docs for arxiv preprint

Author: MikaelSlevinsky

Project.toml

@@ -1,6 +1,6 @@
 name = "HypergeometricFunctions"
 uuid = "34004b35-14d8-5ef3-9330-4cdb6864b03a"
-version = "0.3.18"
+version = "0.3.19"
 
 [deps]
 DualNumbers = "fa6b7ba4-c1ee-5f82-b5fc-ecf0adba8f74"

README.md

@@ -2,7 +2,7 @@
 
 [![Build Status](https://github.com/JuliaMath/HypergeometricFunctions.jl/workflows/CI/badge.svg)](https://github.com/JuliaMath/HypergeometricFunctions.jl/actions?query=workflow%3ACI) [![codecov](https://codecov.io/gh/JuliaMath/HypergeometricFunctions.jl/branch/master/graph/badge.svg)](https://codecov.io/gh/JuliaMath/HypergeometricFunctions.jl) [![](https://img.shields.io/badge/docs-stable-blue.svg)](https://JuliaMath.github.io/HypergeometricFunctions.jl/stable) [![](https://img.shields.io/badge/docs-dev-blue.svg)](https://JuliaMath.github.io/HypergeometricFunctions.jl/dev)
 
-This package implements the generalized hypergeometric function `pFq((α1,…,αp), (β1,…,βq), z)`. Please refer to the documentation for more information.
+This package provides an implementation of the generalized hypergeometric function `pFq((α1,…,αp), (β1,…,βq), z)`.
 
 ```julia
 julia> using HypergeometricFunctions
@@ -23,3 +23,11 @@ julia> pFq((1, 2+im), (3.5, ), exp(im*π/3)) # ₂F₁ at that special point in
 0.6786952632946592 + 0.4523504929285015im
 
 ```
+
+# References
+
+[1] N. Michel and M. V. Stoitsov, [Fast computation of the Gauss hypergeometric function with all its parameters complex with application to the Pöschl–Teller–Ginocchio potential wave functions](https://doi.org/10.1016/j.cpc.2007.11.007), *Comp. Phys. Commun.*, **178**:535–551, 2008.
+
+[2] J. W. Pearson, S. Olver and M. A. Porter, [Numerical methods for the computation of the confluent and Gauss hypergeometric functions](https://doi.org/10.1007/s11075-016-0173-0), *Numer. Algor.*, **74**:821–866, 2017.
+
+[3] R. M. Slevinsky, [Fast and stable rational approximation of generalized hypergeometric functions](https://arxiv.org/abs/2307.06221), arXiv:2307.06221, 2023.

docs/src/index.md

@@ -22,28 +22,17 @@ z = x' .+ im*y
 
 import Logging # To avoid printing warnings
 Logging.with_logger(Logging.SimpleLogger(Logging.Error)) do
-    fz = map(z->pFq((), (), z), z)
-    img = portrait(fz, ctype = "nist")
+    img = portrait(map(z->pFq((), (), z), z), ctype = "nist")
     save("0F0.png", img)
-
-    fz = map(z->pFq((), (1.0, ), z), z)
-    img = portrait(fz, ctype = "nist")
+    img = portrait(map(z->pFq((), (1.0, ), z), z), ctype = "nist")
     save("0F1.png", img)
-
-    fz = map(z->pFq((0.5, ), (0.75, ), z), z)
-    img = portrait(fz, ctype = "nist")
+    img = portrait(map(z->pFq((0.5, ), (0.75, ), z), z), ctype = "nist")
     save("1F1.png", img)
-
-    fz = map(z->pFq((3.5+7.5im, ), (), z), z)
-    img = portrait(fz, ctype = "nist")
+    img = portrait(map(z->pFq((3.5+7.5im, ), (), z), z), ctype = "nist")
     save("1F0.png", img)
-
-    fz = map(z->pFq((1.0, 3.5+7.5im), (0.75, ), z), z)
-    img = portrait(fz, ctype = "nist")
+    img = portrait(map(z->pFq((1.0, 3.5+7.5im), (0.75, ), z), z), ctype = "nist")
     save("2F1.png", img)
-
-    fz = map(z->pFq((1.0, 1.5+7.5im), (), z), z)
-    img = portrait(fz, ctype = "nist")
+    img = portrait(map(z->pFq((1.0, 1.5+7.5im), (), z), z), ctype = "nist")
     save("2F0.png", img)
 end
 nothing # hide
@@ -71,6 +60,8 @@ HypergeometricFunctions.U
 HypergeometricFunctions._₂F₁positive
 HypergeometricFunctions._₂F₁general
 HypergeometricFunctions._₂F₁general2
+HypergeometricFunctions.pFqdrummond
+HypergeometricFunctions.pFqweniger
 HypergeometricFunctions.pFqcontinuedfraction
 HypergeometricFunctions.pochhammer
 HypergeometricFunctions.@clenshaw

