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Uniform bounds for the complementary incomplete gamma function, Preprint at http://locutus.cs.dal.ca:8088/archive/00000335
"... Abstract. We prove upper and lower bounds for the complementary incomplete gamma function Γ(a, z) with complex parameters a and z. Our bounds are refined within the circular hyperboloid of one sheet {(a, z) : z > ca − 1} with a real and z complex. Our results show that within the hyperboloid, ..."
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Abstract. We prove upper and lower bounds for the complementary incomplete gamma function Γ(a, z) with complex parameters a and z. Our bounds are refined within the circular hyperboloid of one sheet {(a, z) : z > ca − 1} with a real and z complex. Our results show that within the hyperboloid, Γ(a, z)  is of order z  a−1 e − Re(z) , and extends an upper estimate of Natalini and Palumbo to complex values of z.
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 exam ..."
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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.
Interval Computation of Gamma Probabilities and their Inverses
"... A new method for computing the gamma cumulative distribution functions and their inverses is presented in this paper. This method uses two continued fractions for computation, one for the incomplete gamma function and the other for the complement of the incomplete gamma function. An improved interva ..."
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A new method for computing the gamma cumulative distribution functions and their inverses is presented in this paper. This method uses two continued fractions for computation, one for the incomplete gamma function and the other for the complement of the incomplete gamma function. An improved interval method for computation and implemented it using C++ language classes is used. This is a selfvalidated computation. We developed programming techniques to speed up the increment in the iterative loops for finding the inverse of the gamma cumulative distribution function for a given probability. In fact, the inverses can be considered random gamma variates if a uniform random number generator is used to generate the probabilities over the interval [0, 1). The entire computation only involves two simple algebraic functions. There is no use of transcendental functions, auxiliary functions, power series, or Newton's method in the computation. Therefore, one can expect it is easy to implement. 1.
Uniform asymptotic analysis for waves in an incompressible elastic rod I. Disturbances superimposed on an initially stressfree state
, 1996
"... This paper studies the propagation of disturbances in an initially stressfree elastic circular rod. Starting from the MindlinHermann equations, we derive a fourthorder partial differential equation as the governing equation for small axialradial deformations in a rod composed of an incompressibl ..."
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This paper studies the propagation of disturbances in an initially stressfree elastic circular rod. Starting from the MindlinHermann equations, we derive a fourthorder partial differential equation as the governing equation for small axialradial deformations in a rod composed of an incompressible material. Then, we consider an initialvalue problem with an initial singularity in the shear strain. Using the technique of Fourier transforms, we manage to express the physical quantities in terms of integrals. The classical method of stationary phase is first used to obtain some important information. However, for material points in a neighbourhood behind the shearwave front, the phase function of the integrals has a stationary point which approaches positive infinity. Consequently, the classical method of stationary phase fails completely. Here, instead, we use a new method developed by us (Dai & Wong 1994 Wave Motion 19, 293308) to handle this case. An asymptotic expansion for the shear strain, which is uniformly valid in a neighbourhood behind the shearwave front, is derived. This uniform asymptotic expansion reveals that for the shear strain there is a transition from 0(1) disturbance to O(t~*) disturbance as the distance to the shearwave front increases. We also find that the shear strain has three other jumps in terms of asymptotic orders. The first jump is from a larger O(t~$) disturbance to a smaller O(t ~ ) disturbance behind and ahead of the barwave front. The second jump is from a larger 0(1) disturbance to a smaller O(t~l) disturbance behind and ahead of the barwave front. This also implies that in an asymptotic sense the initial singularity in the shear strain will be preserved and propagates with the shearwave speed as time progresses. The third jump is from a smaller O(f~') disturbance to a larger O(t~i) disturbance behind and ahead of a third wave front. 1.
unknown title
"... 1.1. Background. Lfunctions and modular forms underlie much of twentieth century number theory and are connected to the practical applications of number theory in cryptography. The fundamental importance of these functions in mathematics is supported by the fact that two of the seven Clay Mathemati ..."
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1.1. Background. Lfunctions and modular forms underlie much of twentieth century number theory and are connected to the practical applications of number theory in cryptography. The fundamental importance of these functions in mathematics is supported by the fact that two of the seven Clay Mathematics Million Dollar Millennium Problems [20] deal with properties of these functions, namely the
4 Examples of Point Processes Associated to Weights
, 2008
"... We give a closed form for the correlation functions of ensembles of a class of asymmetric real matrices in terms of the Pfaffian of an antisymmetric matrix formed from a 2×2 matrix kernel associated to the ensemble. We apply this result to the real Ginibre ensemble and compute the bulk and edge scal ..."
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We give a closed form for the correlation functions of ensembles of a class of asymmetric real matrices in terms of the Pfaffian of an antisymmetric matrix formed from a 2×2 matrix kernel associated to the ensemble. We apply this result to the real Ginibre ensemble and compute the bulk and edge scaling limits of its correlation functions as the size of the matrices becomes large.