We first recap on odd and even functions. We say that f is even on [-L, L] if f(-x)=f(x) for . This means that the graph of f from –L to zero is the reflection across the y-axis of the graph from zero to L, as shown below.

Examples of even functions are . Replacing x by –x has no effect on f(x).

A function is odd on [-L, L] if f(-x)=-f(x) for . For such a case, the graph from –L to zero is the reflection across the y-axis, and then across the x-axis, of the graph from zero to L, as shown below.

Examples of an odd function are , where replacing x by –x results in obtaining the negative of the value obtained for x.

Most functions are neither even nor odd, for example  and  are neither even nor odd. However, a product of two even or two odd functions is even, and a product of an even function with an odd function is odd:

The product of  is odd because  changes sign if x is replaced by –x.

To take advantage of odd and even functions in finding the Fourier coefficients, we exploit the following facts from calculus.

LEMMA 1
Let  be integrable on .
1. If  is even on , then .
2. If  is odd on , then
A sketch of the graphs of odd and even functions as a proof of the above lemma should suffice for now. A proper proof, while not difficult, may be read up in any calculus textbook by the reader. We now turn our focus back to the Fourier coefficients.

The above lemma makes the computation of the Fourier coefficients of f easier when f is even or odd. If f  is even on ,  is odd for any positive integer n; hence, all the ’s are zero by conclusion 2 of the lemma. This also allows us to write a simpler formula for ’s by conclusion 1. If f is odd,  is odd and each  is zero. The following theorem gives us the definite details.

Let  be integrable on .
1. If  is even, the Fourier series of  on  is

where

2. If  is odd, the Fourier series of  on  is

where

Here is a short outline of the proof if you are interested.
If  is even,

and

,

by conclusion 1 of the lemma. By conclusion 2,

We can use a similar argument to show the results of when  is odd. When applying such formulas for even and odd functions, it pays to point out that the terms in front of the integrals are different as compared to the corresponding terms in the usual definitions for ‘any’ function. Roughly speaking, we multiply the term by 2 when finding the Fourier coefficients for odd or even functions.

So whenever you are asked to find the Fourier Series of an even function, you can save yourself the trouble by omitting  and likewise omitting  and  when the function is odd.