Package 'hypergeo' reference manual

Title:	The Gauss Hypergeometric Function
Description:	The Gaussian hypergeometric function for complex numbers.
Authors:	Robin K. S. Hankin [aut, cre] , Richard D. Morey [ctb]
Maintainer:	Robin K. S. Hankin <[email protected]>
License:	GPL-2
Version:	1.2-15
Built:	2024-11-11 04:45:23 UTC
Source:	https://github.com/robinhankin/hypergeo

The Gauss hypergeometric function

Description

The Gaussian hypergeometric function for complex numbers.

Details

The DESCRIPTION file:

Package:	hypergeo
Title:	The Gauss Hypergeometric Function
Version:	1.2-15
Authors@R:	c( person(c("Robin", "K. S. "), "Hankin", role=c("aut","cre"), email="[email protected]", comment = c(ORCID = "0000-0001-5982-0415")), person("Richard D.", "Morey", role = "ctb", email = "[email protected]", comment = c(ORCID = "0000-0001-9220-3179")) )
Maintainer:	Robin K. S. Hankin <[email protected]>
Depends:	R (>= 3.1.0)
Imports:	elliptic (>= 1.3-5), contfrac (>= 1.1-9), deSolve, Rcpp (>= 0.12.3)
Suggests:	magrittr,testthat
Description:	The Gaussian hypergeometric function for complex numbers.
LinkingTo:	Rcpp, RcppArmadillo
License:	GPL-2
URL:	https://github.com/RobinHankin/hypergeo
BugReports:	https://github.com/RobinHankin/hypergeo/issues
Repository:	https://robinhankin.r-universe.dev
RemoteUrl:	https://github.com/robinhankin/hypergeo
RemoteRef:	HEAD
RemoteSha:	e64c4148abc06b22c28243f3011ac9816bf28ac3
Author:	Robin K. S. Hankin [aut, cre] (<https://orcid.org/0000-0001-5982-0415>), Richard D. Morey [ctb] (<https://orcid.org/0000-0001-9220-3179>)

Index of help topics:

buhring                 Evaluation of the hypergeometric function using
                        Buhring's method
complex_gamma           Gamma function for complex arguments
complex_gamma_cpp       Low-level C functions
f15.3.1                 Hypergeometric function using Euler's integral
                        representation
f15.3.10                Transformations of the hypergeometric function
f15.3.3                 Various transformation formulae for the
                        hypergeometric function
f15.5.1                 Hypergeometric functions via direct numerical
                        integration
genhypergeo             The generalized hypergeometric function
hypergeo                The hypergeometric function
hypergeo-package        The Gauss hypergeometric function
hypergeo_A_nonpos_int   Hypergeometric functions for nonpositive
                        integer arguments
hypergeo_contfrac       Continued fraction expansion of the
                        hypergeometric function
hypergeo_cover1         Hypergeometric functions for special values of
                        the parameters
hypergeo_gosper         Evaluation of the hypergeometric function using
                        Gosper's method
hypergeo_powerseries    The hypergeometric function as determined by
                        power series
hypergeo_residue        Evaluation of the hypergeometric function using
                        the residue theorem
i15.3.6                 Helper functions
is.nonpos               Various utilities
shanks                  Evaluation of the hypergeometric function using
                        Shanks's method
wolfram                 Various functions taken from the Wolfram
                        Functions Site

Author(s)

Robin K. S. Hankin [aut, cre] (<https://orcid.org/0000-0001-5982-0415>), Richard D. Morey [ctb] (<https://orcid.org/0000-0001-9220-3179>) Original R code written by Robin Hankin, C++ by Richard Morey.

Maintainer: Robin K. S. Hankin <[email protected]>

References

M. Abramowitz and I. A. Stegun 1965. Handbook of mathematical functions. New York: Dover
R. K. S. Hankin 2015. “Numerical Evaluation of the Gauss Hypergeometric Function with the hypergeo Package”. The R Journal 7:2, pp81-88

Examples

hypergeo(1.1,2.3,1.9 , 1+6i)

options(showHGcalls = TRUE)    # any non-null value counts as TRUE
hypergeo(4.1, 3.1, 5.1, 1+1i)  # shows trace back
options(showHGcalls = FALSE)   # reset


hypergeo(1.1,2.3,1.9 , 1+6i)

options(showHGcalls = TRUE)    # any non-null value counts as TRUE
hypergeo(4.1, 3.1, 5.1, 1+1i)  # shows trace back
options(showHGcalls = FALSE)   # reset

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)
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.

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.

Note

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

Author(s)

Robin K. S. Hankin

References

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

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


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)

# 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)

Gamma function for complex arguments

Description

Gamma and factorial functions for complex arguments

Usage

complex_gamma(z, log = FALSE)
complex_factorial(z, log = FALSE)
lanczos(z,log=FALSE)
complex_gamma(z, log = FALSE)
complex_factorial(z, log = FALSE)
lanczos(z,log=FALSE)

Arguments

`z`	Primary argument, a complex vector
`log`	Boolean, with default `FALSE` meaning to return the function value and `TRUE` meaning to return its logarithm

Details

Method follows that of Lanczos, coefficients identical to those of the GSL

Author(s)

Robin K. S. Hankin

References

Lanczos, C. 1964. “A precision approximation of the gamma function”. Journal of the society for industrial and applied mathematics series B, Volume 1, pp86-96

M. Galassi et al, GNU Scientific Library Reference Manual (3rd Ed.), ISBN 0954612078.

Examples


complex_gamma(5)  # should be 4!=24

complex_gamma(1+1i)   # takes complex arguments

complex_gamma(-5/2) + sqrt(pi)*8/15  # should be small

z <- pi + 1i*sqrt(2)
complex_gamma(z+1)-z*complex_gamma(z)   # should be small


complex_gamma(z)*complex_gamma(1-z) - pi/sin(pi*z)  # small


complex_gamma(5)  # should be 4!=24

complex_gamma(1+1i)   # takes complex arguments

complex_gamma(-5/2) + sqrt(pi)*8/15  # should be small

z <- pi + 1i*sqrt(2)
complex_gamma(z+1)-z*complex_gamma(z)   # should be small


complex_gamma(z)*complex_gamma(1-z) - pi/sin(pi*z)  # small

Low-level C functions

Description

Various low-level functions written in C for efficiency

Usage

complex_gamma_cpp(real,imag)
lanczos_cpp(real,imag)
genhypergeo_series_cpp(real_U, imag_U, real_L, imag_L,
    real_z, imag_z, tol, maxiter, check_mod, polynomial)
complex_gamma_cpp(real,imag)
lanczos_cpp(real,imag)
genhypergeo_series_cpp(real_U, imag_U, real_L, imag_L,
    real_z, imag_z, tol, maxiter, check_mod, polynomial)

Arguments

`real`, `imag`, `real_U`, `imag_U`, `real_L`, `imag_L`, `real_z`, `imag_z`	Numeric vectors giving the real and imaginary components of arguments; “U” and “L” stand for Upper or Lower arguments respectively
`tol`, `maxiter`, `check_mod`, `polynomial`	As for `genhypergeo_series()`

Note

NOTE HERE

Author(s)

Robin K. S. Hankin

Examples

1+1
1+1

Hypergeometric function using Euler's integral representation

Description

Hypergeometric function using Euler's integral representation, evaluated using numerical contour integrals.

