Mathematical Methods for Physics and Engineering : A Comprehensive Guide

24.4 SOME ELEMENTARY FUNCTIONS

real-variable counterpart it is called theexponential function; also like its real

counterpart it is equal to its own derivative.

The multiplication of two exponential functions results in a further exponential

function, in accordance with the corresponding result for real variables.

Show thatexpz 1 expz 2 =exp(z 1 +z 2 ).

From the series expansion (24.15) of expz 1 and a similar expansion for expz 2 ,itisclear
that the coefficient ofzr 1 z 2 sin the corresponding series expansion of expz 1 expz 2 is simply
1 /(r!s!).
But, from (24.15) we also have

exp(z 1 +z 2 )=

∑∞

n=0

(z 1 +z 2 )n

n!

.

In order to find the coefficient ofzr 1 z 2 sin this expansion, we clearly have to consider the
term in whichn=r+s,namely

(z 1 +z 2 )r+s

(r+s)!

=

1

(r+s)!

(r+s

C 0 zr 1 +s+···+r+sCszr 1 zs 2 +···+r+sCr+sz 2 r+s

)

.

The coefficient ofz 1 rzs 2 in this is given by

r+sCs^1

(r+s)!

=

(r+s)!

s!r!

1

(r+s)!

=

1

r!s!

.

Thus, since the corresponding coefficients on the two sides are equal, and all the series
involved are absolutely convergent for allz, we can conclude that expz 1 expz 2 =exp(z 1 +
z 2 ).

As an extension of (24.15) we may also define the complex exponent of a real

numbera>0 by the equation

az= exp(zlna), (24.16)

where lnais the natural logarithm ofa. The particular casea=eand the fact

that lne= 1 enable us to write expzinterchangeably withez.Ifzis real then the

definition agrees with the familiar one.

The result forz=iy,

expiy=cosy+isiny, (24.17)

has been met already in equation (3.23). Its immediate extension is

expz=(expx)(cosy+isiny). (24.18)

Aszvaries over the complex plane, the modulus of expztakes all real positive

values, except that of 0. However, two values ofzthat differ by 2πki, for any

integerk, produce the same value of expz, as given by (24.18), and so expzis

periodic with period 2πi. If we denote expzbyt, then the strip−π<y≤πin

thez-plane corresponds to the whole of thet-plane, except for the pointt=0.

The sine, cosine, sinh and cosh functions of a complex variable are defined from

the exponential function exactly as are those for real variables. The functions