# Fourier Series: Basic Results

Recall that the mathematical expression

is called a Fourier series.
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 a0, ai and bi, , are called the coefficients of Fn(x).

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 Fourier series associated to the function f(x). We will use the notation

Example. Find the Fourier series of the function

Answer. Since f(x) is odd, then an = 0, for . We turn our attention to the coefficients bn. For any , we have

We deduce

Hence

Example. Find the Fourier series of the function

and

We obtain b2n = 0 and

Therefore, the Fourier series of f(x) is

Example. Find the Fourier series of the function function

Answer. Since this function is the function of the example above minus the constant . So Therefore, the Fourier series of f(x) is

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 F(x) is defined and integrable on . Consider the Fourier series of F(x)

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

Answer. Since L = 2, we obtain

for . Therefore, we have

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Author: M.A. Khamsi