6
Confluent Hypergeometric Function
Abstract
Topics of this chapter are the confluent hypergeometric function of first and
second kind, the hypergeometric function 2 F 0 (a 1 , a 2 ; ; z), the confluent hypergeometric limit function 0 F 1 (; b; z), and the Whittaker functions. The evaluation
of the functions will be based either on a series expansion or on a path integration
technique in dependence of the function argument and the functions parameters.
The computations support the complex domain and complex function parameters. The programming code is free available for download.
Hypergeometric functions play an important role for many applications in physics.
In this chapter we will focus on confluent hypergeometric functions and in Chap. 8
on Gaussian hypergeometric functions.
For hypergeometric series we use the following notation:
p F q (a 1 , · · · , a p ; b 1 , · · · , b q ; z) =
∞
k=0
(a 1 ) k · · ·
a p
k
(b 1 ) k · · ·
b q
k
z k
k!
,
(6.1)
with (x) n the Pochhammer Symbol (1.8), and p refers to the number of parameters
in the numerator and q in the denominator. The radius of convergence R for the
series representation is
R =
⎧
⎨
⎩
∞ : p ≤ q
1 : p = q + 1
0 : p > q + 1.
(6.2)
Even if the hypergeometric series does not convergence, there exists an analytic
continuation for the corresponding hypergeometric function, and even if the func© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
W. Schweizer, Special Functions in Physics with MATLAB,
https://doi.org/10.1007/978-3-030-64232-7_6
91
Confluent Hypergeometric Function
Abstract
Topics of this chapter are the confluent hypergeometric function of first and
second kind, the hypergeometric function 2 F 0 (a 1 , a 2 ; ; z), the confluent hypergeometric limit function 0 F 1 (; b; z), and the Whittaker functions. The evaluation
of the functions will be based either on a series expansion or on a path integration
technique in dependence of the function argument and the functions parameters.
The computations support the complex domain and complex function parameters. The programming code is free available for download.
Hypergeometric functions play an important role for many applications in physics.
In this chapter we will focus on confluent hypergeometric functions and in Chap. 8
on Gaussian hypergeometric functions.
For hypergeometric series we use the following notation:
p F q (a 1 , · · · , a p ; b 1 , · · · , b q ; z) =
∞
k=0
(a 1 ) k · · ·
a p
k
(b 1 ) k · · ·
b q
k
z k
k!
,
(6.1)
with (x) n the Pochhammer Symbol (1.8), and p refers to the number of parameters
in the numerator and q in the denominator. The radius of convergence R for the
series representation is
R =
⎧
⎨
⎩
∞ : p ≤ q
1 : p = q + 1
0 : p > q + 1.
(6.2)
Even if the hypergeometric series does not convergence, there exists an analytic
continuation for the corresponding hypergeometric function, and even if the func© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
W. Schweizer, Special Functions in Physics with MATLAB,
https://doi.org/10.1007/978-3-030-64232-7_6
91