Hypergeometric functions for special values of the parameters
Hypergeometric functions for special values of the parameters
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)
A,B,C |
parameters for the hypergeometric function |
m,n |
Integers (positive or negative) |
z |
Primary complex argument |
tol,maxiter |
Numerical arguments passed to |
method |
Method, passed to |
give |
Boolean with |
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]
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.
Robin K. S. Hankin
M. Abramowitz and I. A. Stegun 1965. Handbook of mathematical functions. New York: Dover
# 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)
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