94
6 Confluent Hypergeometric Function
is first tried, where the upper sign is taken if −
1
2 π < arg(z) <
3
2 π, and the lower
sign if −
3
2 π < arg(z) < −
1
2 π. For small absolute values of the phase angle both
ansätze result in the same numerical value. Thus, the plus sign is taken for 0 <
arg(z) ≤ π otherwise the minus sign. To minimize cancellation errors, the two
sums are combined to one single sum and then treated in a similar way as described
above for the hypergeometric series. In case of no convergency the path integration
technique will be used. For some special parameters elementary functions can be
evaluated:
1 F 1 (a; a; z) = exp(z) and
(6.9a)
1 F 1 (1; 2; 2z) =
exp(z)
z
sinh(z) .
(6.9b)
A comprehensive list of special cases can be found, e.g., in [1] or [2].
6.1.2 The Function U and 2F 0
The functions 1 F 1 (a; b; z) and U(a, b, z) are independent solutions of Kummer’s
equation (6.3), with
U(a, b, z) =
Γ (1 − b)
Γ (1 + a − b)
1 F 1 (a; b; z)
+
Γ (b − 1)
Γ (a)
z
1−b
1 F 1 (1 + a − b; 2 − b; z),
(6.10a)
respectively,
U(a, b, z) = z
−a
2 F 0 (a, 1 + a − b; ; −1/z).
(6.10b)
The Kummer transformation reads
U(a, b, z) = z
1−b U(1 + a − b, 2 − b, z) .
(6.11)
The series representation of the hypergeometric function 2 F 0 is
2 F 0 (a, b; ; z) =
∞
k=0
(a) k (b) k
z k
k!
.
(6.12)
Unless a or b are negative integers the series representation of 2 F 0 is divergent.
6 Confluent Hypergeometric Function
is first tried, where the upper sign is taken if −
1
2 π < arg(z) <
3
2 π, and the lower
sign if −
3
2 π < arg(z) < −
1
2 π. For small absolute values of the phase angle both
ansätze result in the same numerical value. Thus, the plus sign is taken for 0 <
arg(z) ≤ π otherwise the minus sign. To minimize cancellation errors, the two
sums are combined to one single sum and then treated in a similar way as described
above for the hypergeometric series. In case of no convergency the path integration
technique will be used. For some special parameters elementary functions can be
evaluated:
1 F 1 (a; a; z) = exp(z) and
(6.9a)
1 F 1 (1; 2; 2z) =
exp(z)
z
sinh(z) .
(6.9b)
A comprehensive list of special cases can be found, e.g., in [1] or [2].
6.1.2 The Function U and 2F 0
The functions 1 F 1 (a; b; z) and U(a, b, z) are independent solutions of Kummer’s
equation (6.3), with
U(a, b, z) =
Γ (1 − b)
Γ (1 + a − b)
1 F 1 (a; b; z)
+
Γ (b − 1)
Γ (a)
z
1−b
1 F 1 (1 + a − b; 2 − b; z),
(6.10a)
respectively,
U(a, b, z) = z
−a
2 F 0 (a, 1 + a − b; ; −1/z).
(6.10b)
The Kummer transformation reads
U(a, b, z) = z
1−b U(1 + a − b, 2 − b, z) .
(6.11)
The series representation of the hypergeometric function 2 F 0 is
2 F 0 (a, b; ; z) =
∞
k=0
(a) k (b) k
z k
k!
.
(6.12)
Unless a or b are negative integers the series representation of 2 F 0 is divergent.