hypergeo
The Gauss Hypergeometric Function
The Gaussian hypergeometric function for complex numbers.
The hypergeo package: the Gauss hypergeometric function in R
To cite the package in publications, please use Hankin 2015.
Overview
The hypergeo package provides functionality for working with the Gauss
hypergeometric function. The package uses a range of transformations to
ensure rapid numerical evaluation of the Gauss hypergeometric function,
and generalized hypergeometric functions. It is vectorized in argument
. If
we
have
where
is the rising Pochhammer symbol; if
the series may be analytically continued.
Installation
You can install the released version of hypergeo from
CRAN with:
# install.packages("hypergeo") # uncomment this to install the package
library("hypergeo")
The hypergeo package in use
The main workhorse of the package is hypergeo():
hypergeo(1/2,1/3,1/5, 1+8i)
#> [1] -0.0901118+0.2390856i
Observe that the hypergeometric series has radius of convergence one,
but analytic continuation is used to evaluate the function outside the
unit circle. The package works well with the visualization suite
provided by the elliptic package (Hankin 2008):
suppressMessages(library("elliptic"))
x <- seq(from=-2, to=2, len=100)
y <- x
z <- outer(x,1i*y, "+")
f <- hypergeo(1/2,1/3,1/5,z)
par(pty="s")
view(x,y,limit(f),code=0,drawlabel=FALSE,axes=FALSE,xlab="Re(z)",ylab="Im(z)")
References
- M. Abramowitz and I. A. Stegun 1965. Handbook of Mathematical Functions, New York: Dover.
- R. K. S. Hankin 2006. “Introducing
elliptic, anRpackage for elliptic and modular functions”, Journal of Statistical Software, 15(7). - R. K. S. Hankin 2015. “Numerical evaluation of the Gauss
hypergeometric function with the
hypergeopackage”. The R journal, 7(2): 81–88