He also obtained a mellin barnes integral representation for 1t h ef a ct or4 is ndu x p l v. Paris, asymptotics of the gauss hypergeometric function with large parameters, ii,j. Fractional integrals and derivatives, theory and applications. Asymptotics and mellin barnes integrals, first published in, provides an account of the use and properties of a type of complex integral representation that. Asymptotics and mellin barnes integrals encyclopedia of mathematics and its applications 85 by r.
A unified approach to the integrals of mellinbarneshecke. In this paper, we provide a unified approach to a family of integrals of mellin barnes type using distribution theory and fourier transforms. On convergent series representations of mellinbarnes. As a result, the mellinbarnes integral representations of the twoloop vertex feynman. After developing the properties of these integrals. As shown in 22, 29, see also 10, 16, 32, 36, a powerful method to evaluate integrals can be obtained based on the mellin setting. In this talk, we describe part of our recent work 1 see also 2, 3 that gives new results in the context of asymptotic expansions of feynman diagrams using the mellin barnes representation.
Mceliece the theory of information and coding, 2nd edn 87 b. The corresponding algorithm is implemented as a mathematica code mbresolve. Mbsumsa mathematica package for the representation of mellin. Asymptotics of the gauss hypergeometric function with. Repeated integration by parts will often lead to an asymptotic expansion.
Finally, the numerical integration of multidimensional mb integrals in minkowskian regions is carried out. Asymptoticsandmellinbarnesintegralsisacomprehensiveaccountoftheproper tiesofmellinbarnesintegralsandtheirapplicationtoproblemsinvolvingspecial functions. We apply the mellinbarnes integral representation to. Kaminski, asymptotics and mellin barnes integrals, cambridge university press, 2001. Kaminski asymptotics and mellin barnes integrals 86 r. Feynman integrals may be represented by the mathematica packages ambre and mb as multiple mellinbarnes integrals. See also other reference books cited in 2, 21, and 24. Further, making use of the lemma 4 and interpreting the resulting mellin barnes multiple integrals contour as. To address this problem we introduce and discuss in detail a. Methods of generating such expansions include the eulermaclaurin summation formula and integral transforms such as the laplace and mellin transforms. Introduction to mellinbarnes representations for feynman integrals tord riemann, desy, zeuthen calc, july 2006, dubna introduction. Asymptotics and mellin barnes integrals, first published in 2001, provides an account of the use and properties of a type of complex integral representation that arises frequently in the study of special functions typically of interest in classical analysis and mathematical physics.
Barnes integral representation the mellin transform of a function is given by fs. We consider the asymptotic expansion of the singleparameter mittagleffler function ea. Tables of mellin transforms download ebook pdf, epub. We consider the asymptotic expansion of the generalised exponential integral involving the mittagleffler function introduced recently by mainardi and masina fract. The efficiency of application of the proposed contour and the quadratic approximation to the contour of the stationary phase is compared by the example of the inverse mellin transform for the. At present the analytic evaluation for most of these integrals is out of reach and purely. Paris the asymptotics of a generalised beta function mathematica aeterna, vol. Asymptotics and mellinbarnes integral pdf asymptotics and mellinbarnes integrals, first published in, provides an account of the use and properties of a type of complex integral representation skip to content. Volume 96 basic hypergeometric series second edition.
The resurgence properties of the large order asymptotics. The fundamental solution is interpreted as a probability density function and its entropy in the shannon sense is calculated. This integral transform is closely connected to the theory of dirichlet series, and is often. University of abertay dundee, dundee dd1 1hg, uk division of computing and mathematics, r. First, we define integral over a curve depending on and apply the residue theorem. This says that the asymptotics of the mirror oscillatory integral is described in terms of the bclass of x.
