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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.
On optimal truncation of divergent series solutions of nonlinear differential systems; Berry smoothing.
 Proc. Roy. Soc. London A 452
"... We prove that for divergent series solutions of nonlinear (or linear) differential systems near a generic irregular singularity, the common prescription of summation to the least term is, if properly interpreted, meaningful and correct, and we extend this method to transseries solutions. In every di ..."
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Cited by 11 (5 self)
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We prove that for divergent series solutions of nonlinear (or linear) differential systems near a generic irregular singularity, the common prescription of summation to the least term is, if properly interpreted, meaningful and correct, and we extend this method to transseries solutions. In every direction in the complex plane at the singularity (Stokes directions not excepted) there exists a nonempty set of solutions whose difference from the "optimally" (i.e., near the least term) truncated asymptotic series is of the same (exponentially small) order of magnitude as the least term of the series. There is a family of generalized Borel summation formulas B which commute with the usual algebraic and analytic operations (addition, multiplication, differentiation, etc). We show that there is exactly one of them, B0 , such that for any formal series solution ~ f, B0 ( ~ f) differs from the optimal truncation of ~ f by at most the order of the least term of ~ f . We show in addition that the Berry (1989) smoothing phenomenon is universal within this class of differential systems. Whenever the terms "beyond all orders" change in crossing a Stokes line, these terms vary smoothly on the Berry scale arg(x) jxj \Gamma1=2 and the transition is always given by the error function; under the same conditions we show that Dingle's rule of signs for Stokes transitions holds. 1
The DLMF Project: A New Initiative in Classical Special Functions
 International Workshop on Special Functions  Asymptotics, Harmonic Analysis and Mathematical Physics. Hong Kong
, 2000
"... that aims to produce a successor to Abramowitz and Stegun’s Handbook of Mathematical Functions, published by the National Bureau of Standards in 1964 and reprinted by Dover in 1965. Both editions continue to sell briskly and are widely cited in the scientific literature. However, with the many advan ..."
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that aims to produce a successor to Abramowitz and Stegun’s Handbook of Mathematical Functions, published by the National Bureau of Standards in 1964 and reprinted by Dover in 1965. Both editions continue to sell briskly and are widely cited in the scientific literature. However, with the many advances in the theory, computation and application of special functions that have occurred since 1960, a new standard reference is badly needed. NIST intends to satisfy this need by providing a Digital Library of Mathematical Functions (DLMF) as a free Web site together with an associated book and CDROM. The Web site will provide many capabilities that are impossible to provide in print media alone. 1
On the theory of Complex Rays
 SIAM Review
, 1997
"... The article surveys the application of complex ray theory to the scalar Helmholtz equation in two dimensions. The first objective is to motivate a framework within which complex rays may be used to make predictions about wavefields in a wide variety of geometrical configurations. A crucial ingredien ..."
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The article surveys the application of complex ray theory to the scalar Helmholtz equation in two dimensions. The first objective is to motivate a framework within which complex rays may be used to make predictions about wavefields in a wide variety of geometrical configurations. A crucial ingredient in this framework is the role played by Stokes' phenomenon in determining the regions of existence of complex rays. The identification of the Stokes surfaces emerges as a key step in the approximation procedure, and this leads to the consideration of the many characterisations of Stokes surfaces, including the adaptation and application of recent developments in exponential asymptotics to the complex wkb expansion of these wavefields. Examples will be given for several cases of physical importance. 1 Introduction The objective of this paper is to lay down a systematic complexified theory of monochromatic highfrequency wave propagation. The computational and analytical usefulness of such ...