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The Incomplete Gamma Functions Since Tricomi
 In Tricomi's Ideas and Contemporary Applied Mathematics, Atti dei Convegni Lincei, n. 147, Accademia Nazionale dei Lincei
, 1998
"... The theory of the incomplete gamma functions, as part of the theory of conuent hypergeometric functions, has received its rst systematic exposition by Tricomi in the early 1950s. His own contributions, as well as further advances made thereafter, are surveyed here with particular emphasis on asy ..."
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The theory of the incomplete gamma functions, as part of the theory of conuent hypergeometric functions, has received its rst systematic exposition by Tricomi in the early 1950s. His own contributions, as well as further advances made thereafter, are surveyed here with particular emphasis on asymptotic expansions, zeros, inequalities, computational methods, and applications.
Uniform Asymptotics for the Incomplete Gamma Functions Starting From Negative Values of the Parameters
"... We consider the asymptotic behavior of the incomplete gamma functions fl(\Gammaa; \Gammaz) and \Gamma(\Gammaa; \Gammaz) as a !1. Uniform expansions are needed to describe the transition area z a, in which case error functions are used as main approximants. We use integral representations of the i ..."
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Cited by 5 (1 self)
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We consider the asymptotic behavior of the incomplete gamma functions fl(\Gammaa; \Gammaz) and \Gamma(\Gammaa; \Gammaz) as a !1. Uniform expansions are needed to describe the transition area z a, in which case error functions are used as main approximants. We use integral representations of the incomplete gamma functions and derive a uniform expansion by applying techniques used for the existing uniform expansions for fl(a; z) and \Gamma(a; z). The result is compared with Olver's uniform expansion for the generalized exponential integral. A numerical verification of the expansion is given in a final section.
Recent Problems from Uniform Asymptotic Analysis of Integrals In Particular in Connection with Tricomi's $Psi$function
, 1998
"... The paper discusses asymptotic methods for integrals, in particular uniform approximations. We discuss several examples, where we apply the results to Tricomi's \Psi\Gammafunction, in particular we consider an expansion of TricomiCarlitz polynomials in terms of Hermite polynomials. We mention othe ..."
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The paper discusses asymptotic methods for integrals, in particular uniform approximations. We discuss several examples, where we apply the results to Tricomi's \Psi\Gammafunction, in particular we consider an expansion of TricomiCarlitz polynomials in terms of Hermite polynomials. We mention other recent expansions for orthogonal polynomials that do not satisfy a differential equation, and for which methods based on integral representations produce powerful uniform asymptotic expansions. 1991 Mathematics Subject Classification: 41A60, 33B20, 33C10, 33C45, 11B73, 30E15. Keywords and Phrases: uniform asymptotic expansions, Tricomi's \Psi\Gammafunction, Kummer functions, confluent hypergeometric functions, Whittaker functions, Hermite polynomials, TricomiCarlitz polynomials. Note: Work carried out under project MAS2.8 Exploratory research. Extended version of a paper presented at the Conference Tricomi's Ideas and Contemporary Applied Mathematics to celebrate the 100th anniversary o...
Uniform Asymptotic Expansions of Integrals: A Selection of Problems
, 1995
"... On the occasion of the conference we mention examples of Stieltjes' work on asymptotics of special functions. The remaining part of the paper gives a selection of asymptotic methods for integrals, in particular on uniform approximations. We discuss several "standard" problems and examples, in which ..."
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On the occasion of the conference we mention examples of Stieltjes' work on asymptotics of special functions. The remaining part of the paper gives a selection of asymptotic methods for integrals, in particular on uniform approximations. We discuss several "standard" problems and examples, in which known special functions (error functions, Airy functions, Bessel functions, etc.) are needed to construct uniform approximations. Finally, we discuss the recent interest and new insights in the Stokes phenomenon. An extensive bibliography on uniform asymptotic methods for integrals is given, together with references to recent papers on the Stokes phenomenon for integrals and related topics.
Symmetry, Integrability and Geometry: Methods and Applications NonPerturbative Asymptotic Improvement of Perturbation Theory and Mellin–Barnes Representation
"... Abstract. Using a method mixing Mellin–Barnes representation and Borel resummation we show how to obtain hyperasymptotic expansions from the (divergent) formal power series which follow from the perturbative evaluation of arbitrary “Npoint ” functions for the simple case of zerodimensional φ 4 fie ..."
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Abstract. Using a method mixing Mellin–Barnes representation and Borel resummation we show how to obtain hyperasymptotic expansions from the (divergent) formal power series which follow from the perturbative evaluation of arbitrary “Npoint ” functions for the simple case of zerodimensional φ 4 field theory. This hyperasymptotic improvement appears from an iterative procedure, based on inverse factorial expansions, and gives birth to interwoven nonperturbative partial sums whose coefficients are related to the perturbative ones by an interesting resurgence phenomenon. It is a nonperturbative improvement in the sense that, for some optimal truncations of the partial sums, the remainder at a given hyperasymptotic level is exponentially suppressed compared to the remainder at the preceding hyperasymptotic level. The Mellin–Barnes representation allows our results to be automatically valid for a wide range of the phase of the complex coupling constant, including Stokes lines. A numerical analysis is performed to emphasize the improved accuracy that this method allows to reach compared to the usual perturbative approach, and the importance of hyperasymptotic optimal truncation schemes. Key words: exactly and quasiexactly solvable models; Mellin–Barnes representation; hyperasymptotics; resurgence; nonperturbative effects; field theories in lower dimensions 2010 Mathematics Subject Classification: 41A60; 30E15 1