Usage

f15.3.1(A, B, C, z, h = 0)
f15.3.1(A, B, C, z, h = 0)

Arguments

`A`, `B`, `C`	Parameters
`z`	Primary complex argument
`h`	specification for the path to be taken; see details

Details

Argument h specifies the path to be taken (the path has to avoid the point $1/z$ ). If h is real and of length 1, the path taken comprises two straight lines: one from $0$ to $0.5+hi$ and one from $0.5+hi$ to $1$ (if $h=0$ the integration is performed over a single segment).

Otherwise, the integration is performed over length(h)+1 segments: $0$ to h[1], then h[i] to h[i+1] for $1\leq i\leq n-1$ and finally h[n] to 1.

See examples and notes sections below.

Note

The Mellin-Barnes form is not yet coded up.

Author(s)

Robin K. S. Hankin

References

M. Abramowitz and I. A. Stegun 1965. Handbook of mathematical functions. New York: Dover

Examples



# For |z| <1 the path can be direct:
f15.3.1(2,1,2,-1/2) -2/3

# cf identity 07.23.03.0046.01 of Hypergeometric2F1.pdf with b=1




f <- function(h){f15.3.1(1,2,3, z=2, h=h)}

# Winding number [around 1/z] matters:
f(0.5)
f(c(1-1i, 1+1i, -2i))

# Accuracy isn't too bad; compare numerical to analytical result :
f(0.5) - (-1+1i*pi/2)

# For |z| <1 the path can be direct:
f15.3.1(2,1,2,-1/2) -2/3

# cf identity 07.23.03.0046.01 of Hypergeometric2F1.pdf with b=1




f <- function(h){f15.3.1(1,2,3, z=2, h=h)}

# Winding number [around 1/z] matters:
f(0.5)
f(c(1-1i, 1+1i, -2i))

# Accuracy isn't too bad; compare numerical to analytical result :
f(0.5) - (-1+1i*pi/2)

Transformations of the hypergeometric function

Description

Transformations of the hypergeometric function detailed in AMS-55, page 559-560.

Usage

f15.3.10       (A, B,    z, tol = 0, maxiter = 2000, method = "a")
f15.3.10_a     (A, B,    z, tol = 0, maxiter = 2000              )
f15.3.10_b     (A, B,    z, tol = 0, maxiter = 2000              )
f15.3.11       (A, B, m, z, tol = 0, maxiter = 2000, method = "a")
f15.3.11_bit1  (A, B, m, z, tol = 0                              )
f15.3.11_bit2_a(A, B, m, z, tol = 0, maxiter = 2000              )
f15.3.11_bit2_b(A, B, m, z, tol = 0, maxiter = 2000              )
f15.3.12       (A, B, m, z, tol = 0, maxiter = 2000, method = "a")
f15.3.12_bit1  (A, B, m, z, tol = 0                              )
f15.3.12_bit2_a(A, B, m, z, tol = 0, maxiter = 2000              )
f15.3.12_bit2_b(A, B, m, z, tol = 0, maxiter = 2000              )
f15.3.13       (A, C,    z, tol = 0, maxiter = 2000, method = "a")
f15.3.13_a     (A, C,    z, tol = 0, maxiter = 2000              )
f15.3.13_b     (A, C,    z, tol = 0, maxiter = 2000              )
f15.3.14       (A, C, m, z, tol = 0, maxiter = 2000, method = "a")
f15.3.14_bit1_a(A, C, m, z, tol = 0, maxiter = 2000              )
f15.3.14_bit1_b(A, C, m, z, tol = 0, maxiter = 2000              )
f15.3.14_bit2  (A, C, m, z, tol = 0                              )
f15.3.13_14    (A, C, m, z, tol = 0, maxiter = 2000, method = "a")
f15.3.10_11_12 (A, B, m, z, tol = 0, maxiter = 2000, method = "a")
f15.1.1        (A, B, C, z, tol = 0, maxiter = 2000              )
f15.3.10       (A, B,    z, tol = 0, maxiter = 2000, method = "a")
f15.3.10_a     (A, B,    z, tol = 0, maxiter = 2000              )
f15.3.10_b     (A, B,    z, tol = 0, maxiter = 2000              )
f15.3.11       (A, B, m, z, tol = 0, maxiter = 2000, method = "a")
f15.3.11_bit1  (A, B, m, z, tol = 0                              )
f15.3.11_bit2_a(A, B, m, z, tol = 0, maxiter = 2000              )
f15.3.11_bit2_b(A, B, m, z, tol = 0, maxiter = 2000              )
f15.3.12       (A, B, m, z, tol = 0, maxiter = 2000, method = "a")
f15.3.12_bit1  (A, B, m, z, tol = 0                              )
f15.3.12_bit2_a(A, B, m, z, tol = 0, maxiter = 2000              )
f15.3.12_bit2_b(A, B, m, z, tol = 0, maxiter = 2000              )
f15.3.13       (A, C,    z, tol = 0, maxiter = 2000, method = "a")
f15.3.13_a     (A, C,    z, tol = 0, maxiter = 2000              )
f15.3.13_b     (A, C,    z, tol = 0, maxiter = 2000              )
f15.3.14       (A, C, m, z, tol = 0, maxiter = 2000, method = "a")
f15.3.14_bit1_a(A, C, m, z, tol = 0, maxiter = 2000              )
f15.3.14_bit1_b(A, C, m, z, tol = 0, maxiter = 2000              )
f15.3.14_bit2  (A, C, m, z, tol = 0                              )
f15.3.13_14    (A, C, m, z, tol = 0, maxiter = 2000, method = "a")
f15.3.10_11_12 (A, B, m, z, tol = 0, maxiter = 2000, method = "a")
f15.1.1        (A, B, C, z, tol = 0, maxiter = 2000              )

Arguments

`A`, `B`, `C`	Parameters of the hypergeometric function
`m`	Integer linking `A`, `B`, `C` as set out in AMS-55, page 559,560
`z`	primary complex argument
`tol`, `maxiter`	numerical parameters
`method`	Length 1 character vector specifying the method. See details

Details

Naming scheme (functions and arguments) follows AMS-55, pages 559-560.

The method argument to (eg) f15.3.14() specifies whether to use psigamma() directly (method “a”), or the recurrence 6.3.5 (method “b”). Press et al recommend method “b”, presumably on the grounds of execution speed. I'm not so sure (method “a” seems to be more accurate in the sense that it returns values closer to those of Maple).

