# 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   [Geometry] [Algebra] [Trigonometry ]
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