Page 536 - Electromagnetics

Integrals

x ν+1 J ν (x) dx = x ν+1 J ν+1 (x) + C (E.104)

[bZ ν (ax)Z ν−1 (bx) − aZ ν−1 (ax)Z ν (bx)]
Z ν (ax)Z ν (bx)xdx = x + C, a = b (E.105)
2
a − b 2
2
2 x 2
xZ (ax) dx = Z (ax) − Z ν−1 (ax)Z ν+1 (ax) + C (E.106)
ν
ν
2
∞ 1

J ν (ax) dx = , ν > −1, a > 0 (E.107)
0 a
Fourier–Bessel expansion of a function
∞
ρ
f (ρ) = a m J ν p νm , 0 ≤ ρ ≤ a, ν > −1 (E.108)
a
m=1
2 a
ρ
a m = f (ρ)J ν p νm ρ dρ (E.109)
2
a J 2 (p νm ) a
ν+1 0
∞
ρ
f (ρ) = b m J ν p , 0 ≤ ρ ≤ a, ν > −1 (E.110)
νm
m=1 a
2 a p νm
b m =
f (ρ)J ν ρ ρ dρ (E.111)
2
a 2 1 − ν 2 2 J (p ) 0 a

νm
ν
p νm
Series of Bessel functions
∞

k
e jz cos φ = j J k (z)e jkφ (E.112)
k=−∞
∞
k
jz cos φ
e = J 0 (z) + 2 j J k (z) cos φ (E.113)
k=1
∞
k
sin z = 2 (−1) J 2k+1 (z) (E.114)
k=0
∞
k
cos z = J 0 (z) + 2 (−1) J 2k (z) (E.115)
k=1
E.2 Legendre functions

Notation

x, y,θ = real numbers; l, m, n = integers;
m
P (cos θ) = associated Legendre function of the first kind
n

© 2001 by CRC Press LLC