Method “c” means to use a totally dull, slow, direct (but clearly correct) summation, for the purposes of debugging. This is only used for the functions documented under wolfram.Rd

Functions f15.3.13_14() and f15.3.10_11_12() are convenience wrappers. For example, function f15.3.13_14() dispatches to either f15.3.13() or f15.3.14() depending on the value of m.

Note

These functions are not really designed to be called by the user: use hypergeo() instead, or hypergeo_cover[123]() for specific cases.

Author(s)

Robin K. S. Hankin

References

M. Abramowitz and I. A. Stegun 1965. Handbook of mathematical functions. New York: Dover

Examples


f15.3.10_11_12(A=1.1, B=pi, m= +3, z=.1+.1i)
f15.3.10_11_12(A=1.1, B=pi, m= -3, z=.1+.1i)

f15.3.10_11_12(A=1.1, B=pi, m= +3, z=.1+.1i)
f15.3.10_11_12(A=1.1, B=pi, m= -3, z=.1+.1i)

Various transformation formulae for the hypergeometric function

Description

Transformations of the hypergeometric function: equations 15.3.3 to 15.3.9

Usage

f15.3.3(A, B, C, z, tol = 0, maxiter = 2000)
f15.3.4(A, B, C, z, tol = 0, maxiter = 2000)
f15.3.5(A, B, C, z, tol = 0, maxiter = 2000)
f15.3.6(A, B, C, z, tol = 0, maxiter = 2000)
f15.3.7(A, B, C, z, tol = 0, maxiter = 2000)
f15.3.8(A, B, C, z, tol = 0, maxiter = 2000)
f15.3.9(A, B, C, z, tol = 0, maxiter = 2000)
f15.3.3(A, B, C, z, tol = 0, maxiter = 2000)
f15.3.4(A, B, C, z, tol = 0, maxiter = 2000)
f15.3.5(A, B, C, z, tol = 0, maxiter = 2000)
f15.3.6(A, B, C, z, tol = 0, maxiter = 2000)
f15.3.7(A, B, C, z, tol = 0, maxiter = 2000)
f15.3.8(A, B, C, z, tol = 0, maxiter = 2000)
f15.3.9(A, B, C, z, tol = 0, maxiter = 2000)

Arguments

`A`, `B`, `C`	Parameters of the hypergeometric function
`z`	Primary complex argument
`tol`, `maxiter`	parameters passed to `genhypergeo()`

Details

The naming scheme follows that of Abramowitz and Stegun

Author(s)

Robin K. S. Hankin

References

M. Abramowitz and I. A. Stegun 1965. “Handbook of mathematical functions”. New York: Dover

Examples

f15.3.4(1.1,2.2,3.4,-1+0.1i)
f15.3.4(1.1,2.2,3.4,-1+0.1i)

Hypergeometric functions via direct numerical integration

Description

The hypergeometric function may be evaluated using Gauss's differential equation 15.5.1:

$z(1-z)w''+(c-(a+b+1)z)w'-abw=0$

using a start value away from the three singular points. This page documents a suite of related functionality.

Usage

hypergeo_press(A,B,C,z, ...)
f15.5.1(A, B, C, z, startz, u, udash, give=FALSE, ...)
hypergeo_func(Time, State, Pars, u, udash)
to_real(o)
to_complex(p)
complex_ode(y, times, func, parms=NA, method=NULL, u, udash, ...)
semicircle(t,z0,z1,clockwise=TRUE)
semidash(t,z0,z1,clockwise=TRUE)
straight(t,z0,z1)
straightdash(t,z0,z1)
hypergeo_press(A,B,C,z, ...)
f15.5.1(A, B, C, z, startz, u, udash, give=FALSE, ...)
hypergeo_func(Time, State, Pars, u, udash)
to_real(o)
to_complex(p)
complex_ode(y, times, func, parms=NA, method=NULL, u, udash, ...)
semicircle(t,z0,z1,clockwise=TRUE)
semidash(t,z0,z1,clockwise=TRUE)
straight(t,z0,z1)
straightdash(t,z0,z1)

Arguments

`A`, `B`, `C`, `z`	Standard parameters for the hypergeometric function
`u`, `udash`	Functions to specify the path of integration, and its derivative
`give`	In function `f15.5.1()`, Boolean with `TRUE` meaning to return extra information from `ode()` and default `FALSE` meaning to return only the evaluated function
`startz`	In function `f15.5.1()`, the start position of the path
`...`	Further arguments passed to `ode()`
`o`, `p`	Real and complex objects to be coerced to each other in `to_real()` and `to_complex()`
`y`, `times`, `func`, `parms`, `method`	In function `complex_ode()`, arguments matching those of `ode()`
`t`, `z0`, `z1`, `clockwise`	Arguments for the standard path functions `semicircle()` et seq: `u` is the primary argument (real, $0\leq u\leq 1$ ); `z0` and `z1` are the start and end points of the path; and `clockwise` is Boolean, indicating whether the path proceeds clockwise or not
`Time`, `State`, `Pars`	arguments matchin those of standard examples in the `deSolve` package

Details

Function hypergeo_press() is the most user-friendly of the functions documented here. It performs integration of Gauss's ODE, along a straight line path from the start-point to z. It follows Press et al's suggestion of start-point.

Function f15.5.1() is a little more flexible in that it allows the user to choose a start point and an integration path.

Function complex_ode() is a complex generalization of ode() of package deSolve; function hypergeo_func is an internal function, designed for use with complex_ode(), that specifies the Gauss ODE which is satisified by the hypergeometric function.

Functions to_real() and to_complex() are internal functions which coerce from real to complex and back; they are needed because ode() deals only with real values.

Functions semicircle() and straight() are helper functions which specify straight or semicircular paths from z0 to z1; note that f(0)=z0 and f(1)=z1. Functions semidash() and straightdash() provide the differential of the path.

Note

Accuracy is low compared with the other methods in the package.

