Wikimore

Description

This is a graph of x³, periodic on (-π,π), with the Fourier Series Approximations drawn in at k=7 (red) and k=15 (blue). The approximation is given by

$f\left(x\right)\approx \sum \limits _{n=1}^{k}{{{-2\left({-1}\right)^{n}\left({\pi ^{2}n^{2}-6}\right)} \over {n^{3}}}\sin \left({nx}\right)}$ .

This is the counterpart to this, which has just the original function drawn in.

Date

22 February 2008

Source

Own Drawing, Plotted in Mathematica, edited in Inkscape.

Author

Inductiveload

Permission
(Reusing this file)

I, the copyright holder of this work, release this work into the public domain. This applies worldwide.
In some countries this may not be legally possible; if so:
I grant anyone the right to use this work for any purpose, without any conditions, unless such conditions are required by law.

Category:Self-published work#XCubed%20Fourier%20Series%20Approximation%20n=7,15.svg Category:PD-self#XCubed%20Fourier%20Series%20Approximation%20n=7,15.svg

Other versions

image:XCubed Periodic (-pi, pi).svg

Mathematica Code

f[x_] = x^3;

ffour[x_, k_] =
  Hold[
   Sum[
    ((-2*(-1)^n*(Pi^2*n^2 - 6))/(n^3))*Sin[n*x],
    {n, 1, k}]
   ];

Plot[
 {f[Mod[x, 2 \[Pi], -\[Pi]]],
  ReleaseHold[ffour[x, 7]],
  ReleaseHold[ffour[x, 15]]},
 {x, -2 Pi, 2 Pi}]

Category:Fourier series Category:Order 15

Wikimore

File:XCubed Fourier Series Approximation n=7,15.svg

Mathematica Code