1550251515-Classical_Complex_Analysis__Gonzalez_

32 Chapter 1

(1.8-4)
The two preceding equations imply that (Ji = 82 + 2k7r, since there is just
one value of 8 on any semiopen interval of length 27r where (1.8-4) holds.

1.9 Exponential Form of the Complex Number

In Section 1.14 we define the exponential ez for any complex z. Presently

we shall consider only the special definition

eilJ = cos8 + isini9 (1.9-1)

for any real 8. This definition is justified by the fact that the complex
exponential thus defined obeys the same rules as ex for x real, namely,

1. e^0 == 1


2. eilleill' = ei(ll+IJ')


3. e-ill = 1/eilJ

4. eill + eill' = ei(ll-11')

5. ( ei^11 )n = einlJ ( n a positive integer)

Property 1 is obvious since cos 0 = 1 and sin 0 = 0. As for property 2,
we have

ei^9 ei^9 ' = (cos B + i sin 8)( cos B' + i sin B')

= (cos B cos B' -sin B sin B') + i(sin B cos B' +cos B sin B')

= cos( B + B') + i sin( B + B')

= ei(IJ+li')

To check property 3, we have

e-ill = cos(-B) + i sin(-B) =cos B - i sin B

1 1

=

cosB+isinB eill

Property 4 follows easily from properties 2 and 3, and property 5 follows
by repeated application of property 2.
In view of (1.9-1) the polar form (1.8-2) of the complex number can be
expressed as follows:

(1.9-2)

This is called the exponential form of the complex number. It will be found

to be a very useful representation of the complex number.