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Euler relation∗
rm50†
2013-03-21 12:45:50
Euler’s relation (also known as Euler’s formula) is considered the first bridge
between the fields of algebra and geometry, as it relates the exponential function
to the trigonometric sine and cosine functions.
Euler’s relation states that
eix = cos x + i sin x
Start by noting that


1

i
ik =
−1



−i
if
if
if
if
k
k
k
k
≡0
≡1
≡2
≡3
(mod
(mod
(mod
(mod
4)
4)
4)
4)
Using the Taylor series expansions of ex , sin x and cos x (see the entries on
the complex exponential function and the complex sine and cosine), it follows
that
P∞ n n
eix
= n=0 i n!x
P∞ x4n
ix4n+1
x4n+2
ix4n+3
= n=0 (4n)!
+ (4n+1)!
− (4n+2)!
− (4n+3)!
Because the series expansion above is absolutely convergent for all x, we can
rearrange the terms of the series as
eix
=
n x2n
n=0 (−1) (2n)!
P∞
+i
n x2n+1
n=0 (−1) (2n+1)!
P∞
= cos x + i sin x
As a special case, we get the beautiful and well-known identity, often called
Euler’s identity:
eiπ = −1
∗ hEulerRelationi
created: h2013-03-21i by: hrm50i version: h30714i Privacy setting: h1i
hDefinitioni h30B10i
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