# Fourier Series for Three Periodic Functions

Fourier Series for Three Periodic Functions

Periodic phenomena occur frequently in nature. Fourier series approximate periodic functions using trigonometric functions. This Demonstration shows three functions and their approximations using Fourier series. The functions are an even function, , an odd function, , and a function that is neither even nor odd.

g(x)=g(-x)

h(x)=-h(x)

For a function defined on the interval , the Fourier coefficients are defined by:

f

[-L,L]

a

0

1

L

L

∫

-L

a

n

1

L

L

∫

-L

nπx

L

b

n

1

L

L

∫

-L

nπx

L

With those coefficients and for suitably well behaved, .

f

f(x)=+cos+sin

1

2

a

0

∞

∑

n=1

a

n

nπx

L

b

n

nπx

L