eix = cos x + i sin x
MA 222
K. Rotz
Example. Use Maclaurin Series to show that eix = cos x + i sin x, where i2 = −1.
Solution: Recall the Maclaurin series for ex , cos x and sin x:
x
e =
sin x =
cos x =
∞
X
xn
n=0
∞
X
n=0
∞
X
n=0
n!
=1+x+
x2 x3 x4 x5 x6 x7
+
+
+
+
+
+ ···
2!
3!
4!
5!
6!
7!
(−1)n x2n+1
(2n + 1)!
=x−
x3 x5 x7
+
−
+ ···
3!
5!
7!
(−1)n x2n
(2n)!
=1−
x2 x4 x6
+
− .
2!
4!
6!
Using the series for ex , we get
(ix)2 (ix)3 (ix)4 (ix)5 (ix)6 (ix)7
+
+
+
+
+
+ ···
2!
3!
4!
5!
6!
7!
x2
x3
x4
x5
x6
x7
= 1 + ix + i2 + i3 + i4 + i5 + i6 + i7 + · · ·
2!
3!
4!
5!
6!
7!
eix = 1 + ix +
Notice that i2 = −1, i3 = −i, i4 = 1, i5 = i, and the pattern repeats from there. So
eix = 1 + ix −
x3 x4
x5 x6
x7
x2
−i +
+i −
− i + ···
2!
3!
4!
5!
6!
7!
Now group by terms with an i (the odd terms) and terms without an i (the even terms):
x3
x5
x7
x2 x4 x6
+
−
+ · · · + ix − i + i − i + · · ·
eix = 1 −
5!
7!
2! 2 4! 4 6! 6
3!
x
x
x
x3 x5 x7
= 1−
+
−
+ ··· + i x −
+
−
+ ···
2!
4!
6!
3!
5!
7!
Finally, we can recognize the first term in the parentheses as cos x, and the second set of parentheses as
sin x. In conclusion,
eix = cos x + i sin x.
Neat fact: now you can talk about exponentials of complex numbers. For example,
π
π
π
eiπ = cos π + i sin π = −1 and ei 2 = cos + i sin = i.
2
2
This identity will be used later in the semester when we talk about second order linear differential
equations with complex eigenvalues.
1