The integration cycle here should be a lagrangian section of the stromingeryauzaslow bration 53 i. He also obtained a mellin barnes integral representation for 1t h ef a ct or4 is ndu x p l v. Methods of generating such expansions include the eulermaclaurin summation formula and integral transforms such as the laplace and mellin transforms. Applications of the mellin barnes integral representation. Barnes integral representation the mellin transform of a function is given by fs. At present the analytic evaluation for most of these integrals is out of reach and purely. Kaminski, asymptotics and mellin barnes integrals, cambridge university press, cambridge, 2001. Feynman integrals may be represented by the mathematica packages ambre and mb as multiple mellinbarnes integrals. Paris, asymptotics of the gauss hypergeometric function with large parameters, ii,j. Asymptoticsandmellinbarnesintegralsisacomprehensiveaccountoftheproper tiesofmellinbarnesintegralsandtheirapplicationtoproblemsinvolvingspecial functions. On convergent series representations of mellinbarnes integrals samuel friot yand david greynat z yinstitutdephysiquenucleairedorsay universiteparissud11,91405orsaycedex,france zdepartamentodefisicateorica universidaddezaragoza,clpedrocerbuna12,e50009zaragoza,spain. Mellinbarnes representations for multiloop feynman integrals with. 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. 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.
Asymptotics and mellinbarnes integrals encyclopedia. A unified approach to the integrals of mellinbarneshecke. We calculate massive 5propagator 2loop integrals for operator matrix elements in the lightcone expansion, using mellin barnes techniques and representations through generalized hypergeometric functions. Nemes, the resurgence properties of the largeorder asymptotics of the hankel and bessel func. Asymptotics and mellin barnes integrals encyclopedia of mathematics and its applications 85 by r. In mathematics, a barnes integral or mellinbarnes integral is a contour integral involving a product of gamma functions. 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.
Interesting features arise in many of the cases which call for the application of pullbacks of distributions. Mceliece the theory of information and coding, 2nd edn 87 b. Mellin barnes integrals in the dimensional regularization parameter, or a parameter of the analytic regularization, is formulated in a modi. 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. After developing the properties of these integrals. We extend the definition of this function using the twoparameter mittagleffler function. Asymptotic expansions of feynman diagrams and the mellin. Asymptotic expansion of the modified exponential integral.
Feynman integrals mellinbarnes representations sums desy. Volume 96 basic hypergeometric series second edition. As an illustration, suppose that we are interested in the properties of a function fn as n becomes very large. Further, making use of the lemma 4 and interpreting the resulting mellin barnes multiple integrals contour as. Repeated integration by parts will often lead to an asymptotic expansion. 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 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.
They are closely related to generalized hypergeometric series. Finally, the numerical integration of multidimensional mb integrals in minkowskian regions is carried out. The sum of the residues converges to an expression containing. Kaminski, asymptotics and mellin barnes integrals, cambridge university press, 2001. Kaminski asymptotics and mellin barnes integrals 86 r. It is shown that the integral representation of feynman diagrams in terms of the traditional feynman parameters, when combined with properties of the mellin barnes representation and the socalled converse mapping theorem, provide a very simple and efficient way to obtain the analytic asymptotic behaviours in both the large and small ratios of mass scales. Nuclear physics institute of moscow state university abstract one of the two existing strategies of resolving singularities of multifold mellin barnes integrals in the dimensional regularization parameter, or a parameter of the analytic regularization, is formulated in a modi. We apply the mellinbarnes integral representation to. The corresponding algorithm is implemented as a mathematica code mbresolve. We consider the asymptotic expansion of the singleparameter mittagleffler function ea. First, some integral representations of its fundamental solution including the mellin barnes integral representation are derived. 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. This integral transform is closely connected to the theory of dirichlet series, and is often.
Asymptotics of feynman diagrams and the mellinbarnes. In the last section, we will introduce mellin barnes integral representations for the. The fundamental solution is interpreted as a probability density function and its entropy in the shannon sense is calculated. As a result, the mellinbarnes integral representations of the twoloop vertex feynman. Numerical evaluation of mellinbarnes integrals in minkowskian. 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. In mathematical analysis, asymptotic analysis, also known as asymptotics, is a method of describing limiting behavior. This says that the asymptotics of the mirror oscillatory integral is described in terms of the bclass of x. We apply the mellinbarnes integral representation to several situations of interest in mathematicalphysics. The mellin barnes mb integrals are widely used in highenergy physics. In this paper, we provide a unified approach to a family of integrals of mellin barnes type using distribution theory and fourier transforms. See also other reference books cited in 2, 21, and 24.
On convergent series representations of mellinbarnes. 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. Highprecision numerical estimates of the mellinbarnes. Asymptotics and mellin barnes integrals, first published in, provides an account of the use and properties of a type of complex integral representation that. 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.
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. To address this problem we introduce and discuss in detail a. 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. Asymptotics of the gauss hypergeometric function with. 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.
Mbsumsa mathematica package for the representation of mellin. They were introduced by ernest william barnes 1908, 1910. Applications of the mellinbarnes integral representation. We consider the asymptotic expansion of the generalised exponential integral involving the mittagleffler function introduced recently by mainardi and masina fract.
Fractional integrals and derivatives, theory and applications. The second kummer function with matrix parameters and its. First, we define integral over a curve depending on and apply the residue theorem. Mceliece the theory of information and coding 2 ed. Introduction to mellinbarnes representations for feynman integrals tord riemann, desy, zeuthen calc, july 2006, dubna introduction. 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. Analogously to the case with realvalued parameters see 21, p. Since a convergent taylor series fits the definition of asymptotic expansion as well. The mellin barnes integrals are strongly related with the mellin transform, in particular with its inverse transformation, in the framework of the socalled melling setting. Paris the asymptotics of a generalised beta function mathematica aeterna, vol. Encyclopedia of mathematics and its applications pdf. Tables of mellin transforms download ebook pdf, epub. On the contrary, many asymptotic methods have been designed.
The expansions of the similarly extended sine and cosine integrals are also. Here we concentrate on the form of this exponentially small. Fractional integral formulae involving the srivastava. In mathematics, the mellin transform is an integral transform that may be regarded as the multiplicative version of the twosided laplace transform. The hankel integral, arises naturally in many fields of application in physics 15. 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. 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. Evaluating twoloop massive operator matrix elements with.
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