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

Author

Robin K. S. Hankin

Details

The hypergeometric function as defined by Abramowitz and Stegun, equation 15.1.1, page 556 is $_{2} F_{1} (a, b; c; z) = \sum_{n = 0}^{\infty} \frac{(a)_{n} (b)_{n}}{(c)_{n}} \cdot \frac{z^{n}}{n!}$

where $(a)_{n} = a (a + 1) \dots (a + n - 1) = Γ (a + n) / Γ (a)$ 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

Examples

Run this code

#  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()

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