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HypergeometricFunctions.jl Documentation

Introduction

HypergeometricFunctions.jl provides a numerical computation of generalized hypergeometric functions. The main exported function and recommended interface is pFq, but there are a few others for specialists' convenience.

HypergeometricFunctions.pFq — Function
pFq(α, β, z)

Compute the generalized hypergeometric function, defined by

\[{}_pF_q(α, β, z) = \sum_{k=0}^\infty \dfrac{(\alpha_1)_k\cdots(\alpha_p)_k}{(\beta_1)_k\cdots(\beta_q)_k}\dfrac{z^k}{k!},\]

where the series converges and elsewhere by analytic continuation.

External links: DLMF, Wikipedia.

Examples

julia> pFq((), (), 0.1) # ≡ exp(0.1)
1.1051709180756477

julia> pFq((0.5, ), (), 1.0+0.001im) # ≡ exp(-0.5*log1p(-1.0-0.001im))
22.360679774997894 + 22.36067977499789im

julia> pFq((), (1.5, ), -π^2/4) # A root of a spherical Bessel function
4.042865030283967e-17

julia> pFq((), (1.5, ), -big(π)^2/4) # In extended precision
8.674364372518869408017614476652675180406967418943475242812160199356160822272727e-78

julia> pFq((1, ), (2, ), 0.01) # ≡ expm1(0.01)/0.01
1.0050167084168058

julia> pFq((1/3, ), (2/3, ), -1000) # A confluent hypergeometric with large argument
0.050558053946448855

julia> pFq((1, 2), (4, ), 1) # a well-poised ₂F₁
2.9999999999999996

julia> pFq((1, 2+im), (3.5, ), exp(im*π/3)) # ₂F₁ at that special point in ℂ
0.6786952632946592 + 0.4523504929285015im

julia> pFq((1, 2+im), (3.5, ), exp(im*big(π)/3)) # More digits, you say?
0.6786952632946589823300834090168381068073515492901393549193461972311801512528996 + 0.4523504929285013648194489713901658143893464679689810112119412310631860619947939im

julia> pFq((1, 2+im, 2.5), (3.5, 4), exp(im*π/3)) # ₃F₂ because why not
0.8434434031615691 + 0.34175507615463174im

julia> pFq((1, 2+im, 2.5), (3.5, 4), exp(im*big(π)/3)) # Also in extended precision
0.8434434031615690763389963048175253868863156451003855955719081209861492349268002 + 0.3417550761546319732614495656712509723030350666571102474299311122586948108413206im

julia> pFq((1, 1), (), -1) # A divergent series
0.5963473623231942

julia> pFq((1, 1), (), -big(1))
0.5963473623231940743410784993692793760741778601525487815734849104823272191142015
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Complex phase portraits

Broadly speaking, there are three classes of generalized hypergeometric functions: when $p\le q$ they are entire functions of the complex variable $z$; when $p = q+1$, they are analytic functions in the cut plane $\mathbb{C}\setminus[1,\infty)$; and, when $p > q+1$, they are analytic functions in the cut plane $\mathbb{C}\setminus[0,\infty)$.

Examples of each of these classes are illustrated over $\left\{z\in\mathbb{C} : -10<\Re z<10, -10<\Im z<10\right\}$ with complex phase portraits, a beautiful tool in computational complex analysis.

using ComplexPhasePortrait, HypergeometricFunctions, Images
x = range(-10, stop=10, length=300)
y = range(-10, stop=10, length=300)
z = x' .+ im*y

import Logging # To avoid printing warnings
Logging.with_logger(Logging.SimpleLogger(Logging.Error)) do
    img = portrait(map(z->pFq((), (), z), z), ctype = "nist")
    save("0F0.png", img)
    img = portrait(map(z->pFq((), (1.0, ), z), z), ctype = "nist")
    save("0F1.png", img)
    img = portrait(map(z->pFq((0.5, ), (0.75, ), z), z), ctype = "nist")
    save("1F1.png", img)
    img = portrait(map(z->pFq((3.5+7.5im, ), (), z), z), ctype = "nist")
    save("1F0.png", img)
    img = portrait(map(z->pFq((1.0, 3.5+7.5im), (0.75, ), z), z), ctype = "nist")
    save("2F1.png", img)
    img = portrait(map(z->pFq((1.0, 1.5+7.5im), (), z), z), ctype = "nist")
    save("2F0.png", img)
end
p\q01
0₀F₀₀F₁
1₁F₀₁F₁
2₂F₀₂F₁

Library

HypergeometricFunctions._₃F₂ — Function

Compute the generalized hypergeometric function ₃F₂(a₁, 1, 1, b₁, 2, z).

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Compute the generalized hypergeometric function ₃F₂(a₁, a₂, a₃, b₁, b₂; z).

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Internals

HypergeometricFunctions.@clenshaw — Macro
@clenshaw(x, c...)

Evaluate the Chebyshev polynomial series $\displaystyle \sum_{k=1}^N c[k] T_{k-1}(x)$ by the Clenshaw algorithm.

External links: DLMF, Wikipedia.

Examples

julia> HypergeometricFunctions.@clenshaw(1, 1, 2, 3)
6

julia> HypergeometricFunctions.@clenshaw(0.5, 1, 2, 3)
0.5
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HypergeometricFunctions.@lanczosratio — Macro
@lanczosratio(z, ϵ, c₀, c...)

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, Comp. Phys. Commun., 178:535–551, 2008.

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