src/drummond.jl

@@ -1,5 +1,10 @@
-# Rational approximations to generalized hypergeometric functions
-# using Drummond's sequence transformation
+@doc raw"""
+    pFqdrummond(α, β, z; kmax)
+Compute the generalized hypergeometric function [`pFq`](@ref) by rational approximations of type (k, k) generated by Drummond's sequence transformation described in
+
+> R. M. Slevinsky, [Fast and stable rational approximation of generalized hypergeometric functions](https://arxiv.org/abs/2307.06221), arXiv:2307.06221, 2023.
+"""
+pFqdrummond
 
 # ₀F₀(;z)
 function pFqdrummond(::Tuple{}, ::Tuple{}, z::T; kmax::Int = KMAX) where T

src/gauss.jl

@@ -71,7 +71,7 @@ end
 Compute the Gauss hypergeometric function `₂F₁(a, b, c; z)` with general parameters a, b, and c.
 This polyalgorithm is designed based on the paper
 
-> N. Michel and M. V. Stoitsov, [Fast computation of the Gauss hypergeometric function with all its parameters complex with application to the Pöschl–Teller–Ginocchio potential wave functions](https://doi.org/10.1016/j.cpc.2007.11.007), Comp. Phys. Commun., 178:535–551, 2008.
+> N. Michel and M. V. Stoitsov, [Fast computation of the Gauss hypergeometric function with all its parameters complex with application to the Pöschl–Teller–Ginocchio potential wave functions](https://doi.org/10.1016/j.cpc.2007.11.007), *Comp. Phys. Commun.*, **178**:535–551, 2008.
 """
 function _₂F₁general(a, b, c, z; kwds...)
     if z == 1
@@ -101,7 +101,7 @@ end
 Compute the Gauss hypergeometric function `₂F₁(a, b, c; z)` with general parameters a, b, and c.
 This polyalgorithm is designed based on the review
 
-> J. W. Pearson, S. Olver and M. A. Porter, [Numerical methos for the computation of the confluent and Gauss hypergeometric functions](https://doi.org/10.1007/s11075-016-0173-0), Numer. Algor., 74:821–866, 2017.
+> J. W. Pearson, S. Olver and M. A. Porter, [Numerical methods for the computation of the confluent and Gauss hypergeometric functions](https://doi.org/10.1007/s11075-016-0173-0), *Numer. Algor.*, **74**:821–866, 2017.
 """
 function _₂F₁general2(a, b, c, z)
     T = promote_type(typeof(a), typeof(b), typeof(c), typeof(z))

src/specialfunctions.jl

@@ -51,7 +51,7 @@ ogamma(x::Number) = (isinteger(x) && x<0) ? zero(float(x)) : inv(gamma(x))
 """
     @clenshaw(x, c...)
 
-Evaluate the Chebyshev polynomial series ``\\sum_{k=1}^N c[k] T_{k-1}(x)`` by the Clenshaw algorithm.
+Evaluate the Chebyshev polynomial series ``\\displaystyle \\sum_{k=1}^N c[k] T_{k-1}(x)`` by the Clenshaw algorithm.
 
 External links: [DLMF](https://dlmf.nist.gov/3.11#ii), [Wikipedia](https://en.wikipedia.org/wiki/Clenshaw_algorithm).
 
@@ -179,11 +179,11 @@ unsafe_gamma(z) = gamma(z)
 """
     @lanczosratio(z, ϵ, c₀, c...)
 
