Page 530 - Electromagnetics

Appendix E

Properties of special functions

E.1 Bessel functions

Notation

z = complex number; ν, x = real numbers; n = integer
J ν (z) = ordinary Bessel function of the first kind
N ν (z) = ordinary Bessel function of the second kind
I ν (z) = modified Bessel function of the first kind
K ν (z) = modified Bessel function of the second kind
H ν (1) = Hankel function of the first kind
H ν (2) = Hankel function of the second kind
j n (z) = ordinary spherical Bessel function of the first kind
n n (z) = ordinary spherical Bessel function of the second kind
(1)
h (z) = spherical Hankel function of the first kind
n
(2)
h (z) = spherical Hankel function of the second kind
n
f (z) = df (z)/dz = derivative with respect to argument

Differential equations
2
d Z ν (z) 1 dZ ν (z) ν 2
+ + 1 − Z ν (z) = 0 (E.1)
dz 2 z dz z 2


 J ν (z)

N ν (z)

Z ν (z) = (1) (E.2)
 H ν (z)

H (z)
 (2)
ν
cos(νπ)J ν (z) − J −ν (z)
N ν (z) = , ν = n, | arg(z)| <π (E.3)
sin(νπ)

H (1) (z) = J ν (z) + jN ν (z) (E.4)
ν
H (2) (z) = J ν (z) − jN ν (z) (E.5)
ν

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