# Hypergeometric function using Euler's integral representation

### Description

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

### Usage

1 |

### 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

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)
``` |

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