4.3 Bessel Functions and Related Functions
73
path method will be used. Note, half-integer orders are especially of importance for
spherical Bessel functions.
For complex orders ν we use directly the series expansion (4.11). For ν negative
integers (4.11) will diverge and thus we map −n → n due to Eq. (4.13). The
series expansion (4.11) will be split in four subseries: real or imaginary, positive
or negative, and reordered as described above. A second source of errors is the
truncation of the series. The number of terms will be estimated by the absolute
value of the argument z. Therefore we have to check the results with respect to both
effects on convergency. In case no convergency could be reached, either we use an
approximate equation [2] or the path integral technique.
The approximate equation for |z| > 1 is given by
J ±ν (z) =
2
πz
cos
z ∓
π
2
ν −
π
4
·
n−1
k=0
(−1) k
(2z) 2k
Γ (ν + 2k +
1
2 )
(2k)!Γ (ν − 2k +
1
2 )
(4.20)
− sin
z ∓
π
2
ν −
π
4
·
n−1
k=0
(−1) k
(2z) 2k+1
Γ (ν + 2k +
3
2 )
(2k + 1)!Γ (ν − 2k +
1
2 )
,
with | arg z| < π. Again, by splitting the total series in four subseries errors are
minimized and convergency carefully controlled.
In case none of the methods listed above the path integral method will be used.
Thus based on differential equation (4.10) a path will be defined via z(s) = z 0 +
s(z 1 − z0) to compute J ν (z 1 ). The initial conditions at position z 0 will be computed
and for the derivative Eq. (4.17) will be used.
Computational Aspects: Bessel Function Y ν (z)
For ν half-integer, the following equations will be used:
Y 0.5 (z) = −
2
πz
cos(z) , Y −0.5 (z) =
2
πz
sin(z) ,
(4.21)
and Eq. (4.12) reduces to
Y n+
1
2
= (−1)
n−1 J −n−
1
2
(z) , Y −n−
1
2
= (−1)
n J n+
1
2
(z) ,
(4.22)
with n positive integer.
For general orders ν as a first Eq. (4.12) will be tried. In case no convergency
could be reached the path integral technique for Y ν (z) will be used as described
above.
73
path method will be used. Note, half-integer orders are especially of importance for
spherical Bessel functions.
For complex orders ν we use directly the series expansion (4.11). For ν negative
integers (4.11) will diverge and thus we map −n → n due to Eq. (4.13). The
series expansion (4.11) will be split in four subseries: real or imaginary, positive
or negative, and reordered as described above. A second source of errors is the
truncation of the series. The number of terms will be estimated by the absolute
value of the argument z. Therefore we have to check the results with respect to both
effects on convergency. In case no convergency could be reached, either we use an
approximate equation [2] or the path integral technique.
The approximate equation for |z| > 1 is given by
J ±ν (z) =
2
πz
cos
z ∓
π
2
ν −
π
4
·
n−1
k=0
(−1) k
(2z) 2k
Γ (ν + 2k +
1
2 )
(2k)!Γ (ν − 2k +
1
2 )
(4.20)
− sin
z ∓
π
2
ν −
π
4
·
n−1
k=0
(−1) k
(2z) 2k+1
Γ (ν + 2k +
3
2 )
(2k + 1)!Γ (ν − 2k +
1
2 )
,
with | arg z| < π. Again, by splitting the total series in four subseries errors are
minimized and convergency carefully controlled.
In case none of the methods listed above the path integral method will be used.
Thus based on differential equation (4.10) a path will be defined via z(s) = z 0 +
s(z 1 − z0) to compute J ν (z 1 ). The initial conditions at position z 0 will be computed
and for the derivative Eq. (4.17) will be used.
Computational Aspects: Bessel Function Y ν (z)
For ν half-integer, the following equations will be used:
Y 0.5 (z) = −
2
πz
cos(z) , Y −0.5 (z) =
2
πz
sin(z) ,
(4.21)
and Eq. (4.12) reduces to
Y n+
1
2
= (−1)
n−1 J −n−
1
2
(z) , Y −n−
1
2
= (−1)
n J n+
1
2
(z) ,
(4.22)
with n positive integer.
For general orders ν as a first Eq. (4.12) will be tried. In case no convergency
could be reached the path integral technique for Y ν (z) will be used as described
above.