8
Hypergeometric Functions
Abstract
Focus of this chapter is the hypergeometric function 2 F 1 (a, b; c; z) in the complex domain with complex function parameters. Corresponding programming
code for download is available. With various coordinate transformations the
hypergeometric function will be mapped on arguments outside the unit circle.
The numerical techniques used and discussed are direct series expansions,
path integral methods and recurrence formulas in dependence of the function
argument and function parameters.
In this chapter we focus on the hypergeometric function 2 F 1 or Gauss hypergeometric function for complex parameters and arguments. Many functions have
representations as hypergeometric function [3]; applications can be found, e.g., in
quantum dynamics [1]. (For some additional remarks see also Sect. 6).
The Gauss hypergeometric function 2 F 1 (a, b; c; z) can be evaluated with the
SPECFUNPHYS-class gausshyp.
8.1
The Hypergeometric Series
The Gauss hypergeometric series
2 F 1 (a, b; c; z) =
∞
k
(a) k (b) k
(c) k
z n
n!
,
(8.1)
© 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_8
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