hypergeo (version 1.2-9)

buhring: Evaluation of the hypergeometric function using Buhring's method

Description

Expansion of the hypergeometric function using the residue theorem; useful for when the primary argument is close to the critical points $1/2\pm i\sqrt{3}/2$

Usage

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)

Arguments

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 $1/\left(z-z_0\right)^n$

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()

Details

The functions are direct transcriptions of Buhring 1987. The basic idea is to expand the hypergeometric function in powers of $(z-z_0)^{-1}$.

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

References

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

See Also

residue

Examples

# 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


# following command fails due to numerical instability:
hypergeo_buhring(1/2,1/3,pi,z=1/2 + 1i*sqrt(3)/2)