-Evaluate ``\\dfrac{\\sum_{k=0}^{N-1} \\frac{c[k+1]}{(z+k)(z+k+\\epsilon)}}{c_0 + \\sum_{k=0}^{N-1} \\frac{c[k+1]}{z+k}}`.
+Evaluate ``\\dfrac{\\displaystyle \\sum_{k=0}^{N-1} \\frac{c[k+1]}{(z+k)(z+k+\\epsilon)}}{\\displaystyle c_0 + \\sum_{k=0}^{N-1} \\frac{c[k+1]}{z+k}}``.
 
 This ratio is used in the Lanczos approximation of ``\\log\\frac{\\Gamma(z+\\epsilon)}{\\Gamma(z)}`` in
 
-> N. Michel and M. V. Stoitsov, [Fast computation of the Gauss hypergeometric function with all its parameters complex with application to the Pöschl–Teller–Ginocchio potential wave functions](https://doi.org/10.1016/j.cpc.2007.11.007), Comp. Phys. Commun., 178:535–551, 2008.
+> N. Michel and M. V. Stoitsov, [Fast computation of the Gauss hypergeometric function with all its parameters complex with application to the Pöschl–Teller–Ginocchio potential wave functions](https://doi.org/10.1016/j.cpc.2007.11.007), *Comp. Phys. Commun.*, **178**:535–551, 2008.
 """
 macro lanczosratio(z, ϵ, c₀, c...)
     ex_num = :(zero(z))
@@ -228,7 +228,7 @@ end
 """
 Compute the function ``\\dfrac{\\frac{1}{\\Gamma(z)}-\\frac{1}{\\Gamma(z+\\epsilon)}}{\\epsilon}`` by the method dscribed in
 
-> N. Michel and M. V. Stoitsov, [Fast computation of the Gauss hypergeometric function with all its parameters complex with application to the Pöschl–Teller–Ginocchio potential wave functions](https://doi.org/10.1016/j.cpc.2007.11.007), Comp. Phys. Commun., 178:535–551, 2008.
+> N. Michel and M. V. Stoitsov, [Fast computation of the Gauss hypergeometric function with all its parameters complex with application to the Pöschl–Teller–Ginocchio potential wave functions](https://doi.org/10.1016/j.cpc.2007.11.007), *Comp. Phys. Commun.*, **178**:535–551, 2008.
 """
 function G(z::Union{Float64, ComplexF64, Dual128, DualComplex256}, ϵ::Union{Float64, ComplexF64, Dual128, DualComplex256})
     n, zpϵ = round(Int, real(z)), z+ϵ
@@ -260,7 +260,7 @@ G(z::Number, ϵ::Number) = G(promote(z, ϵ)...)
 """
 Compute the function ``\\dfrac{(z+\\epsilon)_m-(z)_m}{\\epsilon}`` by the method dscribed in
 
-> N. Michel and M. V. Stoitsov, [Fast computation of the Gauss hypergeometric function with all its parameters complex with application to the Pöschl–Teller–Ginocchio potential wave functions](https://doi.org/10.1016/j.cpc.2007.11.007), Comp. Phys. Commun., 178:535–551, 2008.
+> N. Michel and M. V. Stoitsov, [Fast computation of the Gauss hypergeometric function with all its parameters complex with application to the Pöschl–Teller–Ginocchio potential wave functions](https://doi.org/10.1016/j.cpc.2007.11.007), *Comp. Phys. Commun.*, **178**:535–551, 2008.
 """
 function P(z::Number, ϵ::Number, m::Int)
     n₀ = -round(Int, real(z))

src/weniger.jl

@@ -1,5 +1,10 @@
-# Rational approximations to generalized hypergeometric functions
-# using Weniger's sequence transformation
+@doc raw"""
+    pFqweniger(α, β, z; kmax, γ = 2)
+Compute the generalized hypergeometric function [`pFq`](@ref) by rational approximations of type (k, k) generated by a factorial Levin-type sequence transformation described in
+
+> R. M. Slevinsky, [Fast and stable rational approximation of generalized hypergeometric functions](https://arxiv.org/abs/2307.06221), arXiv:2307.06221, 2023.
+"""
+pFqweniger
 
 # ₀F₀(;z), γ = 2.
 function pFqweniger(::Tuple{}, ::Tuple{}, z::T; kmax::Int = KMAX) where T