Hypergeometric function using Euler's integral representation
Hypergeometric function using Euler's integral representation, evaluated using numerical contour integrals.
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
# 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)Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.