Understanding Engineering Mathematics

Ifm,nare any integers, then:

∫π

−π

sinmtsinntdt=0ifm=n

∫π

−π

sin^2 ntdt=π

∫π

−π

cosmtcosntdt=0ifm=n

∫π

−π

cos^2 ntdt=π

∫π

−π

sinmtcosntdt=0forallm,n

∫π

−π

sinmtdt=

∫π

−π

cosmtdt= 0

These are called theorthogonality relations for sine and cosine.The limits−π,πon
the integrals may in fact be replaced byanyintegral of length 2π, or integer multiple of
2 π. The orthogonality relations should not be thought of as simply specific integral rela-
tions – many other functions satisfy similar orthogonality relations (often calledspecial,
ororthogonal functions) and can be used in a similar way to expand in a ‘generalised
Fourier series’.

Exercise on 17.8

Prove the orthogonality relations.

17.9 The Fourier series expansion

A Fourier series is essentially a means of expressing any periodic functionf(t)as a sum
(possibly infinite) of sines or cosines of different frequencies. The point is of course that
sine and cosine are relatively simple functions to deal with.
Letf(t)be a function with period 2π. It may be a square wave, or triangular wave,
for example. If we compare it with a sine or cosine wave with the same period there may
be little resemblance, on the face of it. However, it turns out that if we combine a large
enough number of sine and/or cosine waves of appropriate amplitudes and frequencies
then it is possible, under certain conditions, to approximate to any function of period
2 π. This is certainly not obvious, but is mathematically well established and so here we
will simplyassumethatf(t)can be expanded in a series of sines and cosines in the
form

f(t)=

a 0

2

+

∑∞

n= 1

ancosnt+

∑∞

n= 1

bnsinnt

This is called aFourier (series) expansionforf(t). Note that each term on the right-
hand side has period 2π, likef(t), but asnincreases cosnt and sinnt oscillate an