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.
Definition
Let f(x) be a real periodic square integrable function with period 2L. The Fourier Series is defined as
inf inf
f(x) = a_{0} + Σ a_{n}cos(nπx/L) + Σ b_{n}sin (nπx/L).
n=1 n=1
where the coefficients are determined by the
integrals
a_{n} = 1/L integral(f(x)cos(nπx/L)dx, x=0...2L)
b_{n} = 1/L integral(f(x)sin(nπx/L)dx, x=0...2L).
Justification
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 δ_{mn}
integral(cos(nπx/L)cos(mπx/L)dx, x=0...2L) = L δ_{mn}
integral(cos(nπx/L)sin(mπx/L)dx, x=0...2L) = 0
where δ
_{mn} is the
Kronecker Delta. Now assume we have an expression of the form as above, and we wish to determine the coefficient b
_{m}. Multiply the relation by sin(mπx/L) to obtain
inf inf
f(x) sin(mπx/L)= a_{0}sin(mπx/L) + Σ a_{n}cos(nπx/L)sin(mπx/L) + Σ b_{n}sin (nπx/L)sin(mπx/L).
n=1 n=1
We now integrate both sides from 0 to 2L. By the orthogonality relations, all the integrals vanish except for the term involving b
_{m}. Hence
integral(f(x)sin(mπx/L)dx, x=0...2L) = L b_{m}
which gives the required result. The a
_{m}'s can be determined in a similar way by multiplying through by cos(mπx/L) and integrating.
Example
Consider a sawtooth wave, given by f(x)=x on the open interval (L,L), which is periodic with period 2L.
f(x)

/  / /
/  / /
/  / /
/ / /
++++ x
L / L /2L
/  /
/  /
/  /

Then by above we have
a_{n} = 1/L integral(x cos(nπx/L)dx, x=L...L)
= 0,
by symmetry. Using
integration by parts we obtain
b_{n} = 1/L integral(x sin(nπx/L)dx, x=L...L)
= 2/L integral(x sin (nπx/L)dx, x=0...L)
= 2/L (L/(nπ))(Lcos(nπL/L)  Lcos(nπ0/L)) + 2/(nπ) integral(cos(nπx/L),x=0...L)
= 2L/(nπ)cos(nπ)
= 2L/(nπ) (1)^{n+1}.
From this we deduce that
inf
f(x) = Σ 2L/(nπ) (1)^{n+1} sin (nπx/L)
n=1
= 2L/π (sin(πx/L)  sin(2πx/L)/2 + sin(3πx/L)/3  sin(4πx/L)/4 + ... )
Even for a simple example such as this we have derived a remarkable result: not the sort thing that one could guess.
Complex representation
If the input function f(x) is complex, then we have an alternative formulation
inf
f(x) = Σ c_{n} exp(inπx/L)
n=inf
where the coefficients are determined by
c_{n} = 1/(2L) integral(f(x)exp(inπx/L)dx, x=L...L).