426 Chapter 15Orthogonal expansions. Fourier analysis
The (trigonometric) Fourier series is usually written in the form
2(15.37)
and, because of the orthogonality of the expansion functions, the Fourier coefficients
are given by
(15.38)
(15.39)
EXAMPLE 15.5Confirm the relations (i) (15.32) and (ii) (15.35).
(i) By the trigonometric relations (3.22), whenm 1 ≠ 1 n,
so that
and the zero is obtained because the sine of an integer multiple of πis zero.
(ii) Form 1 = 1 n 1 = 10 , whencos 1 mx 1 = 1 cos 1 mx 1 = 11 ,
Z
−+−+=
=
ππππdx x 2 π
=
−
−
=
−+1
2
0
ππsin(mnx) sin( )
mn
mnx
mn
ZZ
−+−+=++−
ππππcos cosmx nx dx cos(m n x) cos(m n)
1
2
xxdx
cos cosmx nx=++−cos(m n x) cos(m n x)
1
2
bfxnxdx
n=
−+1
π
ππZ ()sin
afxnxdx
n=
−+1
π
ππZ ()cos
=+ +
=∑
a
anxbnx
nnn012
∞(cos sin)
++ + +bxb xb x
12 3sin sin 23 sin
fx
a
( )=+axa xa xcos +cos +cos +
012 32
23
2Jean-Baptiste Joseph Fourier (1768–1830) studied the series in connection with his work on the diffusion of
heat. The work was first presented to the French Academy in 1807, and published in final form in 1822 in the
Théorie analytique de la chaleur(Analytic theory of heat). It generated new developments in the mathematical
theory of functions, and provided a new and powerful tool for the analysis and solution of physical problems.