Fourier Series allow periodic functions to be broken down into
linear combinations sine and cosine waves. Consequently they are
extremely important in many areas of applied mathematics,
particularly in frequency analysis. In many situations they can
be used to solve differential equations, such as the wave
equation.   Let f(x) be a real periodic square integrable
function with period 2L. The Fourier Series is defined as
inf               inf f(x) = a0 +  Σ ancos(nπx/L) +  Σ bnsin
(nπx/L).             n=1               n=1 where the
coefficients are determined by the integrals an = 1/L
integral(f(x)cos(nπx/L)dx, x=0...2L) bn = 1/L
integral(f(x)sin(nπx/L)dx, x=0...2L).   The Fourier series works
because the functions cos(nπx/L) and sin(nπx/L) form a complete
orthogonal set of the space of square integrable functions on
the interval from 0 to 2L. We have the orthogonality relations
for n,m>0 of the form  integral(sin(nπx/L)sin(mπx/L)dx,
x=0...2L) = L δmn integral(cos(nπx/L)cos(mπx/L)dx, x=0...2L) =