diff --git a/dev/index.html b/dev/index.html index f6fdbc3..4c95c3e 100644 --- a/dev/index.html +++ b/dev/index.html @@ -36,7 +36,7 @@ 0.5963473623231942 julia> pFq((1, 1), (), -big(1)) -0.5963473623231940743410784993692793760741778601525487815734849104823272191142015source

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
+0.5963473623231940743410784993692793760741778601525487815734849104823272191142015
source

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
@@ -55,8 +55,8 @@
     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._₁F₁ — Function

Compute Kummer's confluent hypergeometric function ₁F₁(a, b, z).

source
HypergeometricFunctions._₂F₁ — Function

Compute the Gauss hypergeometric function ₂F₁(a, b, c, z).

source
HypergeometricFunctions._₃F₂ — Function

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

source

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

source

Internals

HypergeometricFunctions.M — Function

Compute Kummer's confluent hypergeometric function M(a, b, z) = ₁F₁(a, b, z).

source
HypergeometricFunctions.U — Function

Compute Tricomi's confluent hypergeometric function U(a, b, z) ∼ z⁻ᵃ ₂F₀((a, a-b+1), (), -z⁻¹).

source
HypergeometricFunctions._₂F₁positive — Function

Compute the Gauss hypergeometric function ₂F₁(a, b, c, z) with positive parameters a, b, and c and argument 0 ≤ z ≤ 1. Useful for statisticians.

source
HypergeometricFunctions._₂F₁general — Function

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

source
HypergeometricFunctions._₂F₁general2 — Function

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 methods for the computation of the confluent and Gauss hypergeometric functions, Numer. Algor., 74:821–866, 2017.

source
HypergeometricFunctions.pFqdrummond — Function
pFqdrummond(α, β, z; kmax)

Compute the generalized hypergeometric function pFq 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, arXiv:2307.06221, 2023.

source
HypergeometricFunctions.pFqweniger — Function
pFqweniger(α, β, z; kmax, γ = 2)

Compute the generalized hypergeometric function pFq 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, arXiv:2307.06221, 2023.

source
HypergeometricFunctions.pFqcontinuedfraction — Function

Compute the generalized hypergeometric function pFq(α, β, z) by continued fraction.

source
HypergeometricFunctions.pochhammer — Function

Pochhammer symbol $(x)_n = \frac{\Gamma(x+n)}{\Gamma(x)}$ for the rising factorial.

source
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)
+end
p\q01
0₀F₀₀F₁
1₁F₀₁F₁
2₂F₀₂F₁

Library

HypergeometricFunctions._₃F₂ — Function

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

source

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

source

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
source
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.

source
+0.5source
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.

source
HypergeometricFunctions.G — Function

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

source
HypergeometricFunctions.P — Function

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

source
diff --git a/dev/search/index.html b/dev/search/index.html index 1039dfa..0176e78 100644 --- a/dev/search/index.html +++ b/dev/search/index.html @@ -1,2 +1,2 @@ -Search · HypergeometricFunctions.jl

Loading search...

    +Search · HypergeometricFunctions.jl

    Loading search...