1. Fourier series of function with arbitrary period p=2L
Instead of a period of 2, many functions have an arbitrary period, say a period of 2L. In order to convert the Fourier series defined earlier for these functions, a change of variable is needed:
Replace the variable x by (/L)x: when x=L the new variable equals to ; when x= -L, it equals to - . Therefore, the previous formulas can be used by simply making the change

2. Even and Odd Functions A function f(x) is even when f(x) = f(-x)
On the other hand, if f(x) = -f(-x), the function is an odd function.
An even function
An odd function
f(x)
f(x) x x
Ex: cos(x)
Ex: sin(x)

4. Fourier cosine and sine series

5. Half Range Expansion
Expansion is useful when a function is defined only on a given interval, say between 0 and L. This situation is very common in real life: For example, the vibration of a guitar string occurs only between its bridge and tension peg.
expansion