hypergeo (version 1.2-14)

f15.3.1: Hypergeometric function using Euler's integral representation

Description

Hypergeometric function using Euler's integral representation, evaluated using numerical contour integrals.

Usage

f15.3.1(A, B, C, z, h = 0)

Arguments

A,B,C

Parameters

z

Primary complex argument

h

specification for the path to be taken; see details

Author

Robin K. S. Hankin

Details

Argument h specifies the path to be taken (the path has to avoid the point \(1/z\)). If h is real and of length 1, the path taken comprises two straight lines: one from \(0\) to \(0.5+hi\) and one from \(0.5+hi\) to \(1\) (if \(h=0\) the integration is performed over a single segment).

Otherwise, the integration is performed over length(h)+1 segments: \(0\) to h[1], then h[i] to h[i+1] for \(1\leq i\leq n-1\) and finally h[n] to 1.

See examples and notes sections below.

References

M. Abramowitz and I. A. Stegun 1965. Handbook of mathematical functions. New York: Dover

See Also

hypergeo

Examples

# For |z| <1 the path can be direct:
f15.3.1(2,1,2,-1/2) -2/3

# cf identity 07.23.03.0046.01 of Hypergeometric2F1.pdf with b=1




f <- function(h){f15.3.1(1,2,3, z=2, h=h)}

# Winding number [around 1/z] matters:
f(0.5)
f(c(1-1i, 1+1i, -2i))

# Accuracy isn't too bad; compare numerical to analytical result :
f(0.5) - (-1+1i*pi/2)