4.3 Bessel Functions and Related Functions
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4.3
Bessel Functions and Related Functions
4.3.1 Bessel and Hankel Functions
The Bessel equation is a second order differential equations given by
z
2 d 2
dz 2 + z
d
dz
+ (z
2
− ν
2 )
φ = 0 ,
(4.10)
with ν complex. The general solution of Bessel’s equation yields Bessel functions
of the first kind J ν (z)
J ν (z) =
z
2
ν
∞
k=0
−z 2
4
k
k!Γ (ν + k + 1)
(4.11)
and of the second kind Y ν (z) with
Y ν (z) =
J ν (z) cos(νπ) − J −ν (z)
sin(νπ)
,
(4.12)
where we have to take the limit ν → n for integer values. The Bessel functions are
also called cylinder functions and the Bessel functions of second kind “Y” Neumann
functions N ν (z). In contrast to non-integer orders ν, for integer orders J ν , J −ν are
not linearly independent
J −n (z) = (−1)
n J n (z),
Y −n (z) = (−1)
n Y n (z),
(4.13)
and thus Y is needed as second linearly independent solution to Bessel’s equation.
(For non-integer values J −ν , J ν can serve as independent solution.)
The Bessel functions of third kind or Hankel functions are given as
H
(1)
ν (z) = J ν (z) + iY ν (z),
(4.14)
H
(2)
ν (z) = J ν (z) − iY ν (z),
(4.15)
which provide an alternative pair of solutions to Bessel’s differential equation.
The Bessel functions fulfill various recurrence formulas
J ν+1 (z) =
2ν
z
J ν (z) − J ν−1 (z) , Y ν+1 (z) =
2ν
z
Y ν − Y ν−1 (z) , (4.16)
d
dz
J ν (z) = J ν−1 (z) −
ν
z
J ν (z) ,
d
dz
Y ν (z) = Y ν−1 (z) −
ν
z
Y ν (z) . (4.17)