


Complex Form of Fourier Series


Let the function f (x) be defined on the interval [−π, π].
Using the wellknown Euler's formulas
we can write the Fourier series of the function in complex form:
Here we have used the following notations:
The coefficients c_{n} are called complex Fourier coefficients.
They are defined by the formulas
If necessary to expand a function f (x) of period 2 L,
then we can use the following expressions:
where
The complex form of Fourier series is algebraically simpler and more symmetric. Therefore, it is often used in physics and other sciences.

Example 1


Using complex form, find the Fourier series of the function
Solution.
Calculate the coefficients c_{0} and c_{n} (at n ≠ 0):
If n = 2k, then .
If n = 2k − 1, then .
Hence, the Fourier series of the function in complex form is
We can transform the series and write it in the real form. Rename: .
Then
Graph of the function and its Fourier approximation at n = 5 and n = 50 are shown in Figure 1.

 
Fig.1,
n = 5,
n = 50

 Fig.2,
n = 2,
n = 5


Example 2


Using complex form find the Fourier series of the function ,
defined on the interval [−1, 1].
Solution.
Here the halfperiod is L = 1. Therefore, the coefficient c_{0} is
For n ≠ 0,
Integrating by parts twice, we obtain
Substituting
sin nπ = 0 and cos nπ = (−1)^{n},
we get the compact expression for the coefficients c_{n}:
Thus, the Fourier extension in complex form is given by
Taking into account that , we can finally write:
Graphs of the function and the Fourier series are shown in Figure 2 above.

Example 3


Using complex form find the Fourier series of the function
Solution.
We apply the formulas
This results in the following expression:
Using partial decomposition, we can write:
Calculate the coefficients A,B:
As a result, the function f (x) can be written in the form
We see that
For conjugate, we have the same result:
Expanding the fractions into power series, we get
Thus, the Fourier series of the function f (x) is
Since , the final answer is



