Background. Fourier series are used to expand periodic functions in the trigonometric form.

Theorem 1. (Fourier Expansion) If f(t) has period and f '(t) is piecewise continuous, the Fourier expansion is


Theorem 2. (Fourier Cosine Series) Assume that f(t) is an even function and has period . If f(t) and f '(t) are piecewise continuous, the Fourier series for f(t) involves only the cosine terms, (i.e. ):


Theorem 3. (Fourier Sine Series) Assume that f(t) is an odd function and has period . If f(t) and f '(t) are piecewise continuous, the Fourier series for f(t) involves only the sine terms, (i.e. ):


Load Mathematica's FourierTransform package.

Computer Lab Work.

Example 1. Find the Fourier series expansion for extended periodically with period .

Notice that the function f[t] is odd, so that the coefficients are all zero. The following computations assist with the computation of the coefficients .

The first six coefficients are:

Plot the graph of .

We can use Mathematica's built in Fourier series procedure to perform our computations, for example:

The pattern we have for the Fourier trigonometric polynomials are given by the following summation which we can check out for the case where 5 terms are added in the sum.

Mathematica can sum the infinite Fourier series and obtain a closed form for the "periodic extension" g(t) of f(t).

Example 2. Find the Fourier series expansion for extended periodically with period .

Notice that the function f[t] is even, so that the coefficients are all zero. The following computations assist with the computation of the coefficients .

The first six coefficients are:

Since we will plot the Fourier trigonometric polynomials corresponding to n=1,3,5.

The pattern we have for the Fourier trigonometric polynomials are given by the following summation which we can check out for the case where 3 terms are added in the sum.

Mathematica can sum the infinite Fourier series and obtain a closed form for the "periodic extension" g(t) of f(t).

Example 3. Find the Fourier series expansion for extended periodically with period .

Notice that the function f[t] is odd, so that the coefficients are all zero. The following computations assist with the computation of the coefficients .

The first seven coefficients are:

Since we will plot the Fourier trigonometric polynomials corresponding to n=1,3,5,7.

The pattern we have for the Fourier trigonometric polynomials are given by the following summation which we can check out for the case where 3 terms are added in the sum.

Mathematica can sum the infinite Fourier series and obtain a closed form for the "periodic extension" g(t) of f(t).

(c) John H. Mathews, 1998