Author(s)

Robin K. S. Hankin

References

W. H. Press et al. 1997. Numerical Recipes in C. Cambridge University Press, Second Edition.

Examples


hypergeo_press(A=pi,B=sqrt(2),C=1.4,z=1-2i)
hypergeo      (A=pi,B=sqrt(2),C=1.4,z=1-2i)


jj1 <- 
f15.5.1(
    A=1.1, B=2.2, C=3.3, z=3+0.5i, startz=0.5,
        u    =function(u){semicircle(u,0.5,3+0.5i,FALSE)},
        udash=function(u){semidash(u,0.5,3+0.5i,FALSE)}
        )

jj2 <-
f15.5.1(
    A=1.1, B=2.2, C=3.3, z=3+0.5i, startz=0.5,
        u    =function(u){semicircle(u,0.5,3+0.5i,TRUE)},
        udash=function(u){semidash(u,0.5,3+0.5i,TRUE)}
        )



jj3 <- hypergeo(    A=1.1, B=2.2, C=3.3, z=3+0.5i)
## First one agrees with jj3=hypergeo(...), the second one does not 


# Now try the Airy Ai function;  satisfies f'' =  zf:

pars <- c(kay = 1+1i, ell = 0.1+0.2i)  # not actually used
airy_ai_func <- function(Time, State, Pars, u, udash) {
    with(as.list(c(to_complex(State), to_complex(Pars))), {

      z <- u(Time)
      dz <- udash(Time)
        
      dF <- Fdash*dz
      dFdash <-  z*F*dz # could use kay and ell from pars here if necessary
        
        ## coerce back to real:
        out <- to_real(c(dF,dFdash))
        names(out) <- names(State)
        return(list(out))
    })
}

complex_ode(
    y     = c(F = 1/3^(2/3)/gamma(2/3), Fdash= -1/3^(1/3)/gamma(1/3)),
    times = seq(0,1,by=0.1),
    func  = airy_ai_func,
    parms = pars,
    u     = function(t){straight(t,0,1)},
    udash = function(t){straightdash(t,0,1)}
)

# Look at the last line for the value at 1.
# compare gsl: Ai(1) = 0.1352924 ; Ai'(1) = -0.1591474

# ...although in this case there is actually a hypergeometric series
#  representation:

f <- function(z){
    return(
        +genhypergeo(U=NULL,L=2/3,z^3/9)/3^(2/3)/gamma(2/3)
        -genhypergeo(U=NULL,L=4/3,z^3/9)/3^(1/3)/gamma(1/3)*z
        ) 
}

f(1)

hypergeo_press(A=pi,B=sqrt(2),C=1.4,z=1-2i)
hypergeo      (A=pi,B=sqrt(2),C=1.4,z=1-2i)


jj1 <- 
f15.5.1(
    A=1.1, B=2.2, C=3.3, z=3+0.5i, startz=0.5,
        u    =function(u){semicircle(u,0.5,3+0.5i,FALSE)},
        udash=function(u){semidash(u,0.5,3+0.5i,FALSE)}
        )

jj2 <-
f15.5.1(
    A=1.1, B=2.2, C=3.3, z=3+0.5i, startz=0.5,
        u    =function(u){semicircle(u,0.5,3+0.5i,TRUE)},
        udash=function(u){semidash(u,0.5,3+0.5i,TRUE)}
        )



jj3 <- hypergeo(    A=1.1, B=2.2, C=3.3, z=3+0.5i)
## First one agrees with jj3=hypergeo(...), the second one does not 


# Now try the Airy Ai function;  satisfies f'' =  zf:

pars <- c(kay = 1+1i, ell = 0.1+0.2i)  # not actually used
airy_ai_func <- function(Time, State, Pars, u, udash) {
    with(as.list(c(to_complex(State), to_complex(Pars))), {

      z <- u(Time)
      dz <- udash(Time)
        
      dF <- Fdash*dz
      dFdash <-  z*F*dz # could use kay and ell from pars here if necessary
        
        ## coerce back to real:
        out <- to_real(c(dF,dFdash))
        names(out) <- names(State)
        return(list(out))
    })
}

complex_ode(
    y     = c(F = 1/3^(2/3)/gamma(2/3), Fdash= -1/3^(1/3)/gamma(1/3)),
    times = seq(0,1,by=0.1),
    func  = airy_ai_func,
    parms = pars,
    u     = function(t){straight(t,0,1)},
    udash = function(t){straightdash(t,0,1)}
)

# Look at the last line for the value at 1.
# compare gsl: Ai(1) = 0.1352924 ; Ai'(1) = -0.1591474

# ...although in this case there is actually a hypergeometric series
#  representation:

f <- function(z){
    return(
        +genhypergeo(U=NULL,L=2/3,z^3/9)/3^(2/3)/gamma(2/3)
        -genhypergeo(U=NULL,L=4/3,z^3/9)/3^(1/3)/gamma(1/3)*z
        ) 
}

f(1)

The generalized hypergeometric function

Description

The generalized hypergeometric function, using either the series expansion or the continued fraction expansion.

Usage

genhypergeo(U, L, z, tol=0, maxiter=2000, check_mod=TRUE,
    polynomial=FALSE, debug=FALSE, series=TRUE)
genhypergeo_series(U, L, z, tol=0, maxiter=2000, check_mod=TRUE,
    polynomial=FALSE, debug=FALSE) 
genhypergeo_contfrac(U, L, z, tol = 0, maxiter = 2000)
genhypergeo(U, L, z, tol=0, maxiter=2000, check_mod=TRUE,
    polynomial=FALSE, debug=FALSE, series=TRUE)
genhypergeo_series(U, L, z, tol=0, maxiter=2000, check_mod=TRUE,
    polynomial=FALSE, debug=FALSE) 
genhypergeo_contfrac(U, L, z, tol = 0, maxiter = 2000)

Arguments

`U`, `L`	Upper and lower arguments respectively (real or complex)
`z`	Primary complex argument (see notes)
`tol`	tolerance with default zero meaning to iterate until additional terms to not change the partial sum
`maxiter`	Maximum number of iterations to perform
`check_mod`	Boolean, with default `TRUE` meaning to check that the modulus of `z` is less than 1
`polynomial`	Boolean, with default `FALSE` meaning to evaluate the series until converged, or return a warning; and `TRUE` meaning to return the sum of `maxiter` terms, whether or not converged. This is useful when either `A` or `B` is a nonpositive integer in which case the hypergeometric function is a polynomial
`debug`	Boolean, with `TRUE` meaning to return debugging information and default `FALSE` meaning to return just the evaluate
`series`	In function `genhypergeo()`, Boolean argument with default `TRUE` meaning to return the result of `genhypergeo_series()` and `FALSE` the result of `genhypergeo_contfrac()`

Details

Function genhypergeo() is a wrapper for functions genhypergeo_series() and genhypergeo_contfrac().

Function genhypergeo_series() is the workhorse for the whole package; every call to hypergeo() uses this function except for the (apparently rare—but see the examples section) cases where continued fractions are used.

The generalized hypergeometric function [here genhypergeo()] appears from time to time in the literature (eg Mathematica) as

$F(U,L;z) = \sum_{n=0}^\infty\frac{(u_1)_n(u_2)_n\ldots (u_i)_n}{(l_1)_n(l_2)_n\ldots (l_j)_n}\cdot\frac{z^n}{n!}$

where $U=\left(u_1,\ldots,u_i\right)$ and $L=\left(l_1,\ldots,l_i\right)$ are the “upper” and “lower” vectors respectively. The radius of convergence of this formula is 1.

