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16
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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Cited by 15 (1 self)
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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.
Time development of exponentially small nonadiabatic transitions
 Commun. Math. Phys
, 2004
"... Optimal truncations of asymptotic expansions are known to yield approximations to adiabatic quantum evolutions that are accurate up to exponentially small errors. In this paper, we rigorously determine the leading order non–adiabatic corrections to these approximations for a particular family of two ..."
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Cited by 11 (6 self)
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Optimal truncations of asymptotic expansions are known to yield approximations to adiabatic quantum evolutions that are accurate up to exponentially small errors. In this paper, we rigorously determine the leading order non–adiabatic corrections to these approximations for a particular family of two–level analytic Hamiltonian functions. Our results capture the time development of the exponentially small transition that takes place between optimal states by means of a particular switching function. Our results confirm the physics predictions of Sir Michael Berry in the sense that the switching function for this family of Hamiltonians has the form that he argues is universal.
Aspects of the ODE/IM correspondence
 CONTRIBUTION TO THE PROCEEDINGS “RECENT TRENDS IN EXPONENTIAL ASYMPTOTICS”, JUNE 28 JULY 2 (2004), RIMS, KYOTO
, 2004
"... We review a surprising correspondence between certain twodimensional integrable models and the spectral theory of ordinary differential equations. Particular emphasis is given to the relevance of this correspondence to certain problems in PTsymmetric quantum mechanics. ..."
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Cited by 11 (1 self)
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We review a surprising correspondence between certain twodimensional integrable models and the spectral theory of ordinary differential equations. Particular emphasis is given to the relevance of this correspondence to certain problems in PTsymmetric quantum mechanics.
Optimal uniform estimates and rigorous asymptotics beyond all orders for a class of ordinary differential equations
 Proceedings of the Royal Society. London. Series A
, 1996
"... For first order differential equations of the form y ′ = ∑ P p=0 Fp(x)y p and second order homogeneous linear differential equations y ′ ′ +a(x)y ′ +b(x)y = 0 with locally integrable coefficients having asymptotic (possibly divergent) power series when x  → ∞ on a ray arg(x) =const, under some ..."
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Cited by 7 (2 self)
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For first order differential equations of the form y ′ = ∑ P p=0 Fp(x)y p and second order homogeneous linear differential equations y ′ ′ +a(x)y ′ +b(x)y = 0 with locally integrable coefficients having asymptotic (possibly divergent) power series when x  → ∞ on a ray arg(x) =const, under some further assumptions, it is shown that, on the given ray, there is a onetoone correspondence between true solutions and (complete) formal solutions. The correspondence is based on asymptotic inequalities which are required to be uniform in x and optimal with respect to certain weights. 1 Introduction and Main Results The main purpose of the present paper is to give, in terms of uniform asymptotic estimates, a precise meaning to complete asymptotic expansions (e.g., as power series followed by exponentially small terms) of solutions of a class of differential equations in a neighborhood of an irregular singular point (chosen to be infinity).
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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Cited by 5 (1 self)
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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
The black hole singularity in AdS/CFT
"... We explore physics behind the horizon in eternal AdS Schwarzschild black holes. In dimension d> 3, where the curvature grows large near the singularity, we find distinct but subtle signals of this singularity in the boundary CFT correlators. Building on previous work, we study correlation functions ..."
