where n = 1 , 2 , 3 , ... and T is the period of function f(t). a_{n} and b_{n} are called Fourier coefficients and are given by

__Example :__ Find the Fourier series of the periodic function f(t) defined by

__Solution to the above example__

Coefficient a_{0} is given by

Coefficients a_{n} is given by

And coefficients b_{n} is given by

A computation of the above coefficients gives

a_{0} = 0 , a_{n} = 0 and b_{n} = [ 2 / (n*pi) ] [ 1 - cos (n pi) ]

Note that cos (n pi) may be written as

cos (n pi) = (-1)^{n}

and that b_{n} = 0 whenever n is even.

The given function f(t) has the following Fourier series

Now below is an applet that may be used to explore the Fourier series of f(t) defined above

__Interactive Tutorial__

click on the button above "click here to start" and MAXIMIZE the window obtained.

1 - When the applet is first started, only f(t) (in blue) is displayed.

2 - Use the slider N to increases the number of terms in the series and see how the Fourier series (in red) gives a close approximation to function f(t) as the number of terms N in the series increases. Note also how the amplitude and period of the harmonics (in black) change with N. Although Fourier series are infinite, only few harmonics (terms in the series) are enough for a good approximation of f(t).

3 - You may zoom in and out and see closely what happens where f(t) is discontinuous.

4 - You may use the slider b to change the period of function f(t). Period T is given by T = 2 pi / b.