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19.3 The Exponential and Trigonometric Functions 685

∂u ∂v
x x z
f (z) = + i = e cos(y) + ie sin(y) = e
∂x ∂x
as with the real exponential function.
z
The following properties of e are straightforward consequences of the definition.
0
1. e = 1.
z w
2. e z+w = e e .
z
3. e = 0 for all complex z.
−z
z
4. e = 1/e .
−it
it
5. If t is real, then e = e .
To prove property (3), suppose
x
x
z
e = e cos(y) + ie sin(y) = 0.
Then
x
x
e cos(y) = e sin(y) = 0.
x
Since for real x, e = 0, then
cos(y) = sin(y) = 0.
This is impossible, because the real sine and cosine functions have no common zeros.
For property (4), use properties (1) and (2) to write
z −z
0
e = 1 = e z−z = e e
−z
z
implying that 1/e = e .
To verify property (5), suppose t is a real number. By Euler’s formula,
it
e = cos(t) + i sin(t)
−it
= cos(t) − i sin(t) = e .
z
Perhaps the first surprise we find with the complex exponential function is that e is peri-
odic. This periodicity does not appear in the real exponential function because the period is pure
imaginary.

THEOREM 19.8

z
1. e = 1 if and only if z = 2nπi for some integer n.
z
2. If p is a number such that e z+p = e for all complex z, then for some integer n, p = 2nπi.
z
3. e is periodic with period 2nπi for each nonzero integer n. Furthermore, these are the
only periods of the complex exponential function.
Proof To prove conclusion (1) first observe that, if z = 2nπi for some integer n, then
z
e = e 2nπi = cos(2nπ) + i sin(2nπ) = 1
because cos(2nπ) = 1 and sin(2nπ) = 0.
z
Conversely, if e = e a+ib = 1, then
a
a
e cos(b) + ie sin(b) = 1.
so
a
a
e cos(b) = 1 and e sin(b) = 0.

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October 15, 2010 18:5 THM/NEIL Page-685 27410_19_ch19_p667-694