Euler’s Formula
Where does Euler’s formula
e
iθ
= cos
θ
+
i
sin
θ
come from?
How do we even
define
, for example,
e
i
?
We can’t multiple
e
by itself the
square root of minus one times.
The answer is to use the Taylor series for the exponential function. For any complex
number
z
we define
e
z
by
e
z
=
∞
n
=0
z
n
n
!
.
Since

z
n

=

z

n
, this series converges absolutely:
∞
n
=0

z

n
n
!
is a real series that we already
know converges.
If we multiply the series for
e
z
termbyterm with the series for
e
w
, collect terms of
the same total degree, and use a certain famous theorem of algebra, we find that the law
of exponents
e
z
+
w
=
e
z
·
e
w
continues to hold for complex numbers.
Now for Euler’s formula:
e
iθ
=
∞
n
=0
(
iθ
)
n
n
!
= 1 +
iθ

θ
2
2!

i
θ
3
3!
+
θ
4
4!
+
i
θ
5
5!

θ
6
6!

i
θ
7
7!
+
· · ·
=
1

θ
2
2!
+
θ
4
4!

θ
6
6!
+
· · ·
+
i
θ

θ
3
3!
+
θ
5
5!

θ
7
7!
+
· · ·
= cos
θ
+
i
sin
θ.
The special case
θ
= 2
π
gives
e
2
πi
= 1
.
This celebrated formula links together three numbers of totally different origins:
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '07
 Nelson
 Exponential Function, Taylor Series, Complex number, Euler's formula, cos cos, sin sin

Click to edit the document details