genhypergeo {hypergeo} | R Documentation |
The generalized hypergeometric function, using either the series expansion or the continued fraction expansion.
genhypergeo(U, L, z, tol = 0, maxiter = 2000, check_mod = TRUE, polynomial = FALSE, debug = FALSE, series = TRUE) genhypergeo_series(U, L, z, tol = 0, maxiter = 2000, check_mod = TRUE, polynomial = FALSE, debug = FALSE) genhypergeo_contfrac(U, L, z, tol = 0, maxiter = 2000)
U,L |
Upper and lower arguments respectively (currently real) |
z |
Primary complex argument (see notes) |
tol |
tolerance with default zero meaning to iterate until additional terms to not change the partial sum |
maxiter |
Maximum number of iterations to perform |
check_mod |
Boolean, with default TRUE meaning to check
that the modulus of z is less than 1 |
polynomial |
Boolean, with default FALSE meaning to
evaluate the series until converged, or return a warning; and
TRUE meaning to return the sum of maxiter terms,
whether or not converged. This is useful when either A
orB is a nonpositive integer in which case the hypergeometric
function is a polynomial |
debug |
Boolean, with TRUE meaning to return debugging
information and default FALSE meaning to return just the
evaluate |
series |
In function genhypergeo() , Boolean argument with
default TRUE meaning to return the result of
genhypergeo_series() and FALSE the result of
genhypergeo_contfrac() |
Function genhypergeo()
is a wrapper for functions
genhypergeo_series()
and genhypergeo_contfrac()
.
Function genhypergeo_series()
is the workhorse for the whole
package; every call to hypergeo()
uses this function except for
the (apparently rare—but see the examples section) cases where
continued fractions are used.
The generalized hypergeometric function [here genhypergeo()
]
appears from time to time in the literature (eg Mathematica) as
[omitted; see PDF]
where U=(u_1,...,u_i) and L=(l_1,...,l_i) are the “upper” and “lower” vectors respectively. The radius of convergence of this formula is 1.
For the Confluent Hypergeometric function, use genhypergeo()
with
length-1 vectors for arguments U
and V
.
For the 0F1 function (ie no “upper” arguments), use
genhypergeo(NULL,L,x)
.
See documentation for genhypergeo_contfrac()
for details of
the continued fraction representation.
The radius of convergence for the series is 1 but under some circumstances, analytic continuation defines a function over the whole complex plane (possibly cut along (0,inf)). Further work would be required to implement this.
Robin K. S. Hankin
M. Abramowitz and I. A. Stegun 1965. Handbook of mathematical functions. New York: Dover
genhypergeo(U=c(1.1,0.2,0.3), L=c(10.1,pi*4),z=1.12+0.2i) genhypergeo(U=c(1.1,0.2,0.3), L=c(10.1,pi*4),z=1.12+0.2i,series=FALSE)