1550251515-Classical_Complex_Analysis__Gonzalez_

Complex Numbers 53

while

2(*V-1) = {2i, -2i}

However, for principal values it is true that

vz+ vz=2vz

the symbol+ denoting now ordinary addition of complex numbers.

1.14 POWERS WITH BASE e AND A COMPLEX

EXPONENT

We wish to define ez when z = x+iy so as to preserve the main properties of
the real exponential e", in particular, the law of exponents e" · e"
1
= ex+x'.
Also, we wish to define ez in such a way that ez will reduce to e" when
z = x, and to eiy =cosy+ isiny when z = iy.
For the law of exponents to hold, we must have

But the meaning of each of the factors e"' and eiy is already known.
Hence it seems appropriate to establish the following definition.

Definition 1.8 For every complex z,

Example

ez = ex+iy = e"(cosy + isiny)

ea+(,,./^2 )i = e^3 [cos(7r/2) + isin(7r/2)] = ie^3.

(1.14-1).

We note that for z = x (i.e., for y = 0) the right-hand side of (1.14-1)
reduces to e", and for z = iy (i.e., for x = 0) we obtain eiy =cosy+ i sin y.
The power ez is called the complex exponential, and it is sometimes denoted
expz.
That Definition 1.8 preserves the main properties of e" is established in
the next proposition.

Theorem 1.11 The complex exponential has the following properties:

e^2 k11"i = 1 ( k an integer). In particular, e^0 = 1.

ez. ez' = ez+z'.

e-z = l/ez.

ez /ez I = ez-z. I

JezJ = e", arg ez = y + 2k7r.

ez =f. 0 for every z.

ez+2k,,.i = ez.

(ezr = enz (n an integer).