Numerical evaluation of Appell's F1 hypergeometric function

In this work we present a numerical scheme to compute the two-variable hypergeometric function F1(α, β, β′, γ; x, y) of Appell for complex parameters α, β, β′ and γ, and real values of the variables x and y. We implement a set of analytic continuations that allow us to obtain the F1 function outside...

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Detalles Bibliográficos
Autores principales:	Colavecchia, F.D., Gasaneo, G., Miraglia, J.E.
Formato:	JOUR
Materias:	Appell functions Gauss function Hypergeometric functions Numerical methods Special functions Computational complexity Computational geometry Convergence of numerical methods Integration Ordinary differential equations Partial differential equations Appell's function Horn's function Function evaluation
Acceso en línea:	http://hdl.handle.net/20.500.12110/paper_00104655_v138_n1_p29_Colavecchia
Aporte de:	Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) de Universidad de Buenos Aires

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spelling	todo:paper_00104655_v138_n1_p29_Colavecchia2023-10-03T14:09:06Z Numerical evaluation of Appell's F1 hypergeometric function Colavecchia, F.D. Gasaneo, G. Miraglia, J.E. Appell functions Gauss function Hypergeometric functions Numerical methods Special functions Computational complexity Computational geometry Convergence of numerical methods Integration Ordinary differential equations Partial differential equations Appell's function Horn's function Hypergeometric functions Function evaluation In this work we present a numerical scheme to compute the two-variable hypergeometric function F1(α, β, β′, γ; x, y) of Appell for complex parameters α, β, β′ and γ, and real values of the variables x and y. We implement a set of analytic continuations that allow us to obtain the F1 function outside the region of convergence of the series definition. These continuations can be written in terms of the Horn's G2 function, Appell's F2 function related, and the F1 hypergeometric itself. The computation of the function inside the region of convergence is achieved by two complementary methods. The first one involves a single-index series expansion of the F1 function, while the second one makes use of a numerical integration of a third order ordinary differential equation that represents the system of partial differential equations of the F1 function. We briefly sketch the program and show some examples of the numerical computation. © 2001 Elsevier Science B.V. JOUR info:eu-repo/semantics/openAccess http://creativecommons.org/licenses/by/2.5/ar http://hdl.handle.net/20.500.12110/paper_00104655_v138_n1_p29_Colavecchia
institution	Universidad de Buenos Aires
institution_str	I-28
repository_str	R-134
collection	Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA)
topic	Appell functions Gauss function Hypergeometric functions Numerical methods Special functions Computational complexity Computational geometry Convergence of numerical methods Integration Ordinary differential equations Partial differential equations Appell's function Horn's function Hypergeometric functions Function evaluation
spellingShingle	Appell functions Gauss function Hypergeometric functions Numerical methods Special functions Computational complexity Computational geometry Convergence of numerical methods Integration Ordinary differential equations Partial differential equations Appell's function Horn's function Hypergeometric functions Function evaluation Colavecchia, F.D. Gasaneo, G. Miraglia, J.E. Numerical evaluation of Appell's F1 hypergeometric function
topic_facet	Appell functions Gauss function Hypergeometric functions Numerical methods Special functions Computational complexity Computational geometry Convergence of numerical methods Integration Ordinary differential equations Partial differential equations Appell's function Horn's function Hypergeometric functions Function evaluation
description	In this work we present a numerical scheme to compute the two-variable hypergeometric function F1(α, β, β′, γ; x, y) of Appell for complex parameters α, β, β′ and γ, and real values of the variables x and y. We implement a set of analytic continuations that allow us to obtain the F1 function outside the region of convergence of the series definition. These continuations can be written in terms of the Horn's G2 function, Appell's F2 function related, and the F1 hypergeometric itself. The computation of the function inside the region of convergence is achieved by two complementary methods. The first one involves a single-index series expansion of the F1 function, while the second one makes use of a numerical integration of a third order ordinary differential equation that represents the system of partial differential equations of the F1 function. We briefly sketch the program and show some examples of the numerical computation. © 2001 Elsevier Science B.V.
format	JOUR
author	Colavecchia, F.D. Gasaneo, G. Miraglia, J.E.
author_facet	Colavecchia, F.D. Gasaneo, G. Miraglia, J.E.
author_sort	Colavecchia, F.D.
title	Numerical evaluation of Appell's F1 hypergeometric function
title_short	Numerical evaluation of Appell's F1 hypergeometric function
title_full	Numerical evaluation of Appell's F1 hypergeometric function
title_fullStr	Numerical evaluation of Appell's F1 hypergeometric function
title_full_unstemmed	Numerical evaluation of Appell's F1 hypergeometric function
title_sort	numerical evaluation of appell's f1 hypergeometric function
url	http://hdl.handle.net/20.500.12110/paper_00104655_v138_n1_p29_Colavecchia
work_keys_str_mv	AT colavecchiafd numericalevaluationofappellsf1hypergeometricfunction AT gasaneog numericalevaluationofappellsf1hypergeometricfunction AT miragliaje numericalevaluationofappellsf1hypergeometricfunction
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Numerical evaluation of Appell's F1 hypergeometric function

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