R: Hypergeometric functions for special values of the parameters

hypergeo_cover1 {hypergeo} | R Documentation

## Hypergeometric functions for special values of the parameters

### Description

Hypergeometric functions for special values of the parameters

### Usage

    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)

### Arguments

`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

### Details

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]

### Note

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`.

### Author(s)

Robin K. S. Hankin

### References

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

### See Also

`hypergeo`,`f15.3.10`,`wolfram`

### Examples

    # 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)

[Package hypergeo version 1.2-13 Index]