The mathematician Euler proved an important identity relating complex exponentials to trigonometric functions. Specifically, he discovered the eponymously named identity, Euler's formula, which states that
e
j
x
=
cos
(
x
)
+
j
sin
(
x
)
e
j
x
=
cos
(
x
)
+
j
sin
(
x
)
(2)which can be proven as follows.
In order to prove Euler's formula, we start by evaluating the Taylor series for ezez about z=0z=0, which converges for all complex zz, at z=jxz=jx. The result is
e
j
x
=
∑
k
=
0
∞
(
j
x
)
k
k
!
=
∑
k
=
0
∞
(

1
)
k
x
2
k
(
2
k
)
!
+
j
∑
k
=
0
∞
(

1
)
k
x
2
k
+
1
(
2
k
+
1
)
!
=
cos
(
x
)
+
j
sin
(
x
)
e
j
x
=
∑
k
=
0
∞
(
j
x
)
k
k
!
=
∑
k
=
0
∞
(

1
)
k
x
2
k
(
2
k
)
!
+
j
∑
k
=
0
∞
(

1
)
k
x
2
k
+
1
(
2
k
+
1
)
!
=
cos
(
x
)
+
j
sin
(
x
)
(3)
because the second expression contains the Taylor series for cos(x)cos(x) and sin(x)sin(x) about t=0t=0, which converge for all real xx. Thus, the desired result is proven.
Choosing x=ωtx=ωt this gives the result
e
j
ω
t
=
cos
(
ω
t
)
+
j
sin
(
ω
t
)
e
j
ω
t
=
cos
(
ω
t
)
+
j
sin
(
ω
t
)
(4)which breaks a continuous time complex exponential into its real part and imaginary part. Using this formula, we can also derive the following relationships.
cos
(
ω
t
)
=
1
2
e
j
ω
t
+
1
2
e

j
ω
t
cos
(
ω
t
)
=
1
2
e
j
ω
t
+
1
2
e

j
ω
t
(5)
sin
(
ω
t
)
=
1
2
j
e
j
ω
t

1
2
j
e

j
ω
t
sin
(
ω
t
)
=
1
2
j
e
j
ω
t

1
2
j
e

j
ω
t
(6)
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