96
6 Confluent Hypergeometric Function
The accuracy of a path integral based result depends on the accuracy of the
corresponding initial values, which are tested by the following recurrence relation:
azU (a + 1, b + 1, z) + (z − b + 1)U (a, b, z) − U(a − 1, b − 1, z) = 0. (6.15)
For large arguments z the asymptotic series
U(a, b, z) =
n
k=0
(a) k (1 + a − b) k
k!
z
−k
(6.16)
is tried. In some cases none of the method described above is successful and in some
cases the accuracy is only about 0.001 · · · 0.01. This is especially the case for large
real difference between a and b in combination with intermediate absolute values of
the argument z.
6.2
The Confluent Hypergeometric Limit Function
The confluent hypergeometric limit function is defined by
0 F 1 (; b; z) = lim
a→∞
1 F 1
a; b;
z
a
(6.17)
with series representation
0 F 1 (; b; z) =
∞
k=0
1
(b) k
z k
k!
,
(6.18)
convergent in C. The differential equation for 0 F 1 (; b; z) is
z
d 2 w
dz 2 + b
dw
dz
− w = 0.
(6.19)
The confluent hypergeometric limit function is related to the Bessel functions:
J b (z) =
x
2
b
Γ (b + 1)
0 F 1
; b + 1; −
1
4
z
2
I b (z) =
x
2
b
Γ (b + 1)
0 F 1
; b + 1;
1
4
z
2
,