Search results for "Hypergeometric function"
showing 10 items of 22 documents
Exact Fourier expansion in cylindrical coordinates for the three-dimensional Helmholtz Green function
2009
A new method is presented for Fourier decomposition of the Helmholtz Green Function in cylindrical coordinates, which is equivalent to obtaining the solution of the Helmholtz equation for a general ring source. The Fourier coefficients of the Helmholtz Green function are split into their half advanced+half retarded and half advanced-half retarded components. Closed form solutions are given for these components in terms of a Horn function and a Kampe de Feriet function, respectively. The systems of partial differential equations associated with these two-dimensional hypergeometric functions are used to construct a fourth-order ordinary differential equation which both components satisfy. A s…
Indefinite integrals of special functions from integrating factors
2019
Some general integrals are presented which were obtained from two integrating factors f(x) and fˆ(x) for the first two and last two terms, respectively, of the second-order linear ordinary differen...
Generalizations of Clausen's formula and algebraic transformations of Calabi-Yau differential equations
2011
AbstractWe provide certain unusual generalizations of Clausen's and Orr's theorems for solutions of fourth-order and fifth-order generalized hypergeometric equations. As an application, we present several examples of algebraic transformations of Calabi–Yau differential equations.
Simple connections between generalized hypergeometric series and dilogarithms
1997
AbstractConnections between generalized hypergeometric series and dilogarithms are investigated. Some simple relations of an Appell's function and dilogarithms are found.
Generalized hypergeometric functions and the evaluation of scalar one-loop integrals in Feynman diagrams
2000
Present and future high-precision tests of the Standard Model and beyond for the fundamental constituents and interactions in Nature are demanding complex perturbative calculations involving multi-leg and multi-loop Feynman diagrams. Currently, large effort is devoted to the search for closed expressions of loop integrals, written whenever possible in terms of known - often hypergeometric-type - functions. In this work, the scalar three-point function is re-evaluated by means of generalized hypergeometric functions of two variables. Finally, use is made of the connection between such Appell functions and dilogarithms coming from a previous investigation, to recover well-known results.
Indefinite integrals of products of special functions
2016
ABSTRACTA method is given for deriving indefinite integrals involving squares and other products of functions which are solutions of second-order linear differential equations. Several variations of the method are presented, which applies directly to functions which obey homogeneous differential equations. However, functions which obey inhomogeneous equations can be incorporated into the products and examples are given of integrals involving products of Bessel functions combined with Lommel, Anger and Weber functions. Many new integrals are derived for a selection of special functions, including Bessel functions, associated Legendre functions, and elliptic integrals. A number of integrals o…
Fourier series for elliptic integrals and some generalizations via hypergeometric series
2008
Fourier series are derived for generalizations of the three canonical Legendre incomplete elliptic integrals using a hypergeometric series approach. The Fourier series for the incomplete Epstein–Hubbell integrals are obtained as special cases of the generalization of the Legendre integrals of the first and second kinds. The Fourier series for the integrals of the first and second kinds, and those for the Epstein–Hubbell integrals, were obtained recently using a different approach, but the series obtained for the generalization of the incomplete integral of the third kind is new. All cases of the integral of the third kind are given, with the modulus and the parameter being complex variables…
Special functions for the study of economic dynamics: The case of the Lucas-Uzawa model
2008
The special functions are intensively used in mathematical physics to solve differential systems. We argue that they should be most useful in economic dynamics, notably in the assessment of the transition dynamics of endogenous economic growth models. We illustrate our argument on the famous Lucas-Uzawa model, which we solve by the means of Gaussian hypergeometric functions. We show how the use of Gaussian hypergeometric functions allows for an explicit representation of the equilibrium dynamics of all variables in level. The parameters of the involved hypergeometric functions are identified using the Pontryagin conditions arising from the underlying optimization problems. In contrast to th…
A characterization of the distribution of a weighted sum of gamma variables through multiple hypergeometric functions
2008
Applying the theory on multiple hypergeometric functions, the distribution of a weighted convolution of Gamma variables is characterized through explicit forms for the probability density function, the distribution function and the moments about the origin. The main results unify some previous contributions in the literature on nite convolution of Gamma distributions. We deal with computational aspects that arise from the representations in terms of multiple hypergeometric functions, introducing a new integral representation for the fourth Lauricella function F (n) D and its con uent form (n) 2 , suitable for numerical integration; some graphics of the probability density function and distr…
Approximate survival probability determination of hysteretic systems with fractional derivative elements
2018
Abstract A Galerkin scheme-based approach is developed for determining the survival probability and first-passage probability of a randomly excited hysteretic systems endowed with fractional derivative elements. Specifically, by employing a combination of statistical linearization and of stochastic averaging, the amplitude of the system response is modeled as one-dimensional Markovian Process. In this manner the corresponding backward Kolmogorov equation which governs the evolution of the survival probability of the system is determined. An approximate solution of this equation is sought by employing a Galerkin scheme in which a convenient set of confluent hypergeometric functions is used a…