example of Fourier series

Here we present an example of Fourier series:


Let f:(-π,π) be the “identity” functionMathworldPlanetmath, defined by

f(x)=x, for all x(-π,π).

We will compute the Fourier coefficients for this function. Notice that cos(nx) is an even function, while f and sin(nx) are odd functions.

a0f = 12π-ππf(x)𝑑x=12π-ππx𝑑x=0
anf = 1π-ππf(x)cos(nx)𝑑x=1π-ππxcos(nx)𝑑x=0
bnf = 1π-ππf(x)sin(nx)𝑑x=1π-ππxsin(nx)𝑑x=
= 2π0πxsin(nx)dx=2π([-xcos(nx)n]0π+[sin(nx)n2]0π=)=(-1)n+12n

Notice that a0f,anf are 0 because x and xcos(nx) are odd functions. Hence the Fourier series for f(x)=x is:

f(x)=x = a0f+n=1(anfcos(nx)+bnfsin(nx))=
= n=1(-1)n+12nsin(nx),x(-π,π)

For an application of this Fourier series, see value of the Riemann zeta functionDlmfDlmfMathworldPlanetmath at s=2.

Title example of Fourier series
Canonical name ExampleOfFourierSeries
Date of creation 2013-03-22 13:57:13
Last modified on 2013-03-22 13:57:13
Owner alozano (2414)
Last modified by alozano (2414)
Numerical id 10
Author alozano (2414)
Entry type Example
Classification msc 42A16
Synonym example of Fourier coefficients
Related topic ValueOfTheRiemannZetaFunctionAtS2
Related topic FourierSineAndCosineSeries