Editing Fourier series (section)

===Fourier's motivation===
[[File:Fourier heat in a plate.png|thumb|right|This resulting heat distribution in a metal plate is easily solved using Fourier's method]]
The Fourier series expansion of the sawtooth function (below) looks more complicated than the simple formula <math>s(x)=\tfrac{x}{\pi}</math>, so it is not immediately apparent why one would need the Fourier series. While there are many applications, Fourier's motivation was in solving the [[heat equation]]. For example, consider a metal plate in the shape of a square whose sides measure <math>\pi</math> meters, with coordinates <math>(x,y) \in [0,\pi] \times [0,\pi]</math>. If there is no heat source within the plate, and if three of the four sides are held at 0 degrees Celsius, while the fourth side, given by <math>y=\pi</math>, is maintained at the temperature gradient <math>T(x,\pi)=x</math> degrees Celsius, for <math>x</math> in <math>(0,\pi)</math>, then one can show that the stationary heat distribution (or the heat distribution after a long time has elapsed) is given by
: <math>T(x,y) = 2\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n} \sin(nx) {\sinh(ny) \over \sinh(n\pi)}.</math>
Here, sinh is the [[hyperbolic sine]] function. This solution of the heat equation is obtained by multiplying each term of the equation from [[#Example|Analysis § Example]] by <math>\sinh(ny)/\sinh(n\pi)</math>. While our example function <math>s(x)</math> seems to have a needlessly complicated Fourier series, the heat distribution <math>T(x,y)</math> is nontrivial. The function <math>T</math> cannot be written as a [[closed-form expression]]. This method of solving the heat problem was made possible by Fourier's work.