Mathematical Methods for Physicists: A concise introduction - Site Map

FOURIER SERIES AND INTEGRALS
Fourier series; Euler±Fourier formulas
If the general periodic function f …x† is de®ned in an interval x , the
Fourier series of f …x† in [; ] is de®ned to be a trigonometric series of the form
f …x† ˆ1
2 a 0 ‡ a 1 cos x ‡ a 2 cos 2x ‡‡a n cos nx ‡
‡ b 1 sin x ‡ b 2 sin 2x ‡‡b n sin nx ‡; …4:2†
where the numbers a 0 ; a 1 ; a 2 ; ...; b 1 ; b 2 ; b 3 ; ...are called the Fourier coecients of
f …x† in ‰; Š. If this expansion is possible, then our power to solve physical
problems is greatly increased, since the sine and cosine terms in the series can be
handled individually without diculty. Joseph Fourier (1768±1830), a French
mathematician, undertook the systematic study of such expansions. In 1807 he
submitted a paper (on heat conduction) to the Academy of Sciences in Paris and
claimed that every function de®ned on the closed interval ‰; Š could be represented
in the form of a series given by Eq. (4.2); he also provided integral formulas
for the coecients a n and b n . These integral formulas had been obtained earlier by
Clairaut in 1757 and by Euler in 1777. However, Fourier opened a new avenue by
claiming that these integral formulas are well de®ned even for very arbitrary
functions and that the resulting coecients are identical for di€erent functions
that are de®ned within the interval. Fourier's paper was rejected by the Academy
on the grounds that it lacked mathematical rigor, because he did not examine the
question of the convergence of the series.
The trigonometric series (4.2) is the only series which corresponds to f …x†.
Questions concerning its convergence and, if it does, the conditions under
which it converges to f …x† are many and dicult. These problems were partially
answered by Peter Gustave Lejeune Dirichlet (German mathematician, 1805±
1859) and will be discussed brie¯y later.
Now let us assume that the series exists, converges, and may be integrated term
by term. Multiplying both sides by cos mx, then integrating the result from to
, wehave
Z
f …x† cos mx dx ˆ a0
2
Z
‡ X1
nˆ1
cos mx dx ‡ X1
nˆ1
Z
a n cos nx cos mx dx
Z
b n sin nx cos mx dx:
Now, using the following important properties of sines and cosines:
Z
Z
cos mx dx ˆ
cos mx cos nx dx ˆ
Z
Z
sin mx dx ˆ 0 if m ˆ 1; 2; 3; ...;
(
0 if n 6ˆ m;
sin mx sin nx dx ˆ
if n ˆ m;
…4:3†
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