Mathematical Methods for Physicists: A concise introduction - Site Map

FOURIER SERIES; EULER±FOURIER FORMULAS
Z
sin mx cos nx dx ˆ 0; for all m; n > 0;
we ®nd that all terms on the right hand side of Eq. (4.3) except one vanish:
a n ˆ 1 Z
f …x† cos nx dx; n ˆ integers; …4:4a†
the expression for a 0 can be obtained from the general expression for a n by setting
n ˆ 0.
Similarly, if Eq. (4.2) is multiplied through by sin mx and the result is integrated
from to , all terms vanish save that involving the square of sin nx, and so we
have
b n ˆ 1 Z
f …x† sin nx dx:
…4:4b†
Eqs. (4.4a) and (4.4b) are known as the Euler±Fourier formulas.
From the de®nition of a de®nite integral it follows that, if f …x† is single-valued
and continuous within the interval ‰; Š or merely piecewise continuous (continuous
except at a ®nite numbers of ®nite jumps in the interval), the integrals in
Eqs. (4.4) exist and we may compute the Fourier coecients of f …x† by Eqs. (4.4).
If there exists a ®nite discontinuity in f …x† at the point x 0 (Fig. 4.1), the coecients
a 0 ; a n ; b n are determined by integrating ®rst to x ˆ x 0 and then from x 0 to ,
as
a n ˆ 1 Z x0
Z
f …x† cos nx dx ‡ f …x† cos nx dx ; …4:5a†
b n ˆ 1
Z x0
f …x† sin nx dx ‡
x 0
Z
x 0
f …x† sin nx dx : …4:5b†
This procedure may be extended to any ®nite number of discontinuities.
Example 4.2
Find the Fourier series which represents the function
k