1550251515-Classical_Complex_Analysis__Gonzalez_

54 Chapter^1

Proofs Property 1 follows at once from (1.14-1).
To prove property 2 we have, using (1.14-1), the additivity of real
exponents and that of pure imaginary exponents (Section 1.9),

Since z and z' are arbitrary, we have, in particular, eze-z = ez+(-z) =

e^0 = 1, so that e-z = l/ez, which is property 3. This yields

As to property 5, we have

jezl = je"'lleiyl = e"'

since e"' > 0 for every real x, and jeiYj = 1. Also, since the right-hand side

of (1.14-1) is in polar form, we see that y is a value of argez.

Property 6 follows from property 5, property 7 from properties 1 and 2,

and property 8 by repeated application of property 2 (in combination with

property 3 if n < 0). The case n = 0 is trivial.

1.15 LOGARITHMS TO THE BASE e

Definition 1.9 A natural logarithm of a complex number z f= 0 is any

complex number w such that

(1.15-1)

The case z = 0 must be ruled out by Theorem 1.11(6).

If the equation (1.l{l-1) has a solution at all, it has infinitely many, since

ew = ew+2k1ri,

To see that the equation has solutions whenever z f= O, we proceed as

follows. Let z = rei^9 and w = u +iv. Then (1.15-1) becomes

or

eu( cos v + i sin v) = r( cos B + i sin B)

which implies, by Theorem 1.6, that

and v = e + 2k7r

Hence

u = lnr and v=B+2k1r