Mathematical Methods for Physicists: A concise introduction - Site Map

FOURIER SERIES AND INTEGRALS
but
b n ˆ 1
ˆ 2
Z 0
Z
0
f o …x† sin nx dx ‡
Z
0
f o …x† sin nx dx
f o …x† sin nx dx n ˆ 1; 2; 3; ...: …4:6b†
Here we have made use of the fact that cos(nx† ˆcos nx and sin…nx† ˆ
sin nx. Accordingly, the Fourier series becomes
f o …x† ˆb 1 sin x ‡ b 2 sin 2x ‡:
Similarly, in the Fourier series corresponding to an even function f e …x†, only
cosine terms (and possibly a constant) can be present. Because in this case,
f e …x† sin nx is an odd function and accordingly b n ˆ 0 and the a n are given by
a n ˆ 2 Z
f
e …x† cos nx dx n ˆ 0; 1; 2; ...: …4:7†
0
Note that the Fourier coecients a n and b n , Eqs. (4.6) and (4.7) are computed in
the interval (0, ) which is half of the interval (; ). Thus, the Fourier sine or
cosine series in this case is often called a half-range Fourier series.
Any arbitrary function (neither even nor odd) can be expressed as a combination
of f e …x† and f o …x† as
f …x† ˆ1
2 ‰ f …x†‡f …x† Š‡ 1 2 ‰ f …x†f …x† Š ˆ f e…x†‡f o …x†:
When a half-range series corresponding to a given function is desired, the
function is generally de®ned in the interval (0, ) and then the function is speci®ed
as odd or even, so that it is clearly de®ned in the other half of the interval …; 0†.
Change of interval
A Fourier expansion is not restricted to such intervals as