1550251515-Classical_Complex_Analysis__Gonzalez_

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from which we derive, by Theorem 1.11,

elog z+log z' = zz', elog z-log z' = z

z'

Chapter 1

This shows that a value of log( zz') is log z +log z', and that a value of

log( z / z') is log z - log z' for particular choices of values of log z and log z',

the results differing possibly by a multiple of 27ri.

It is easy to see that if we choose the principal values of log z and log z',

we do not obtain necessarily the principal value of log(zz') or of log(z/ z').

However, we may write

Log( zz') = Log z + Log z' + 2n7ri

Log ( ;, ) = Log z - Log z' + 2n7ri

where n is either -1, 0 or +1, depending on whether the sum (or difference)

of the principal values of the arguments of z and z' is greater than 7r, lies

in (-?r, 1r], or is less than or equal to -?r.

Property 5 follows by noting that if z = rei^8 , then zn = rneinO, so that

{logzn} = {nln r + i(nB + 2k17r)}

while

{n log z} = {n ln r + i(nB + 2k 2 n?r)}

The values in the second set coincide with those in the first only when

kzn = ki, i.e., only when ki contains n as a factor.

1.16 POWERS WITH A COMPLEX BASE AND A

COMPLEX EXPONENT

Having defined powers with base. e and the natural logarithms of complex

numbers, we are in a position to define the power zw with both a complex

base (other than zero) and a complex exponent.

Definition 1.10 For z f= 0 we define zw by means of the equality

The principal value of zw, denoted (p.v.)zw, is defined by

(p.v.)zw = ewLogz

(1.16-1)

(1.16-2)

the meaning of Logz as given in (1.15-5). For z = 0, w f= 0, we define

ow = O, while 0° is not defined.

Since log z has infinitely many values, zw as defined by (1.16-1) may have

also infinitely many values. However, when w is an integer n, the power