1.3. The Exponential Function 17
Solution: By Euler’s identity, we have
e2πi
= cos + i sin = 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
.
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