R: Hypergeometric function using Euler's integral representation

f15.3.1 {hypergeo} | R 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 <= i <= 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)

[Package hypergeo version 1.2-13 Index]