hypergeo_cover1: Hypergeometric functions for special values of the parameters
In hypergeo: The Gauss Hypergeometric Function

Description Usage Arguments Details Note Author(s) References See Also Examples

Hypergeometric functions for special values of the parameters

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hypergeo_cover1(A, B, m, z, tol = 0, maxiter = 2000, method = "a", give = FALSE)
hypergeo_cover2(A, C, m, z, tol = 0, maxiter = 2000, method = "a", give = FALSE)
hypergeo_cover3(A, n, m, z, tol = 0, maxiter = 2000, method = "a", give = FALSE)

`A,B,C`	parameters for the hypergeometric function
`m,n`	Integers (positive or negative)
`z`	Primary complex argument
`tol,maxiter`	Numerical arguments passed to `genhypergeo()`
`method`	Method, passed to `f15.3.10()` (qv)
`give`	Boolean with `TRUE` meaning to return the choice of helper function used (eg `f15.3.7()`), and default `FALSE` meaning to return the hypergeometric function's value

These functions deal with the exceptional cases listed on page 559-560.

Function hypergeo_cover1() deals with the case C=A+B +/- m, m=0,1,2,...
Function hypergeo_cover2() deals with the case B=A +/- m, m=0,1,2,...
Function hypergeo_cover3() deals with the case C-A=0,-1,-2,... [elementary] and C-A=1,2,... [not covered by AMS-55]

Function hypergeo_cover3() is required because the “limiting process” mentioned on p560, just after 15.3.14, is not explicit. Which is why it dispatches to w07.23.06.0026.01() and w07.23.06.0031.01(), documented at wolfram.

Robin K. S. Hankin

M. Abramowitz and I. A. Stegun 1965. Handbook of mathematical functions. New York: Dover

hypergeo,f15.3.10,wolfram

# Test hypergeo_cover1():
jjR <- hypergeo(pi,pi/2,3*pi/2-4, z=0.1+0.2i)
jjM <- 0.53745229690249593045 + 1.8917456473240515664i


# Test hypergeo_cover2():
jjM <- -0.15888831928748121465e-5 + 0.40339599711492215912e-4i
jjR <- hypergeo(pi,pi+2, 1.1 , 1+10i)  # This is 15.3.13
stopifnot(Mod(jjR-jjM)<1e-10)


# Test hypergeo_cover3()
jjM <- -0.24397135980533720308e-1 + 0.28819643319432922231i
jjR <- hypergeo(pi, 1.4, pi+4, 1+6i)
stopifnot(Mod(jjR-jjM)<1e-10)