8.5 Euler’s formula 237
This relation between the exponential function and the trigonometric functions is
called Euler’s formula.
5It forms the basis for the unified theory of the elementary
functions, and it is one of the important relations in mathematics.
The function z 1 = 1 e
iθhas modulus and argument
arg z 1 = 1 θ. For each value of θ, the number lies on the unit circle of the complex plane,
and as θvaries from 0 to 2 πthe function defines the unit circle (Figure 8.6). The
complex conjugate ofz 1 = 1 e
iθis
z* 1 = 1 e
−iθ1 = 1 cos 1 θ 1 − 1 i 1 sin 1 θ (8.34)
and this is also the inverse,z
− 11 = 1 z* 1 = 1 e
−iθ. Then
zz* 1 = 1 zz
− 11 = 1 e
iθe
−iθ1 = 1 e
01 = 11
The pair of equations (8.33) and (8.34) can be inverted to
give relations for the trigonometric functions in terms of
the exponentials:
(8.35)
(8.36)
From the polar form of a complex number, equation (8.15), it now follows that every
complex number can be written as
z 1 = 1 x 1 + 1 iy 1 = 1 re
iθ(8.37)
wherer 1 = 1 |z|andθ 1 = 1 arg 1 z. The complex conjugate and inverse functions of zare
(8.38)
EXAMPLE 8.9The numberz 1 = 111 + 1 i.
It was shown in Example 8.5 that
zi=+= +i
12
44
cos sin
ππ
zxiyre z
xiy r
e
ii*=− = , =
=
−−θθ 1 −11
sinθ
θθ=−
−1
2 i
ee
iicosθ
θθ=+
−1
2
ee
ii|| cos sinz =+=
22θθ 1
5The formula, and others for the logarithmic and trigonometric functions of z, appeared in Euler’s Introductio
of 1748.
.....................
..........
......
......
......
....
....
.....
....
...
....
...
....
...
...
...
...
...
...
...
..
...
...
...
..
...
..
...
...
..
...
..
...
..
...
..
...
..
...
..
...
..
...
...
..
...
..
...
...
..
...
...
...
...
..
....
...
...
...
...
...
....
....
....
....
....
.....
.....
......
......
..........
.................
.............................................................................................................................................................................................................................................................................................................................
..
...
..
...
...
...
..
....
.o
x
y
θ
−θ
e
iθe
−iθ1
1
.
.......
.........
.......
........
........
........
.......
.........
.......
.......
.........
.......
........
........
........
.......
.........
.......
.......
.........
.......
........
........
.
.......
.........
.......
........
........
........
.......
.........
.......
........
........
.......
........
........
........
.......
.........
.......
........
........
.......
........
........
..
Figure 8.6