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

f15.3.1R Documentation

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

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.

Note

The Mellin-Barnes form is not yet coded up.

Author(s)

Robin K. S. Hankin

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)


RobinHankin/hypergeo documentation built on July 18, 2024, 9:49 p.m.