.18)

which does not contain exponential terms.

For oddn, the planarn-dimensional cosexponential functionfnk(y)is related to then-dimensional cosexponential functiongnk(y)also by the relation

fnk(y)=(−1)kgnk(−y), (3.19)

as can be seen by comparing the series for the two classes of functions.

Addition theorems for the planarn-dimensional cosexponential functions can be obtained from the relation exph1(y+z)=exph1y·exph1z, by substituting the ex-pression of the exponentials as given byeh1y₌n−1

p=0hpfnp(y),

fnk(y+z)=fn0(y)fnk(z)+fn1(y)fn,k−1(z)+···+fnk(y)fn0(z)

−fn,k+1(y)fn,n−1(z)−fn,k+2(y)fn,n−2(z)−···−fn,n−1(y)fn,k+1(z), (3.20)

fork=0,1,...,n−1. It can also be shown that

fn0(y)+h1fn1(y)+···+hn−1fn,n−1(y)l=fn0(ly)+h1fn1(ly)+···+hn−1fn,n−1(ly). (3.21) Forn=2, equation (3.13) has the formeh1y_{=f20+h1f21, and the planar} cosexpo-nential functions aref20(y)=cosyandf21(y)=siny.

The planarn-dimensional cosexponential functions are solutions of thenth-order differential equation

dn_ζ

dun= −ζ (3.22)

whose solutions are of the formζ(u)=A0fn0(u)+A1fn1(u)+ ··· +An−1fn,n−1(u). It can be checked that the derivatives of the planar cosexponential functions are related by

dfn0

du = −fn,n−1, dfn1

du =fn0, ...,

dfn,n−2

du =fn,n−3,

dfn,n−1

du =fn,n−2. (3.23) 3.4. Exponential and trigonometric forms of planarn-complex numbers. In order to obtain the exponential and trigonometricforms of planarn-complex numbers, a new set of hypercomplex bases will be introduced by the relations

ek=_n2 n−1

p=0 hpcos

_π(2k−1)p

n

, ˜ek=_n2 n−1

p=0 hpsin

_π(2k−1)p

n

fork=1,...,n/2. The multiplication relations for the new hypercomplex bases are

e2

k=ek, e˜k2= −ek, ek˜ek=˜ek, ekel=0, ek˜el=0, e˜k˜el=0, k=l, (3.25)

fork,l=1,...,n/2. It can be shown that

x0+h1x1+···+hn−1xn−1= n/2

k=1

ekvk+˜ekv˜k. (3.26)

The exponential form of the planarn-complex numberuis

u=ρexp

    

n−1

p=1 hp

 ₋2

n n/2

k=2

cosπ(2k−1)p n

lntanψk−1

 ₊n/2

k=1 ˜ ekφk

   

, (3.27)

where the amplitude is

ρ=ρ2

1···ρ2n/21/n. (3.28)

The trigonometricform of the planarn-complex numberuis

u=d

_n

2

1/2

1+_tan1_2ψ1+_tan1_2ψ2+···+_tan2ψ1 n/2−1

−1/2

×

 _e1+n/2

k=2 ek tanψk−1

 _exp

 n/2

k=1 ˜ ekφk

 _.

(3.29)

Forn=2,e1=1, ˜e1=h1,ρ=d, there is no planar angle, and (3.27) and (3.29) have both the formu=ρexp(h1φ1).

3.5. Elementary functions of a planarn-complex variable. The logarithmu1 of the planarn-complex numberu,u1=lnu, can be defined as the solution of the equa-tionu=eu1_{. The logarithm exists as a planarn-complex function with real} compo-nents for all values ofx0,...,xn−1for whichρ=0. The expression of the logarithm is

lnu=lnρ+ n−1

p=1 hp

 ₋2

n n/2

k=2 cos

_π(2k−1)p

n

lntanψk−1

 ₊n/2

k=1 ˜

ekφk. (3.30)

The function lnuis multivalued because of the presence of the terms ˜ekφk.

The power functionum_{of the planarn-complex variableucan be defined for real} values ofmasum_=emlnu_{. It can be shown that}

um₌n/2 k=1

ρm k

ekcosmφk+˜eksinmφk. (3.31)

The power function is multivalued unlessmis an integer.

3.6. Power series of planarn-complex numbers. A power series of the planarn -complex variableuis a series of the form

a0+a1u+a2u2_+···+a

Using the inequality

_{uu ≤} n

2 u u , (3.33)

which replaces the relation of equality extant for 2-dimensional complex numbers, it can be shown that the series (3.32) is absolutely convergent for |u|< c, where c=liml→∞|al|/%n/2|al+1|.

