Page 531 - Electromagnetics

2 ¯
¯
d Z ν (x) 1 dZ ν (z) ν 2
¯
+ − 1 + Z ν = 0 (E.6)
dz 2 z dz z 2

¯ I ν (z)
Z ν (z) = (E.7)
K ν (z)

I ν (z)
L(z) = jνπ (E.8)
e K ν (z)
π
− jνπ/2
jπ/2
I ν (z) = e J ν (ze ), −π< arg(z) ≤ (E.9)
2
π
j3νπ/2
− j3π/2
I ν (z) = e J ν (ze ), < arg(z) ≤ π (E.10)
2
jπ jνπ/2 (1) jπ/2 π
K ν (z) = e H ν (ze ), −π< arg(z) ≤ (E.11)
2 2
jπ − jνπ/2 (2) − jπ/2 π
K ν (z) =− e H ν (ze ), − < arg(z) ≤ π (E.12)
2 2
I n (x) = j −n J n ( jx) (E.13)
π n+1 (1)
K n (x) = j H n ( jx) (E.14)
2
2
d z n (z) 2 dz n (z) n(n + 1)
+ + 1 − z n (z) = 0, n = 0, ±1, ±2,... (E.15)
dz 2 z dz z 2

 j n (z)

n n (z)

z n (z) = (1) (E.16)
 h (z)
n

h (z)
 (2)
n

π
j n (z) = J 1 (z) (E.17)
2z n+ 2

π
n n (z) = N 1 (z) (E.18)
2z n+ 2
π

(1) (1)
h (z) = H 1 (z) = j n (z) + jn n (z) (E.19)
n
2z n+ 2

π
(2) (2)
h (z) = H 1 (z) = j n (z) − jn n (z) (E.20)
n
2z n+ 2
n n (z) = (−1) n+1 j −(n+1) (z) (E.21)
Orthogonality relationships
a

2 2
p νm p νn a 2 a 2
J ν ρ J ν ρ ρ dρ = δ mn J ν+1 (p νn ) = δ mn J (p νn ) , ν > −1
ν
a a 2 2
0
(E.22)
© 2001 by CRC Press LLC