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

## Description

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

## Usage

 `1` ```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

`hypergeo`
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16``` ```# 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) ```