Fractional integral formulae involving the srivastava. Since a convergent taylor series fits the definition of asymptotic expansion as well. Asymptotics and mellinbarnes integrals encyclopedia of. The hankel integral, arises naturally in many fields of application in physics 15. Highprecision numerical estimates of the mellinbarnes. Description asymptotics and mellin barnes integrals, first published inprovides an account of the use and properties of a type of complex integral representation that arises frequently in the study of special functions typically of interest in classical analysis and mathematical. Asymptotics and mellinbarnes integral pdf asymptotics and mellinbarnes integrals, first published in, provides an account of the use and properties of a type of complex integral representation that. The integration cycle here should be a lagrangian section of the stromingeryauzaslow bration 53 i. Then a series representation and asymptotics of the fundamental solution are discussed. Asymptotics and mellin barnes integrals, first published in 2001, provides an account of the use and properties of a type of complex integral representation that arises frequently in the study.
Applications of the mellinbarnes integral representation. The expansions of the similarly extended sine and cosine integrals are also. On the contrary, many asymptotic methods have been designed. Here we concentrate on the form of this exponentially small. Asymptotics and mellinbarnes integrals, first published in 2001, provides an account of the use and properties of a type of complex integral representation that arises frequently in the study of special functions typically of interest in classical analysis and mathematical physics. In mathematics, a barnes integral or mellinbarnes integral is a contour integral involving a product of gamma functions. Asymptotic expansions of feynman diagrams and the mellin.
Asymptotics and mellinbarnes integral pdf february 28, 2020 admin asymptotics and mellinbarnes integrals, first published in, provides an account of the use and properties of a type of complex integral representation that. The book also fills a gap in the literature on asymptotic analysis and special functions by providing a thorough account of the use of mellin barnes integrals that. Asymptotics and mellin barnes integrals provides an account of the use and properties of a type of complex integral representation that arises frequently in the study of special functions. The second kummer function with matrix parameters and its. The sum of the residues converges to an expression containing. They are closely related to generalized hypergeometric series. Analogously to the case with realvalued parameters see 21, p.
A recent example, which has served as the motivation behind this work, arises in the quantum tunnelling problem where the traversal time across a potential barrier appears in the form of a hankel integral of the zeroth order. Asymptotics and mellinbarnes integrals provides an account of the use and properties of a type of complex integral representation that arises frequently in the study of special functions typically of interest in classical analysis and mathematical physics. In mathematics, the mellin transform is an integral transform that may be regarded as the multiplicative version of the twosided laplace transform. They were introduced by ernest william barnes 1908, 1910. Mellin barnes integrals in the dimensional regularization parameter, or a parameter of the analytic regularization, is formulated in a modi. Asymptotics and mellinbarnes integrals encyclopedia. Kaminski, asymptotics and mellin barnes integrals, cambridge university press, cambridge, 2001. On convergent series representations of mellinbarnes integrals samuel friot yand david greynat z yinstitutdephysiquenucleairedorsay universiteparissud11,91405orsaycedex,france zdepartamentodefisicateorica universidaddezaragoza,clpedrocerbuna12,e50009zaragoza,spain. As an illustration, suppose that we are interested in the properties of a function fn as n becomes very large. Evaluating twoloop massive operator matrix elements with. Asymptotics and mellinbarnes integral pdf posted on january 26, 2019 by admin asymptotics and mellinbarnes integrals, first published in, provides an account of the use and properties of a type of complex integral representation that. We calculate massive 5propagator 2loop integrals for operator matrix elements in the lightcone expansion, using mellin barnes techniques and representations through generalized hypergeometric functions. Asymptotics of feynman diagrams and the mellinbarnes.
Mceliece the theory of information and coding 2 ed. Asymptotic expansion of the modified exponential integral. Nemes, the resurgence properties of the largeorder asymptotics of the hankel and bessel func. Asymptotics and mellin barnes integral pdf asymptotics and mellin barnes integrals, first published in, provides an account of the use and properties of a type of complex integral representation.
We extend the definition of this function using the twoparameter mittagleffler function. Mellinbarnes representations for multiloop feynman integrals with. Applications of the mellin barnes integral representation. Numerical evaluation of mellinbarnes integrals in minkowskian. Interesting features arise in many of the cases which call for the application of pullbacks of distributions. First, some integral representations of its fundamental solution including the mellin barnes integral representation are derived. In mathematical analysis, asymptotic analysis, also known as asymptotics, is a method of describing limiting behavior.
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