For the Confluent Hypergeometric function, use genhypergeo() with length-1 vectors for arguments U and V.

For the ${}_0\!F_1$ function (ie no “upper” arguments), use genhypergeo(NULL,L,x).

See documentation for genhypergeo_contfrac() for details of the continued fraction representation.

Note

The radius of convergence for the series is 1 but under some circumstances, analytic continuation defines a function over the whole complex plane (possibly cut along $(1,\infty)$ ). Further work would be required to implement this.

Function genhypergeo() tests for its return value being either on (or close to) the real axis, and if so, coerces its value to numeric:

if(all(zapsmall(Im(out)) == 0)){out <- Re(out)}

Thus the code is sensitive, via zapsmall(), to the value of getOption("digits").

Author(s)

Robin K. S. Hankin

References

M. Abramowitz and I. A. Stegun 1965. Handbook of mathematical functions. New York: Dover

Examples


genhypergeo(U=c(1.1,0.2,0.3), L=c(10.1,pi*4), check_mod=FALSE, z=1.12+0.2i)
genhypergeo(U=c(1.1,0.2,0.3), L=c(10.1,pi*4),z=4.12+0.2i,series=FALSE)

genhypergeo(U=c(1.1,0.2,0.3), L=c(10.1,pi*4), check_mod=FALSE, z=1.12+0.2i)
genhypergeo(U=c(1.1,0.2,0.3), L=c(10.1,pi*4),z=4.12+0.2i,series=FALSE)

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)
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 $\le 12$ 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.

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())


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())

The hypergeometric function

Description

The Hypergeometric and generalized hypergeometric functions as defined by Abramowitz and Stegun. Function hypergeo() is the user interface to the majority of the package functionality; it dispatches to one of a number of subsidiary functions.

Usage

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

Arguments

`A`, `B`, `C`	Parameters for `hypergeo()`
`z`	Primary argument, complex
`tol`	absolute tolerance; default value of zero means to continue iterating until the result does not change to machine precision; strictly positive values give less accuracy but faster evaluation
`maxiter`	Integer specifying maximum number of iterations

Details

The hypergeometric function as defined by Abramowitz and Stegun, equation 15.1.1, page 556 is

${}_2F_1(a,b;c;z) = \sum_{n=0}^\infty\frac{(a)_n(b)_n}{(c)_n}\cdot\frac{z^n}{n!}$

where $(a)_n=a(a+1)\ldots(a+n-1)=\Gamma(a+n)/\Gamma(a)$ is the Pochammer symbol (6.1.22, page 256).

Function hypergeo() is the front-end for a rather unwieldy set of back-end functions which are called when the parameters A, B, C take certain values.

The general case (that is, when the parameters do not fall into a “special” category), is handled by hypergeo_general(). This applies whichever of the transformations given on page 559 gives the smallest modulus for the argument z.

Sometimes hypergeo_general() and all the transformations on page 559 fail to converge, in which case hypergeo() uses the continued fraction expansion hypergeo_contfrac().

If this fails, the function uses integration via f15.3.1().

Note

Abramowitz and Stegun state:

“The radius of convergence of the Gauss hypergeometric series $\ldots$ is $\left|z\right|=1$ ” (AMS-55, section 15.1, page 556).

This reference book gives the correct radius of convergence; use the ratio test to verify it. Thus if $|z|>1$ , the hypergeometric series will diverge and function genhypergeo() will fail to converge.

However, the hypergeometric function is defined over the whole of the complex plane, so analytic continuation may be used if appropriate cut lines are used. A cut line must join $z=1$ to (complex) infinity; it is conventional for it to follow the real axis in a positive direction from $z=1$ but other choices are possible.

Note that in using the package one sometimes draws a “full precision not achieved” warning from gamma(); and complex arguments are not allowed. I would suggest either ignoring the warning (the error of gamma() is unlikely to be large) or to use one of the bespoke functions such as f15.3.4() and tolerate the slower convergence, although this is not always possible.

Author(s)

Robin K. S. Hankin

References

Abramowitz and Stegun 1955. Handbook of mathematical functions with formulas, graphs and mathematical tables (AMS-55). National Bureau of Standards

Examples

#  equation 15.1.3, page 556:
f1 <- function(x){-log(1-x)/x}
f2 <- function(x){hypergeo(1,1,2,x)}
f3 <- function(x){hypergeo(1,1,2,x,tol=1e-10)}
x <- seq(from = -0.6,to=0.6,len=14)
f1(x)-f2(x)
f1(x)-f3(x)  # Note tighter tolerance

# equation 15.1.7, p556:
g1 <- function(x){log(x + sqrt(1+x^2))/x}
g2 <- function(x){hypergeo(1/2,1/2,3/2,-x^2)}
g1(x)-g2(x)  # should be small 
abs(g1(x+0.1i) - g2(x+0.1i))  # should have small modulus.

# Just a random call, verified by Maple [ Hypergeom([],[1.22],0.9087) ]:
genhypergeo(NULL,1.22,0.9087)


# Little test of vectorization (warning: inefficient):
hypergeo(A=1.2+matrix(1:10,2,5)/10, B=1.4, C=1.665, z=1+2i)


# following calls test for former bugs:
hypergeo(1,2.1,4.1,1+0.1i)
hypergeo(1.1,5,2.1,1+0.1i)
hypergeo(1.9, 2.9, 1.9+2.9+4,1+0.99i) # c=a+b+4; hypergeo_cover1()


#  equation 15.1.3, page 556:
f1 <- function(x){-log(1-x)/x}
f2 <- function(x){hypergeo(1,1,2,x)}
f3 <- function(x){hypergeo(1,1,2,x,tol=1e-10)}
x <- seq(from = -0.6,to=0.6,len=14)
f1(x)-f2(x)
f1(x)-f3(x)  # Note tighter tolerance

# equation 15.1.7, p556:
g1 <- function(x){log(x + sqrt(1+x^2))/x}
g2 <- function(x){hypergeo(1/2,1/2,3/2,-x^2)}
g1(x)-g2(x)  # should be small 
abs(g1(x+0.1i) - g2(x+0.1i))  # should have small modulus.

# Just a random call, verified by Maple [ Hypergeom([],[1.22],0.9087) ]:
genhypergeo(NULL,1.22,0.9087)


# Little test of vectorization (warning: inefficient):
hypergeo(A=1.2+matrix(1:10,2,5)/10, B=1.4, C=1.665, z=1+2i)


# following calls test for former bugs:
hypergeo(1,2.1,4.1,1+0.1i)
hypergeo(1.1,5,2.1,1+0.1i)
hypergeo(1.9, 2.9, 1.9+2.9+4,1+0.99i) # c=a+b+4; hypergeo_cover1()

Hypergeometric functions for nonpositive integer arguments

Description

Hypergeometric functions for A and/or B being nonpositive integers

Usage

hypergeo_A_nonpos_int(A, B, C, z, tol = 0)
hypergeo_AorB_nonpos_int(A, B, C, z, tol = 0)
hypergeo_A_nonpos_int(A, B, C, z, tol = 0)
hypergeo_AorB_nonpos_int(A, B, C, z, tol = 0)

Arguments

`A`, `B`, `C`	Parameters for the hypergeometric function
`z`	Primary complex argument
`tol`	tolerance

Details

The “point” of these functions is that if A and C (or B and C) are identical nonpositive integers, a warning needs to be given because the function is defined as the appropriate limit and one needs to be sure that both A and C approach that limit at the same speed.

