Recall that the mathematical expression

is called a

Since this expression deals with convergence, we start by defining a similar expression when the sum is finite.

**Definition.** A **Fourier polynomial** is an expression of the form

which may rewritten as

The constants

The Fourier polynomials are -periodic functions. Using the trigonometric identities

we can easily prove the integral formulas

- (1)
- for ,
we have

for n>0 we have

- (2)
- for
*m*and*n*, we have

- (3)
- for ,
we have

- (4)
- for ,
we have

Using the above formulas, we can easily deduce the following result:

**Theorem.** Let

We have

This theorem helps associate a Fourier series to any -periodic function.

**Definition.** Let *f*(*x*) be a -periodic function which is integrable on
.
Set

The trigonometric series

is called the

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

We deduce

Hence

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

and

We obtain

Therefore, the Fourier series of

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

**Remark.** We defined the Fourier series for functions which are -periodic, one would wonder how to define a similar notion for functions which are *L*-periodic.

Assume that *f*(*x*) is defined and integrable on the interval [-*L*,*L*]. Set

The function

Using the substitution , we obtain the following definition:

**Definition.** Let *f*(*x*) be a function defined and integrable on [-*L*,*L*]. The Fourier series of *f*(*x*) is

where

for .

**Example.** Find the Fourier series of

for . Therefore, we have

**
**

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**Author**: M.A. Khamsi

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