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

while

**Definition.** Let *f*(*x*), *f*_{1}(*x*), and *f*_{2}(*x*) be as defined above.

- (1)
- The Fourier series of
*f*_{1}(*x*) is called the**Fourier Sine series**of the function*f*(*x*), and is given by

where

- (2)
- The Fourier series of
*f*_{2}(*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
.

**Answer.** We have

and

Therefore, we have

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

**Answer.** We have

Hence

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

**Answer.** We have

which gives

Hence

**Special Case of **2*L***-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*b*_{n}= 0, for . Moreover, we have

and

Finally, we have

**2.**- If
*f*(*x*) is odd, then*a*_{n}= 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

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