8.5 Euler’s formula 239
de Moivre’s formula
When θin Euler’s formula, (8.33), is replaced by nθwhere nis an arbitrary number,
the result is
e
inθ1 = 1 cos 1 nθ 1 + 1 i 1 sin 1 nθ
But becausee
inθ1 = 1 (e
iθ)
n, it follows that
(e
iθ)
n1 = 1 (cos 1 θ 1 + 1 i 1 sin 1 θ)
n1 = 1 cos 1 nθ 1 + 1 i 1 sin 1 nθ (8.40)
This is de Moivre’s formula, (8.27), generalized for arbitrary numbers n(that can
themselves be complex).
EXAMPLE 8.12The square root of i:
We havei 1 = 1 e
iπ 22. Therefore
Check:
0 Exercise 43
Rotation operators
When a complex numberz 1 = 1 re
iαis multiplied bye
iθthe product is a number with
the same modulus as zbut with argument increased by θ:
e
iθz 1 = 1 e
iθre
iα1 = 1 re
i(α+θ)Graphically, as shown in Figure 8.7, the multiplication corresponds to the (anti-
clockwise) rotation of the representative point through angle θabout the origin of the
complex plane, from
z 1 = 1 x 1 + 1 iy 1 = 1 r 1 cos 1 α 1 + 1 ir 1 sin 1 α
with cartesian coordinatesx 1 = 1 r 1 cos 1 α 1 ,y 1 = 1 r 1 sin 1 αto
e
iθz 1 = 1 z′ 1 = 1 x′ 1 + 1 iy′ 1 = 1 r 1 cos(α 1 + 1 θ) 1 + 1 ir 1 sin(α 1 + 1 θ)
with coordinates
x′ 1 = 1 r 1 cos(α 1 + 1 θ) 1 = 1 r(cos 1 α 1 cos 1 θ 1 − 1 sin 1 α 1 sin 1 θ)
y′ 1 = 1 r 1 sin(α 1 + 1 θ) 1 = 1 r(sin 1 α 1 cos 1 θ 1 + 1 cos 1 α 1 sin 1 θ)
±+
()
=++=
122122112 iiii()
ie i i
i=± =± +
=± +
πππ
4
12cos 44 1sin
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θ •
z(x,y)
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Figure 8.7