find the Complex form of Fourier series in the function of f(x)= e to the power x in the interval 0<x<2
Answers
Answer:
Let the function
f
(
x
)
be defined on the interval
[
−
π
,
π
]
.
Using the well-known Euler’s formulas
cos
φ
=
e
i
φ
+
e
−
i
φ
2
,
sin
φ
=
e
i
φ
−
e
−
i
φ
2
i
,
we can write the Fourier series of the function in complex form:
f
(
x
)
=
a
0
2
+
∞
∑
n
=
1
(
a
n
cos
n
x
+
b
n
sin
n
x
)
=
a
0
2
+
∞
∑
n
=
1
(
a
n
e
i
n
x
+
e
−
i
n
x
2
+
b
n
e
i
n
x
−
e
−
i
n
x
2
i
)
=
a
0
2
+
∞
∑
n
=
1
a
n
−
i
b
n
2
e
i
n
x
+
∞
∑
n
=
1
a
n
+
i
b
n
2
e
−
i
n
x
=
∞
∑
n
=
−
∞
c
n
e
i
n
x
.
Here we have used the following notations:
c
0
=
a
0
2
,
c
n
=
a
n
−
i
b
n
2
,
c
−
n
=
a
n
+
i
b
n
2
.
The coefficients
c
n
are called complex Fourier coefficients. They are defined by the formulas
c
n
=
1
2
π
π
∫
−
π
f
(
x
)
e
−
i
n
x
d
x
,
n
=
0
,
±
1
,
±
2
,
…
If necessary to expand a function
f
(
x
)
of period
2
L
,
we can use the following expressions:
f
(
x
)
=
∞
∑
n
=
−
∞
c
n
e
i
n
π
x
L
,
where
c
n
=
1
2
L
L
∫
−
L
f
(
x
)
e
−
i
n
π
x
L
d
x
,
n
=
0
,
±
1
,
±
2
,
…
The complex form of Fourier series is algebraically simpler and more symmetric. Therefore, it is often used in physics and other sciences