hypergeo (version 1.2-13)

hypergeo: The hypergeometric function

Description

The Hypergeometric and generalized hypergeometric functions as defined by Abramowitz and Stegun. Function hypergeo() is the user interface to the majority of the package functionality; it dispatches to one of a number of subsidiary functions.

Usage

hypergeo(A, B, C, z, tol = 0, maxiter=2000)

Arguments

A,B,C

Parameters for hypergeo()

z

Primary argument, complex

tol

absolute tolerance; default value of zero means to continue iterating until the result does not change to machine precision; strictly positive values give less accuracy but faster evaluation

maxiter

Integer specifying maximum number of iterations

Details

The hypergeometric function as defined by Abramowitz and Stegun, equation 15.1.1, page 556 is $$ {}_2F_1(a,b;c;z) = \sum_{n=0}^\infty\frac{(a)_n(b)_n}{(c)_n}\cdot\frac{z^n}{n!}$$ where $(a)_n=Gamma(a+n)/Gamma(a)$ is the Pochammer symbol (6.1.22, page 256). Function hypergeo() is the front-end for a rather unwieldy set of back-end functions which are called when the parameters A, B, C take certain values.

The general case (that is, when the parameters do not fall into a “special” category), is handled by hypergeo_general(). This applies whichever of the transformations given on page 559 gives the smallest modulus for the argument z.

Sometimes hypergeo_general() and all the transformations on page 559 fail to converge, in which case hypergeo() uses the continued fraction expansion hypergeo_contfrac(). If this fails, the function uses integration via f15.3.1().

References

Abramowitz and Stegun 1955. Handbook of mathematical functions with formulas, graphs and mathematical tables (AMS-55). National Bureau of Standards

See Also

hypergeo_powerseries, hypergeo_contfrac, genhypergeo

Examples

#  equation 15.1.3, page 556:
f1 <- function(x){-log(1-x)/x}
f2 <- function(x){hypergeo(1,1,2,x)}
f3 <- function(x){hypergeo(1,1,2,x,tol=1e-10)}
x <- seq(from = -0.6,to=0.6,len=14)
f1(x)-f2(x)
f1(x)-f3(x)  # Note tighter tolerance

# equation 15.1.7, p556:
g1 <- function(x){log(x + sqrt(1+x^2))/x}
g2 <- function(x){hypergeo(1/2,1/2,3/2,-x^2)}
g1(x)-g2(x)  # should be small 
abs(g1(x+0.1i) - g2(x+0.1i))  # should have small modulus.

# Just a random call, verified by Maple [ Hypergeom([],[1.22],0.9087) ]:
genhypergeo(NULL,1.22,0.9087)


# Little test of vectorization (warning: inefficient):
hypergeo(A=1.2+matrix(1:10,2,5)/10, B=1.4, C=1.665, z=1+2i)


# following calls test for former bugs:
hypergeo(1,2.1,4.1,1+0.1i)
hypergeo(1.1,5,2.1,1+0.1i)
hypergeo(1.9, 2.9, 1.9+2.9+4,1+0.99i) # c=a+b+4; hypergeo_cover1()