Fourier Series

[pic] Fourier Series: Basic Results

[pic]
Recall that the mathematical expression
[pic]

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
[pic]

which may rewritten as
[pic]

The constants a0, ai and bi, [pic], are called the coefficients of Fn(x).
The Fourier polynomials are [pic]-periodic functions. Using the trigonometric identities
[pic]

we can easily prove the integral formulas
(1)
for [pic], we have [pic]

(2) for m et n, we have [pic]

(3) for [pic], we have [pic]

(4) for [pic], we have [pic]

Using the above formulas, we can easily deduce the following result:
Theorem. Let
[pic]

We have
[pic]

This theorem helps associate a Fourier series to any [pic]-periodic function.
Definition. Let f(x) be a [pic]-periodic function which is integrable on [pic]. Set
[pic]

The trigonometric series
[pic]

is called the Fourier series associated to the function f(x). We will use the notation
[pic]

Example. Find the Fourier series of the function
[pic]

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

We deduce
[pic]

Hence
[pic]

|[pic] |

Example. Find the Fourier series of the function
[pic]

Answer. We have
[pic]

and
[pic]

We obtain b2n = 0 and
[pic]

Therefore, the Fourier series of f(x) is
[pic]

|[pic] |

Example. Find the Fourier series of the function function
[pic]

Answer. Since this function is the