R: Evaluation of the hypergeometric function using Gosper's...

gosper {hypergeo} | R Documentation

## Evaluation of the hypergeometric function using Gosper's method

### Description

Evaluation of the hypergeometric function using Gosper's method

### Usage

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

### Arguments

`A,B,C` | Parameters (real or complex)
`z` | Complex argument
`tol` | tolerance (passed to `GCF()`)
`maxiter` | maximum number of iterations

### Details

Gosper provides a three-term recurrence which converges when z is close to a critical point.

Bill Gosper asserts that the recursion holds for values of z which are inside the cardioid `(sqrt(8)*cos(t)-2*cos(2t), sqrt(8)*sin(t)-2*sin(2t))` (see examples section).

It is suggested that the recursion should only be used when the auxiliary parameters A, B,C are all <=12 in absolute value.

### Author(s)

R code by Robin K. S. Hankin, transcribed from maxima code posted by Richard Fateman, who credited Bill Gosper

### References

Original email was archived at `https://www.ma.utexas.edu/pipermail/maxima/2006/000126.html` but does not appear there now; and the wayback machine doesn't find it either.

### See Also

`hypergeo_contfrac`

### Examples

    hypergeo_gosper(1.1,5.1,3.1,crit())

    # Compare MMA: -0.192225 + 0.692328 I

    t <- seq(from=0,to=2i*pi,len=100)
    plot(exp(t)*(sqrt(8)-exp(t)),asp=1,type='l')
    points(crit())

[Package hypergeo version 1.2-13 Index]