1.3. The Exponential Function 17

Solution: By Euler’s identity, we have

e2πi

= cos 2π + i sin 2π = 1,

eπi

= cos π + i sin π = −1,

while

eπi/2

= cos π/2 + i sin π/2 = i.

Properties of the Exponential Function. The law of exponents and Euler’s

identity immediately imply:

Theorem 1.3.6. If z = x + iy is a complex number, then

ez

=

ex(cos

y + i sin y).

There are a number of properties of the exponential function which follow easily

from this characterization of

ez.

We collect them together in the following theorem

whose proof is left to the exercises.

Theorem 1.3.7. The exponential function has the following properties:

(a)

ez

is never 0;

(b) |

ez

| =

eRe(z);

(c) | ez | ≤ e|z|;

(d) ez is periodic of period 2πi, meaning ez+2πi = ez for every z ∈ C;

(e) ez = 1 if and only if z = 2πni for some integer n.

Complex Trigonometric Functions. If we write out Euler’s identity with y

replaced by θ and then by −θ, we obtain a pair of identities

eiθ

= cos θ + i sin θ,

e−iθ

= cos θ − i sin θ.

If we solve this system of equations for cos θ and sin θ, we obtain

cos θ =

eiθ + e−iθ

2

,

sin θ =

eiθ − e−iθ

2i

.

(1.3.5)

This suggests defining sin z and cos z for a complex variable z in the following way:

Definition 1.3.8. For each z ∈ C, we set

cos z =

eiz + e−iz

2

,

sin z =

eiz − e−iz

2i

.

These are the same functions that one would get by replacing x by z in the

power series expansions of sin x and cos x.

With sin z and cos z defined, it is easy to define complex versions of the other

trigonometric functions. For example,

(1.3.6) tan z =

sin z

cos z

= −i

eiz

−

e−iz

eiz + e−iz

.