Page 504 - Electromagnetics

H (1) (z) and H (2) (z) are the Hankel functions of the first and second kind of order ν, and
ν ν
are related to the Bessel functions by
H (1) (z) = J ν (z) + jN ν (z),
ν
H (2) (z) = J ν (z) − jN ν (z).
ν
The argument z can be complex (as can ν, but this shall not concern us). When z is
imaginary we introduce two new functions I ν (z) and K ν (z), defined for integer order by
−n
I n (z) = j J n ( jz),
π n+1 (1)
K n (z) = j H n ( jz).
2
Expressions for noninteger order are given in Appendix E.1.
Bessel functions cannot be expressed in terms of simple, standard functions. However,
a series solution to (A.124) produces many useful relationships between Bessel functions
of differing order and argument. The recursion relations for Bessel functions serve to
connect functions of various orders and their derivatives. See Appendix E.1.
Of the six possible solutions to (A.124),


A ρ J ν (k c ρ) + B ρ N ν (k c ρ),

 (1)
A ρ J ν (k c ρ) + B ρ H ν (k c ρ),



A ρ J ν (k c ρ) + B ρ H ν (k c ρ),
 (2)

R(ρ) = (1)
A ρ N ν (k c ρ) + B ρ H (k c ρ),
 ν

A ρ N ν (k c ρ) + B ρ H (2) (k c ρ),


ν


A ρ H (k c ρ) + B ρ H (k c ρ),
 (1) (2)
ν ν
which do we choose? Again, we are motivated by convenience and the physical nature
of the problem. If the argument is real or imaginary, we often consider large or small
argument behavior. For x real and large,
)
2 π π
J ν (x) → cos x − − ν ,
πx 4 2
)
2 π π
N ν (x) → sin x − − ν ,
πx 4 2
)
2 j(x− −ν )
π
π
(1)
H (x) → e 4 2 ,
ν
πx
)
2 − j(x− −ν )
π
π
(2)
H ν (x) → e 4 2 ,
πx
)
1 x
I ν (x) → e ,
2πx
)
π −x
K ν (x) → e ,
2x
while for x real and small,
J 0 (x) → 1,
2
N 0 (x) → (ln x + 0.5772157 − ln 2) ,
π
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