gosper: Evaluation of the hypergeometric function using Gosper's...
In hypergeo: The Gauss Hypergeometric Function

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

Evaluation of the hypergeometric function using Gosper's method

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

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

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.

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

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.

hypergeo_contfrac

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())