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   Complex Form of Fourier Series
Let the function f (x) be defined on the interval [−π, π]. Using the well-known Euler's formulas
Euler's formulas
we can write the Fourier series of the function in complex form:
Fourier series in complex form
Here we have used the following notations:
The coefficients cn are called complex Fourier coefficients. They are defined by the formulas
complex Fourier coefficients
If necessary to expand a function f (x) of period 2L, 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 c0 and cn (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 half-period is L = 1. Therefore, the coefficient c0 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 cn:
     
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
     

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