Function hypergeo_AorB_nonpos_int() is a convenience wrapper for hypergeo_A_nonpos_int().

Author(s)

Robin K. S. Hankin

References

M. Abramowitz and I. A. Stegun 1965. Handbook of mathematical functions. New York: Dover

Examples


jjR1 <- hypergeo(-4, pi, 2.2 , 1+6i)
jjR2 <- hypergeo(pi, -4, 2.2 , 1+6i)  # former bug
jjM <- 3464.1890402837334002-353.94143580568566281i  # value given by Mathematica 


jjR1 <- hypergeo(-4, pi, 2.2 , 1+6i)
jjR2 <- hypergeo(pi, -4, 2.2 , 1+6i)  # former bug
jjM <- 3464.1890402837334002-353.94143580568566281i  # value given by Mathematica

Continued fraction expansion of the hypergeometric function

Description

Continued fraction expansion of the hypergeometric and generalized hypergeometric functions using continued fraction expansion.

Usage

hypergeo_contfrac(A, B, C, z, tol = 0, maxiter = 2000)
genhypergeo_contfrac_single(U, L, z, tol = 0, maxiter = 2000)
hypergeo_contfrac(A, B, C, z, tol = 0, maxiter = 2000)
genhypergeo_contfrac_single(U, L, z, tol = 0, maxiter = 2000)

Arguments

`A`, `B`, `C`	Parameters (real or complex)
`U`, `L`	In function `genhypergeo_contfrac()`, upper and lower arguments as in `genhypergeo()`
`z`	Complex argument
`tol`	tolerance (passed to `GCF()`)
`maxiter`	maximum number of iterations

Details

These functions are included in the package in the interests of completeness, but it is not clear when it is advantageous to use continued fraction form rather than the series form.

Note

The continued fraction expression is the RHS identity 07.23.10.0001.01 of Hypergeometric2F1.pdf.

The function sometimes fails to converge to the correct value but no warning is given.

Function genhypergeo_contfrac() is documented under genhypergeo.Rd.

Author(s)

Robin K. S. Hankin

References

M. Abramowitz and I. A. Stegun 1965. Handbook of mathematical functions. New York: Dover
http://functions.wolfram.com/PDF/Hypergeometric2F1.pdf

Examples

hypergeo_contfrac(0.3 , 0.6 , 3.3 , 0.1+0.3i)
# Compare Maple: 1.0042808294775511972+0.17044041575976110947e-1i

genhypergeo_contfrac_single(U=0.2 , L=c(9.9,2.7,8.7) , z=1+10i)
# (powerseries does not converge)
# Compare Maple: 1.0007289707983569879 + 0.86250714217251837317e-2i

hypergeo_contfrac(0.3 , 0.6 , 3.3 , 0.1+0.3i)
# Compare Maple: 1.0042808294775511972+0.17044041575976110947e-1i

genhypergeo_contfrac_single(U=0.2 , L=c(9.9,2.7,8.7) , z=1+10i)
# (powerseries does not converge)
# Compare Maple: 1.0007289707983569879 + 0.86250714217251837317e-2i

Hypergeometric functions for special values of the parameters

Description

Hypergeometric functions for special values of the parameters

Usage

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)
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)

Arguments

`A`, `B`, `C`	parameters for the hypergeometric function
`m`, `n`	Integers (positive or negative)
`z`	Primary complex argument
`tol`, `maxiter`	Numerical arguments passed to `genhypergeo()`
`method`	Method, passed to `f15.3.10()` (qv)
`give`	Boolean with `TRUE` meaning to return the choice of helper function used (eg `f15.3.7()`), and default `FALSE` meaning to return the hypergeometric function's value

Details

These functions deal with the exceptional cases listed on page 559-560.

Function hypergeo_cover1() deals with the case $C=A+B\pm m,m=0,1,2,\ldots$
Function hypergeo_cover2() deals with the case $B=A\pm m,m=0,1,2,\ldots$
Function hypergeo_cover3() deals with the case $C-A=0,-1,-2,\ldots$ [elementary] and $C-A=1,2,\ldots$ [not covered by AMS-55]

Note

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.

Author(s)

Robin K. S. Hankin

References

M. Abramowitz and I. A. Stegun 1965. Handbook of mathematical functions. New York: Dover

Examples



# 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)

# 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)

The hypergeometric function as determined by power series

Description

The hypergeometric function as determined by infinite (hypergeo_powerseries()) or finite (hypergeo_taylor()) power series

Usage

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

Arguments

`A`, `B`, `C`	Parameters of the hypergeometric function
`z`	Primary complex argument
`tol`, `maxiter`	Numerical arguments

Details

Function hypergeo_powerseries() is the primary decision-making function of the package. It is this function that detects degenerate cases of the three parameters and dispatches accordingly. Non-degenerate cases are sent to function hypergeo_general().

Function hypergeo_taylor() deals with cases where the hypergeometric function is a polynomial.

Author(s)

Robin K. S. Hankin

References

M. Abramowitz and I. A. Stegun 1965. Handbook of mathematical functions. New York: Dover

Examples



jjR <- hypergeo(pi,-4,2.2,1+5i)
jjM <- 1670.8287595795885335 - 204.81995157365381258i


jjR <- hypergeo(pi,-4,2.2,1+5i)
jjM <- 1670.8287595795885335 - 204.81995157365381258i

Helper functions

Description

Helper functions for equations 15.3.6-15.3.9

Usage

i15.3.6(A, B, C)
i15.3.7(A, B, C)
i15.3.8(A, B, C)
i15.3.9(A, B, C)
j15.3.6(A, B, C)
j15.3.7(A, B, C)
j15.3.8(A, B, C)
j15.3.9(A, B, C)
i15.3.6(A, B, C)
i15.3.7(A, B, C)
i15.3.8(A, B, C)
i15.3.9(A, B, C)
j15.3.6(A, B, C)
j15.3.7(A, B, C)
j15.3.8(A, B, C)
j15.3.9(A, B, C)

Arguments

A, B, C

Parameters of the hypergeometric function

Details

These functions are named for their equation numbers in Abramowitz and Stegun.

Functions i15.3.?() return the factors at the beginning of equations 15.3.6-9. These functions return zero if the denominator is infinite (because it includes a gamma function of a nonpositive integer).

