Mathematical Methods for Physicists: A concise introduction - Site Map

MULTIPLE FOURIER SERIES
the series on the right hand side is called an orthonormal series; such series are
generalizations of Fourier series. Assuming that the series on the right converges
to f …x†, we can then multiply both sides by ' m …x† and integrate both sides from a
to b to obtain
c m ˆhf …x†j' m …x†i ˆ
Z b
c m can be called the generalized Fourier coecients.
a
f …x†' m …x†dx;
…4:21a†
Multiple Fourier series
A Fourier expansion of a function of two or three variables is often very useful in
many applications. Let us consider the case of a function of two variables, say
f …x; y†. For example, we can expand f …x; y† into a double Fourier sine series
where
f …x; y† ˆX1
B mn ˆ 4
L 1 L 2
Z L1
0
X 1
mˆ1 nˆ1
Z L2
0
B mn sin mx sin ny ;
L 1 L 2
f …x; y† sin mx sin ny dxdy:
L 1 L 2
…4:22†
…4:22a†
Similar expansions can be made for cosine series and for series having both sines
and cosines.
To obtain the coecients B mn , let us rewrite f …x; y† as
where
f …x; y† ˆX1
C m ˆ X1
mˆ1
nˆ1
C m sin mx
L 1
;
B mn sin ny
L 2
:
…4:23†
…4:23a†
Now we can consider Eq. (4.23) as a Fourier series in which y is kept constant
so that the Fourier coecients C m are given by
C m ˆ 2
L 1
Z L1
0
f …x; y† sin mx
L 1
dx: …4:24†
On noting that C m is a function of y, we see that Eq. (4.23a) can be considered as a
Fourier series for which the coecients B mn are given by
B mn ˆ 2
L 2
Z L2
0
163
C m sin ny
L 2
dy: