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

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

Expansion of the hypergeometric function using the residue theorem; useful for when the primary argument is close to the critical points 0.5+/-i.sqrt(3)/2

hypergeo_buhring(A,B,C,z,z0=1/2,tol=0,maxiter=2000,use11=TRUE)
buhring_eqn11(n,S,A,B,C,z0=1/2)
buhring_eqn12(n,S,A,B,C,z0=1/2)
buhring_eqn5_factors(A,B,C,z,z0=1/2)
buhring_eqn5_series(S,A,B,C,z,z0=1/2,use11=FALSE,tol=0,maxiter=2000)

`A,B,C`	Parameters (real)
`S`	Parameter taken to be either `A` or `B`
`n`	Term to calculate in `buhring_eqn11()` or `buhring_eqn12()`
`z`	Primary complex argument
`z0`	Centre of circle of non-convergence; series expressed in powers of 1/(z-z0)^n
`tol,maxiter`	tolerance and maximum number of iterations (as in `hypergeo()`)
`use11`	Boolean with default `TRUE` meaning to use `buhring_eqn11()` and `FALSE` meaning to use `buhring_eqn12()`

The functions are direct transcriptions of Buhring 1987. The basic idea is to expand the hypergeometric function in powers of 1/(z-z0).

Functions buhring_eqn11() and buhring_eqn12() return the coefficients d(n) given by equations 11 and 12 of Buhring 1987.

The series do not converge satisfactorily near the critical points due to some sort of numerical instability. But they seem to work OK if |z-1/2| is large.

There is some issue which prevents the series from converging correctly, also sometimes the sequence converges to a demponstrably incorrect value.

Robin K. S. Hankin

W. Buhring 1987. “An analytic continuation of the hypergeometric series”, Siam J. Math. Anal. 18(3)

residue

# should be identical:
buhring_eqn11(n=0:10,S=1/2,A=1/2,B=1/3,C=pi)
buhring_eqn12(n=0:10,S=1/2,A=1/2,B=1/3,C=pi)
# but differ in one element


a <- hypergeo(1/2,1/3,4,1+8i,maxiter=90)
b <- hypergeo_buhring(1/2,1/3,4,1+8i,maxiter=90)
# should be identical but are not 


# following command fails due to numerical instability:
## Not run: 
hypergeo_buhring(1/2,1/3,pi,z=1/2 + 1i*sqrt(3)/2)

## End(Not run)