1.3. The Exponential Function 15

This is the complex exponential function. It has many important properties,

some expected and some surprising, as we shall see below.

The Law of Exponents. A property that we should expect of an exponential

function is the following.

Theorem 1.3.2. If z, w ∈ C, then

ez+w

=

ez ew.

Proof. The proof uses the complex form of the binomial formula:

(1.3.1) (z +

w)n

=

n

j=0

n!

j!(n − j)!

zjwn−j

.

This is proved using induction on n. We leave it as an exercise (Exercise 1.3.11).

Proceeding with the proof of the theorem, and using (1.3.1), we have

ez+w

=

∞

n=0

(z + w)n

n!

=

∞

n=0

n

j=0

1

j!(n − j)!

zj wn−j.

(1.3.2)

If we make a change of variables by setting k = n −j in the inside summation, then

(1.3.2) becomes

∞

n=0

⎛

⎝

j+k=n

zjwk

j!k!

⎞

⎠

.

This is precisely what we get if we expand the product

ez ew

=

⎛

⎝

∞

j=0

zj

j!

⎞

⎠

∞

k=0

wk

k!

.

and collect terms of degree n. Provided this operation is valid, we conclude that

ez+w

=

ez ew.

It turns out that it is valid to expand the product of two infinite

series in this fashion if they are both absolutely convergent. We prove this in the

following lemma, which will complete the proof of the theorem.

Lemma 1.3.3. Let

∑∞

j=0

aj and

∑∞

k=0

bk be two absolutely convergent series of

complex numbers. Then

(1.3.3)

⎛

⎝

∞

j=0

aj⎠

⎞

∞

k=0

bk =

∞

n=0

⎛

⎝

j+k=n

ajbk⎠

⎞

.

Proof. The partial sums of the series involved here are

sJ =

J

j=0

aj, tK =

K

k=0

bk, uN =

N

n=0

⎛

⎝

j+k=n

ajbk⎠

⎞

.

The left side of (1.3.3) is, by definition, (lim sJ )(lim tK ), while the right side is

lim uN . We know lim sJ and lim tK both exist since the series defining them con-

verge absolutely. We must prove that lim uN exists and equals (lim sJ )(lim tK ).