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

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)

Fourier cosine and sine series

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