Functions j15.3.?() check for the appropriate arguments of the gamma function being nonpositive integers.

Author(s)

Robin K. S. Hankin

References

M. Abramowitz and I. A. Stegun 1965. Handbook of mathematical functions. New York: Dover

Examples

i15.3.6(1.1, 3.2, pi)
i15.3.6(1.1, 3.2, pi)

Various utilities

Description

Various utilities needing nonce functions

Usage

is.near_integer(i, tol=getOption("tolerance"))
is.nonpos(i)
is.zero(i)
isgood(x, tol)
thingfun(z, complex=FALSE)
crit(...)
lpham(x,n)
is.near_integer(i, tol=getOption("tolerance"))
is.nonpos(i)
is.zero(i)
isgood(x, tol)
thingfun(z, complex=FALSE)
crit(...)
lpham(x,n)

Arguments

`i`	Numerical vector of suspected integers
`tol`	Tolerance
`x`	Argument to `isgood()` and `lpham()`
`z`	Complex vector
`complex`	In function `thingfun()`, Boolean with default `FALSE` meaning to return the modulus of the transforms and `TRUE` meaning to return the complex values themselves
`n`	second argument to `lpham()`
`...`	Ignored

Details

Function is.near_integer(i) returns TRUE if i is “near” [that is, within tol] an integer; if the option is unset then 1e-11 is used.
Function is.nonpos() returns TRUE if i is near a nonpositive integer
Function is.zero() returns TRUE if i is, er, near zero
Function isgood() checks for all elements of x having absolute values less than tol
Function thingfun() transforms input vector z by each of the six members of the anharmonic group, viewed as a subgroup of the Mobius group of functions. It returns a real six-column matrix with columns being the modulus of $z,z/(z-1),1-z,1/z,1/(1-z),1-1/z$ . These six columns correspond to the primary argument in equations 15.3.3 to 15.3.9, p551 of AMS-55
Function crit() returns the two critical points, $\frac{1}{2}\pm\frac{\sqrt{3}i}{2}$ . These points have unit modulus as do their six transforms by thingfun()
Function lpham() returns the log of the Pochhammer function $log\left(\Gamma(x+n)/\Gamma(x)\right)$

Note

Function isgood() uses zero as the default tolerance (argument tol passed in from hypergeo()); compare the different meaning of tol used in is.near_integer().

Here, “integer” means one of the sequence $0,\pm 1,\pm 2,\ldots$ [ie not the Gaussian integers].

Author(s)

Robin K. S. Hankin

Examples


is.near_integer(-3)

is.zero(4)
is.near_integer(-3)

is.zero(4)

Evaluation of the hypergeometric function using the residue theorem

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_residue_general(A, B, C, z, r, O=z, tol=0, maxiter=2000)
hypergeo_residue_close_to_crit_single(A, B, C, z, strategy='A', tol=0, maxiter=2000)
hypergeo_residue_close_to_crit_multiple(A, B, C, z, strategy='A', tol=0, maxiter=2000)
hypergeo_residue_general(A, B, C, z, r, O=z, tol=0, maxiter=2000)
hypergeo_residue_close_to_crit_single(A, B, C, z, strategy='A', tol=0, maxiter=2000)
hypergeo_residue_close_to_crit_multiple(A, B, C, z, strategy='A', tol=0, maxiter=2000)

Arguments

`A`, `B`, `C`	Parameters (real or complex)
`z`	Complex argument
`tol`, `maxiter`	tolerance and maximum number of iterations (passed to `hypergeo()`)
`r`, `O`	Radius and center of circle to integrate over
`strategy`	Indicates which strategy to use. Strategy ‘A’ means to use the critical point as the centre of the circle and strategy ‘B’ means to use $z$

Details

These functions are not really intended for the user; hypergeo() uses hypergeo_residue_close_to_crit_multiple() when $\left|z-c\right|$ is less than $0.1$ (hardwired) for $c$ being either of the two critical points. Infinite regress is avoided because the contour is always more than this distance from the critical points.

These functions use the residue theorem $f\left(z_0\right)=\oint_C\frac{f(z)\,dz}{z-z_0}$ to evaluate the hypergeometric function near the two critical points $1/2\pm i\sqrt{3}/2$ . These points are problematic because all of the transformations listed under thingfun() take the points either to themselves or each other.

At these points the ratio of successive terms in the hypergeometric series tends to one and thus numerical summation is difficult.

The hypergeometric function, however, is not at all badly behaved near these critical points (see examples); but OTOH there do not seem to be any identities for the hypergeometric function at these points.

I have not investigated in detail whether strategy ‘A’ or ‘B’ is better. I would expect that ‘A’ is faster but ‘B’ more accurate, on the grounds that ‘A’ uses a contour whose closest approach to the critical point is further than that of ‘B’; but ‘B’ uses a contour which does not vary in distance from $z$ .

But both seem to be fairly accurate and fairly fast, and I have not systematically investigated the pros and cons.

Note

The residue theorem appears to be absurdly accurate for numerical evaluation

Author(s)

Robin K. S. Hankin

References

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

Examples

c1 <- 1/2-sqrt(3)/2i
c2 <- 1/2+sqrt(3)/2i

a1_R <- hypergeo(1/2,1/3,pi,c1)
a1_M <- 1.0154051314906669 + 0.0544835896509068i

x <- y <- seq(from=-0.1,to=0.1,len=100)
elliptic::view(x,y,hypergeo(1/2,1,1/3,outer(x,1i*y,"+")))

c1 <- 1/2-sqrt(3)/2i
c2 <- 1/2+sqrt(3)/2i

a1_R <- hypergeo(1/2,1/3,pi,c1)
a1_M <- 1.0154051314906669 + 0.0544835896509068i

x <- y <- seq(from=-0.1,to=0.1,len=100)
elliptic::view(x,y,hypergeo(1/2,1,1/3,outer(x,1i*y,"+")))

Evaluation of the hypergeometric function using Shanks's method

Description

Evaluation of the hypergeometric function using Shanks transformation of successive sums

Usage

hypergeo_shanks(A,B,C,z,maxiter=20)
genhypergeo_shanks(U,L,z,maxiter=20)
shanks(Last,This,Next)
hypergeo_shanks(A,B,C,z,maxiter=20)
genhypergeo_shanks(U,L,z,maxiter=20)
shanks(Last,This,Next)

Arguments

`A`, `B`, `C`	Parameters (real or complex)
`U`, `L`	Upper and lower vectors
`z`	Primary complex argument
`maxiter`	Maximum number of iterations
`Last`, `This`, `Next`	Three successive convergents

Details

The Shanks transformation of successive partial sums is

$S(n)=\frac{A_{n+1}A_{n-1}-A_n^2}{A_{n+1}-2A_n+A_{n-1}}$

and if the $A_n$ tend to a limit then the sequence $S(n)$ often converges more rapidly than $A_n$ . However, the denominator is susceptible to catastrophic rounding under fixed-precision arithmetic and it is difficult to know when to stop iterating.