The convergence of the series (3.32) can be also studied with the aid of the formulas (3.31), which for integer values ofmare valid for any values ofx0,...,xn−1. Ifal=

n−1

p=0hpalp, and

Alk= n−1

p=0

alpcosπ(2k_n−1)p, A˜lk= n−1

p=0

alpsinπ(2k_n−1)p, (3.34)

fork=1,...,n/2, the series (3.32) can be written as ∞

l=0 n/2

k=1

ekAlk+˜ekA˜lkekvk+˜ekv˜kl. (3.35)

The series in (3.32) is absolutely convergent for

ρk< ck, (3.36)

fork=1,...,n/2, where

ck=lim l→∞

A2 lk+A˜2lk

1/2

A2

l+1,k+A˜2l+1,k

1/2. (3.37)

These relations show that the region of convergence of the series (3.32) is an n -dimensional cylinder.

3.7. Analytic functions of planarn-complex variables. If then-complex function f (u)of the planar n-complex variable uis written in terms of the real functions Pk(x0,...,xn−1),k=0,1,...,n−1 of the real variablesx0,x1,...,xn−1as

f (u)= n−1

k=0

hkPkx0,...,xn−1, (3.38)

then relations of equality exist between the partial derivatives of the functionsPk, ∂Pk

∂x0= ∂Pk+1

∂x1 = ··· = ∂Pn−1 ∂xn−k−1= −

∂P0

∂xn−k= ··· = − ∂Pk−1

∂xn−1, (3.39) fork=0,1,...,n−1. The relations (3.39) are analogous to the Riemann relations for the real and imaginary components of a complex function. It can be shown from (3.39) that the componentsPkfulfill the second-order equations

∂2_P k ∂x0∂xl=

∂2_P k

∂x1∂xl−1= ··· =

∂2_P k

∂x[l/2]∂xl−[l/2] = − ∂2_P

k ∂xl+1∂xn−1

= −_∂x∂2Pk

l+2∂xn−2= ··· = −

∂2_P k

∂xl+1+[(n−l−2)/2]∂xn−1−[(n−l−2)/2],

(3.40)

3.8. Integrals of planar n-complex functions. The singularities of planar n -complex functions arise from terms of the form 1/(u−u0)m_{, withm >0. Functions} containing such terms are singular not only atu=u0, but also at all points of the hypersurfaces passing throughu0and which are parallel to the nodal hypersurfaces. The integral of a planarn-complex function between two pointsA,Balong a path situated in a region free of singularities is independent of path, which means that the integral of an analyticfunction along a loop situated in a region free of singularities

is zero, _!

Γf (u)du=0, (3.41) where it is supposed that a surfacespanning the closed loopΓ is not intersected by any of the hypersurfaces associated with the singularities of the function f (u). Using the expression, equation (3.38), forf (u)and the fact thatdu=kn=−01hkdxk, the explicit form of the integral in (3.41) is

!

Γf (u)du=

!

Γ n−1

k=0 hk

n−1

l=0

(−1)[(n−k−1+l)/n]_P

ldxk−l+n[(n−k−1+l)/n]. (3.42)

If the functionsPkare regular on a surfacespanning the loopΓ, the integral along the loop Γ can be transformed in an integral over the surface of terms of the form∂Pl/∂xk−m+n[(n−k+m−1)/n] −(−1)s∂Pm/∂xk−l+n[(n−k+l−1)/n], wheres=[(n−k+ m−1)/n]−[(n−k+l−1)/n]. The integrals of these terms are equal to zero by (3.39), and this proves (3.41).

The quantitydu/(u−u0)is

du u−u0=

dρ ρ +

n−1

p=1 hp

 ₋2

n n/2

k=2 cos

_2πkp

n

dlntanψk−1

 ₊n/2

k=1 ˜

ekdφk. (3.43)

Sinceρand ln(tanψk−1)are single-valued variables, it follows that"Γdρ/ρ=0, and

"

Γd(lntanψk−1)=0. On the other hand, sinceφkare cyclic variables, they may give contributions to the integral around the closed loopΓ.

Iff (u)is an analyticfunction of a polarn-complex variable which can be expanded in a series which holds on the curveΓ and on a surface spanningΓ, then

!

Γ

f (u)du u−u0 =2πf

u0 n/2

k=1 ˜

ekintu0ξkηk,Γξkηk

. (3.44)

Forn=2, equation (3.44) becomes"Γf (u)du/(u−u0)=2πh1f (u0)int(u0,Γ), which is the theorem of Cauchy for 2-complex numbers.

3.9. Factorization of planarn-complex polynomials. A polynomial of degreem of the planarn-complex variableuhas the form

Pm(u)=um+a1um−1+···+am−1u+am, (3.45)

written as

Pm= n/2

k=1

 

ekvk+e˜kv˜km+ m

l=1

ekAlk+e˜kA˜lkekvk+e˜kv˜km−l

 

, (3.46)

where the constantsAlk,A˜lkare real numbers.

These relations can be written with the aid of (3.24) as

Pm(u)= m

p=1

u−up, (3.47)

where

up= n/2

k=1

ekvkp+e˜kv˜kp, (3.48)

forp=1,...,m. For a givenk, the rootsekvkp+e˜kv˜kpdefined in (3.46) may be ordered arbitrarily. This means that (3.48) gives sets ofmrootsu1,...,umof the polynomial Pm(u), corresponding to the various ways in which the rootsekvkp+˜ekv˜kpare ordered according topfor each value ofk. Thus, while the planar hypercomplex components in (3.46) taken separately have unique factorizations, the polynomial Pn(u)can be written in many different ways as a product of linear factors.

