# Fourier Sine and Cosine Series

Recall that the Fourier series of f(x) is defined by

where

We have the following result:

Theorem. Let f(x) be a function defined and integrable on interval .

(1)
If f(x) is even, then we have

and

(2)
If f(x) is odd, then we have

and

This Theorem helps define the Fourier series for functions defined only on the interval . The main idea is to extend these functions to the interval and then use the Fourier series definition.

Let f(x) be a function defined and integrable on . Set

and

Then f1 is odd and f2 is even. It is easy to check that these two functions are defined and integrable on and are equal to f(x) on . The function f1 is called the odd extension of f(x),
while f2 is called its even extension.

Definition. Let f(x), f1(x), and f2(x) be as defined above.

(1)
The Fourier series of f1(x) is called the Fourier Sine series of the function f(x), and is given by

where

(2)
The Fourier series of f2(x) is called the Fourier Cosine series of the function f(x), and is given by

where

Example. Find the Fourier Cosine series of f(x) = x for .

and

Therefore, we have

Example. Find the Fourier Sine series of the function f(x) = 1 for .

Hence

Example. Find the Fourier Sine series of the function for .

which gives b1 = 0 and for n > 1, we obtain

Hence

Special Case of 2L-periodic functions.
As we did for -periodic functions, we can define the Fourier Sine and Cosine series for functions defined on the interval [-L,L]. First, recall the Fourier series of f(x)

where

for .
1.
If f(x) is even, then bn = 0, for . Moreover, we have

and

Finally, we have

2.
If f(x) is odd, then an = 0, for all , and

Finally, we have

The definitions of Fourier Sine and Cosine may be extended in a similar way.

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Author: M.A. Khamsi