# Fourier Series Coefficients

Cuthbert Nyack

The general form of the Fourier series is shown below:-
Multiplying across by dt and integrating over 1 period
produces the expression below. The second and third terms become
zero when averaged over 1 period.
Integrating the RHS gives the following for a_{o}.
Multiplying across by cos(nwt)dt
and integrating over 1 period gives the following expression.
On the RHS, the first term is zero when integrated over 1 period and
the third term is also zero because of the orthogonality of
sines and cosines. When the second term is summed, it has a finite
value only when n = m.
Integrating the RHS gives the following expression for a_{n}.
Similar calculation gives the following for b_{n}.
As defined here a_{o} is the average value of f(t) and this leads to a factor
1/T in front of the integral for a_{o}. It is also common to have a_{o}/2 in the series
so the factor in front of the integral is 2/T.

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COPYRIGHT © 1996 Cuthbert Nyack.