Page 490 - Electromagnetics

where, with σ(x) = 1 in (A.85), we have

L nπx L
0 f (x) sin L dx 2 nπx
c n = = f (x) sin dx.
L
2 nπx L 0 L
0 sin L dx
Hence we recover the standard Fourier sine series for f (x).
With little extra effort we can examine the eigenfunctions resulting from enforcement
of the periodic boundary conditions

ψ(0) = ψ(L) and ψ (0) = ψ (L).

The general solution (A.88) still holds, so we have the choices ψ(x) = sin kx and ψ(x) =
cos kx. Evidently both
2nπx 2nπx

ψ(x) = sin and ψ(x) = cos
L L
2
satisfy the boundary conditions for n = 1, 2,.... Thus each eigenvalue (2nπ/L) is
associated with two eigenfunctions.

Bessel’s differential equation. Bessel’s equation
d dψ(x) 2 ν 2
x + k x − ψ(x) = 0 (A.90)
dx dx x
occurs when problems are solved in circular-cylindrical coordinates. Comparison with
2
2
(A.77) shows that λ = k , p(x) = x, q(x) =−ν /x, and σ(x) = x. We take [a, b] = [0, L]
along with the boundary conditions
ψ(L) = 0 and |ψ(0)| < ∞. (A.91)
Although the resulting Sturm–Liouville problem is singular, the specified conditions
(A.91) maintain satisfaction of (A.80). The eigenfunctions are orthogonal because (A.80)
is satisfied by having ψ(L) = 0 and p(x) dψ(x)/dx → 0 as x → 0.
As a second-order ordinary differential equation, (A.90) has two solutions denoted by
J ν (kx) and N ν (kx),

and termed Bessel functions. Their properties are summarized in Appendix E.1. The
function J ν (x), the Bessel function of the first kind and order ν, is well-behaved in [0, L].
The function N ν (x), the Bessel function of the second kind and order ν, is unbounded
at x = 0; hence it is excluded as an eigenfunction of the Sturm–Liouville problem.
The condition at x = L shows that the eigenvalues are defined by
J ν (kL) = 0.

We denote the mth root of J ν (x) = 0 by p νm . Then

k νm = λ νm = p νm /L.
The infinitely many eigenvalues are ordered as λ ν1 <λ ν2 <. . .. Associated with eigen-
√
value λ νm is a single eigenfunction J ν ( λ νm x). The orthogonality relation is
L
p νm p νn
J ν x J ν x xdx = 0, m = n.
0 L L

© 2001 by CRC Press LLC