Note

The

Author(s)

Robin K. S. Hankin

References

Shanks, D. (1955). “Non-linear transformation of divergent and slowly convergent sequences”, Journal of Mathematics and Physics 34:1-42

Examples


hypergeo_shanks(1/2,1/3,pi,z= 0.1+0.2i)
hypergeo_shanks(1/2,1/3,pi,z= 0.1+0.2i)

Various functions taken from the Wolfram Functions Site

Description

Various functions taken from the Wolfram Functions Site

Usage

w07.23.06.0026.01(A, n, m, z, tol = 0, maxiter = 2000, method = "a")
w07.23.06.0026.01_bit1(A, n, m, z, tol = 0)
w07.23.06.0026.01_bit2(A, n, m, z, tol = 0, maxiter = 2000)
w07.23.06.0026.01_bit3_a(A, n, m, z, tol = 0)
w07.23.06.0026.01_bit3_b(A, n, m, z, tol = 0)
w07.23.06.0026.01_bit3_c(A, n, m, z, tol = 0)
w07.23.06.0029.01(A, n, m, z, tol = 0, maxiter = 2000)
w07.23.06.0031.01(A, n, m, z, tol = 0, maxiter = 2000)
w07.23.06.0031.01_bit1(A, n, m, z, tol = 0, maxiter = 2000)
w07.23.06.0031.01_bit2(A, n, m, z, tol = 0, maxiter = 2000)
w07.23.06.0026.01(A, n, m, z, tol = 0, maxiter = 2000, method = "a")
w07.23.06.0026.01_bit1(A, n, m, z, tol = 0)
w07.23.06.0026.01_bit2(A, n, m, z, tol = 0, maxiter = 2000)
w07.23.06.0026.01_bit3_a(A, n, m, z, tol = 0)
w07.23.06.0026.01_bit3_b(A, n, m, z, tol = 0)
w07.23.06.0026.01_bit3_c(A, n, m, z, tol = 0)
w07.23.06.0029.01(A, n, m, z, tol = 0, maxiter = 2000)
w07.23.06.0031.01(A, n, m, z, tol = 0, maxiter = 2000)
w07.23.06.0031.01_bit1(A, n, m, z, tol = 0, maxiter = 2000)
w07.23.06.0031.01_bit2(A, n, m, z, tol = 0, maxiter = 2000)

Arguments

`A`	Parameter of hypergeometric function
`m`, `n`	Integers
`z`	Primary complex argument
`tol`, `maxiter`	Numerical arguments as per `genhypergeo()`
`method`	Character, specifying method to be used

Details

The method argument is described at f15.3.10. All functions' names follow the conventions in Hypergeometric2F1.pdf.

Function w07.23.06.0026.01(A, n, m, z) returns ${}_2F_1(A,A+n,A+m,z)$ where $m$ and $n$ are nonnegative integers with $m\geq n$ .
Function w07.23.06.0029.01(A, n, m, z) returns ${}_2F_1(A,A+n,A-m,z)$ .
Function w07.23.06.0031.01(A, n, m, z) returns ${}_2F_1(A,A+n,A+m,z)$ with $m\leq n$ .

Note

These functions use the psigamma() function which does not yet take complex arguments; this means that complex values for A are not supported. I'm working on it.

Author(s)

Robin K. S. Hankin

References

http://functions.wolfram.com/PDF/Hypergeometric2F1.pdf

Examples


# Here we catch some answers from Maple (jjM) and compare it with R's:


jjM <- 0.95437201847068289095 + 0.80868687461954479439i # Maple's answer
jjR <- w07.23.06.0026.01(A=1.1 , n=1 , m=4 , z=1+1i) 
# [In practice, one would type 'hypergeo(1.1, 2.1, 5.1, 1+1i)']

stopifnot(Mod(jjM - jjR) < 1e-10)


jjM <- -0.25955090546083991160e-3 - 0.59642767921444716242e-3i
jjR <- w07.23.06.0029.01(A=4.1 , n=1 , m=1 , z=1+4i)
# [In practice, one would type 'hypergeo(4.1, 3.1, 5.1, 1+1i)']

stopifnot(Mod(jjM - jjR) < 1e-15)

jjM <- 0.33186808222278923715e-1 - 0.40188208572232037363e-1i
jjR <- w07.23.06.0031.01(6.7,2,1,2+1i)
# [In practice, one would type 'hypergeo(6.7, 8.7, 7.7, 2+1i)']
stopifnot(Mod(jjM - jjR) < 1e-10)
# Here we catch some answers from Maple (jjM) and compare it with R's:


jjM <- 0.95437201847068289095 + 0.80868687461954479439i # Maple's answer
jjR <- w07.23.06.0026.01(A=1.1 , n=1 , m=4 , z=1+1i) 
# [In practice, one would type 'hypergeo(1.1, 2.1, 5.1, 1+1i)']

stopifnot(Mod(jjM - jjR) < 1e-10)


jjM <- -0.25955090546083991160e-3 - 0.59642767921444716242e-3i
jjR <- w07.23.06.0029.01(A=4.1 , n=1 , m=1 , z=1+4i)
# [In practice, one would type 'hypergeo(4.1, 3.1, 5.1, 1+1i)']

stopifnot(Mod(jjM - jjR) < 1e-15)

jjM <- 0.33186808222278923715e-1 - 0.40188208572232037363e-1i
jjR <- w07.23.06.0031.01(6.7,2,1,2+1i)
# [In practice, one would type 'hypergeo(6.7, 8.7, 7.7, 2+1i)']
stopifnot(Mod(jjM - jjR) < 1e-10)

Package 'hypergeo'

Help Index

The Gauss hypergeometric function

Description

Details

Author(s)

References

Examples

Evaluation of the hypergeometric function using Buhring's method

Description

Usage

Arguments

Details

Note

Author(s)

References

See Also

Examples

Gamma function for complex arguments

Description

Usage

Arguments

Details

Author(s)

References

Examples

Low-level C functions

Description

Usage

Arguments

Note

Author(s)

Examples

Hypergeometric function using Euler's integral representation

Description

Usage

Arguments

Details

Note

Author(s)

References

See Also

Examples

Transformations of the hypergeometric function

Description

Usage

Arguments

Details

Note

Author(s)

References

See Also

Examples

Various transformation formulae for the hypergeometric function

Description

Usage

Arguments

Details

Author(s)

References

See Also

Examples

Hypergeometric functions via direct numerical integration

Description

Usage

Arguments

Details

Note

Author(s)

References

See Also

Examples

The generalized hypergeometric function

Description

Usage

Arguments

Details

Note

Author(s)

References