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Cited by 5 (0 self)
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We explore physics behind the horizon in eternal AdS Schwarzschild black holes. In dimension d> 3, where the curvature grows large near the singularity, we find distinct but subtle signals of this singularity in the boundary CFT correlators. Building on previous work, we study correlation functions of operators on the two disjoint asymptotic boundaries of the spacetime by investigating the spacelike geodesics that join the boundaries. These dominate the correlators for large mass bulk fields. We show that the Penrose diagram for d> 3 is not square. As a result, the real geodesic connecting the two boundary points becomes almost null and bounces off the singularity at a finite boundary time tc ̸ = 0. If this geodesic were to dominate the correlator there would be a “light cone ” singularity at tc. However, general properties of the boundary theory rule this out. In fact, we argue that the correlator is actually dominated by a complexified geodesic, whose properties yield the large mass quasinormal mode frequencies previously found for this black hole. We find a branch cut in the correlator at small time (in the limit of large mass), arising from coincidence of three geodesics. The tc singularity, a signal of the black hole singularity, occurs on a secondary sheet of the analytically continued correlator. Its properties are computationally accessible. The tc singularity persists to all orders in the 1/m expansion, for finite α ′ , and to all orders in gs. Certain leading nonperturbative effects can also be studied. The behavior of these boundary theory quantities near tc gives, in principle, significant information about stringy and quantum behavior in the vicinity of the black hole singularity. June
Stokes phenomenon and matched asymptotic expansions
 SIAM J. Appl. Math
, 1995
"... Abstract. This paper describes the use of matched asymptotic expansions to illuminate the description of functions exhibiting Stokes phenomenon. In particular the approach highlights the way in which the local structure and the possibility of finding Stokes multipliers explicitly depend on the behav ..."
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Cited by 4 (3 self)
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Abstract. This paper describes the use of matched asymptotic expansions to illuminate the description of functions exhibiting Stokes phenomenon. In particular the approach highlights the way in which the local structure and the possibility of finding Stokes multipliers explicitly depend on the behaviour of the coefficients of the relevant asymptotic expansions. tion Key words. Stokes ’ phenomenon, matched asymptotic expansions, Airy function, error funcAMS subject classifications. 41A60, 33C10, 33B20 1. Introduction. The
Movable singularities of solutions of difference equations in relation to solvability, and study of a superstable fixed point Theoretical and
 Mathematical Physics
, 2002
"... Abstract. We overview applications exponential asymptotics and analyzable function theory to difference equations, in defining an analog of the Painlevé property for them and we sketch the conclusions with respect to the solvability properties of first order autonomous ones. It turns out that if the ..."
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Cited by 2 (1 self)
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Abstract. We overview applications exponential asymptotics and analyzable function theory to difference equations, in defining an analog of the Painlevé property for them and we sketch the conclusions with respect to the solvability properties of first order autonomous ones. It turns out that if the Painlevé property is present the equations are explicitly solvable and in the contrary case, under further assumptions, the integrals of motion develop singularity barriers. We apply the method to the logistic map xn+1 = axn(1 − xn) where it turns out that the only cases with the Painlevé property are a = −2, 0, 2 and 4 for which explicit solutions indeed exist; in the opposite case an associated conjugation map develops singularity barriers. 1.
Gravity Waves in a Horizontal Shear Flow. Part II: Interaction between Gravity Waves and Potential Vorticity Perturbations
, 2007
"... Interaction among potential vorticity perturbations and propagating internal gravity waves in a horizontally sheared zonal flow is investigated. In the strong stratification limit, an initial vorticity perturbation weakly excites two propagating gravity waves while the density component of the poten ..."
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Cited by 2 (1 self)
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Interaction among potential vorticity perturbations and propagating internal gravity waves in a horizontally sheared zonal flow is investigated. In the strong stratification limit, an initial vorticity perturbation weakly excites two propagating gravity waves while the density component of the potential vorticity perturbation is significantly amplified, potentially leading to convective collapse. If stratification is sufficiently weak, a strong coupling between vorticity perturbations and gravity waves is found and spontaneous gravity wave generation occurs. This coupling can be traced to the nonnormal interaction between the potential vorticity and gravity wave manifolds in the weak stratification limit. Vorticity perturbations amplify in energy due to downgradient Reynolds stress when their phase lines tilt against the shear and the large growth attained is transferred to propagating gravity waves. When the flow geometry is such that the excited gravity waves are confined in the vicinity of the vorticity perturbation by their trapping levels, an overall convective collapse of this region can be anticipated. On the other hand, when the flow geometry permits wave propagation, significant gravity wave emission occurs. 1.
INSTITUTE OF PHYSICS PUBLISHING JOURNAL OF OPTICS A: PURE AND APPLIED OPTICS
, 2006
"... refraction ..."