For example,u2_{+1=(u−u1)(u−u2), whereu1= ±˜e1±˜e2±···±e˜}

n/2,u2= −u1, so that there are 2n/2−1_{independent sets of rootsu1,u2ofu}2_{+1. It can be checked} that(±˜e1±˜e2±···±e˜n/2)2= −e1−e2−···−en/2= −1.

3.10. Representation of planarn-complex numbers by irreducible matrices. The planarn-complex numberucan be represented by the matrix

U=

         

x0 x1 x2 ··· xn−1 −xn−1 x0 x1 ··· xn−2 −xn−2 −xn−1 x0 ··· xn−3

..

. ... ... ··· ... −x1 −x2 −x3 ··· x0

         

. (3.49)

The productu=u_{u is, for evenn, represented by the matrix multiplicationU=}

U_{U. It can be shown that the irreducible form [15] of the matrixU, in terms of} matrices with real coefficients, is

      

v+ 0 ··· 0

0 V1 ··· 0

..

. ... ··· ... 0 0 ··· Vn/2

     

, (3.50)

where

Vk=

vk v˜k −v˜k vk

, (3.51)

4. Conclusions. The polar and planar n-complex numbers described in this pa-per have a geometricrepresentation based on modulus, amplitude, and angular vari-ables. Then-complex numbers have exponential and trigonometric forms, which can be expressed with the aid of geometricvariables. The exponential function of ann -complex variable can be developed in terms of the cosexponential functions. Then -complex functions defined by series of powers are analytic, and the partial derivatives of the real components ofn-complex functions are closely related. The integrals of n-complex functions are independent of path in regions where the functions are reg-ular. The fact that the exponential form of then-complex numbers depends on the cyclic azimuthal variables leads to the concept of pole and residue forn-complex in-tegrals on closed paths. The polynomials of polarn-complex variables can be written as products of linear or quadratic factors, and the polynomials of planarn-complex variables can be written as products of linear factors.

References

[1] G. Birkhoff and S. MacLane,Modern Algebra, 3rd ed., The Macmillan Co., New York, 1965. [2] D. M. J. Calderbank and P. Tod,Einstein metrics, hypercomplex structures and the Toda

field equation,http://mentor.lanl.gov/abs/math.DG/9911121.

[3] S. De Leo,Hypercomplex group theory,http://mentor.lanl.gov/abs/physics/9703033. [4] S. De Leo and P. Rotelli, Local hypercomplex analyticity, http://arxiv.org/abs/

funct-an/9703002.

[5] , A new definition of hypercomplex analyticity, http://arxiv.org/abs/ funct-an/9701004.

[6] J. D. E. Grant and I. A. B. Strachan, Hypercomplex integrable systems, Nonlinear-ity 12 (1999), no. 5, 1247–1261, http://mentor.lanl.gov/abs/solv-int/9808019. CMP 1 710 101. Zbl 934.35181.

[7] D. Kaledin, Integrability of the twistor space for a hypercomplex manifold, Selecta Math. (N.S.) 4 (1998), no. 2, 271–278, http://arxiv.org/abs/alg-geom/9612016. MR 2000c:53055. Zbl 906.53048.

[8] S. Olariu, Complex numbers in n dimensions, http://mentor.lanl.gov/abs/math.CV/ 0011044.

[9] L. Ornea and P. Piccinni,On some moment maps and induced Hopf bundles in the quater-nionic projective space,http://mentor.lanl.gov/abs/math.DG/0001066.

[10] K. Scheicher, R. F. Tichy, and K. W. Tomantschger, Elementary inequalities in hyper-complex numbers, Anz. Österreich. Akad. Wiss. Math.-Natur. Kl.134(1997), 3–10 (1998).MR 99j:30053. Zbl 927.30029.

[11] O. Taussky,Algebra, Handbook of Physics (E. U. Condon and H. Odishaw, eds.), McGraw-Hill, New York, 2nd ed., 1958, pp. 1–22.

[12] B. L. van der Waerden,Modern Algebra, 3rd ed., vol. II, Frederick Ungar Publishing Co., New York, 1950.

[13] M. Verbitsky,Hypercomplex varieties, Comm. Anal. Geom. 7 (1999), no. 2, 355–396, http://arxiv.org/abs/alg-geom/9703016.MR 2000m:32029. Zbl 935.32022. [14] E. T. Whittaker and G. N. Watson,A Course of Modern Analysis, 4th ed., Cambridge

Uni-versity Press, Cambridge, 1958.

[15] E. Wigner,Group Theory, Academic Press, New York, 1959.

Silviu Olariu: Institute of Physics and Nuclear Engineering, Tandem Laboratory,