f15.3.1: Hypergeometric function using Euler's integral representation
In hypergeo: The Gauss Hypergeometric Function

Description Usage Arguments Details Note Author(s) References See Also Examples

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

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

`A,B,C`	Parameters
`z`	Primary complex argument
`h`	specification for the path to be taken; see 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 <= i <= n-1 and finally h[n] to 1.

See examples and notes sections below.

The Mellin-Barnes form is not yet coded up.

Robin K. S. Hankin

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

hypergeo

# 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)