The Experts below are selected from a list of 3735 Experts worldwide ranked by ideXlab platform
T M Dunster - One of the best experts on this subject based on the ideXlab platform.
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on the high order coefficients in the uniform asymptotic expansion for the Incomplete Gamma Function
Methods and applications of analysis, 1998Co-Authors: T M Dunster, Richard B. Paris, S CangAbstract:We examine the asymptotic nature as k → ∞ of the coefficientsck(�) appearing in the uniform asymptotic expansion of the Incomplete Gamma Function ( a,z) whereis a variable that depends on the ratio z/a. It is shown that this expansion diverges like the familiar "factorial divided by a power" dependence multiplied by a Function fk(�). For values ofnear the real axis, fk(�) is a slowly varying Function, but in the left half-plane, there are two lobes situated symmetrically about the negative realaxis in which fk(�) becomes large. The asymptotic expansion of fk(�) as k → ∞ is found to reveal a resurgence-type structure in which the high-order coefficients are related tothe low-order coeffi- cients. Numerical examples are given to illustrate the growth of the coefficients ck(�).
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asymptotic solutions of second order linear differential equations having almost coalescent turning points with an application to the Incomplete Gamma Function
Proceedings of The Royal Society A: Mathematical Physical and Engineering Sciences, 1996Co-Authors: T M DunsterAbstract:Uniform asymptotic expansions are derived for solutions of the differential equation d 2 W/dζ 2 = ( u 2 ζ 2 + βu + ψ( u , ζ))W, which are uniformly valid for u real and large, β bounded (real or complex), and £ lying in a well-defined bounded or unbounded complex domain, which contains the origin. The Function ψ(u, ζ) is assumed to be holomorphic in this domain, and is o ( u / ln( u )) uniformly as u —> oo. The approximations involve parabolic cylinder Functions, and include explicit and realistic error bounds. The new theory is then applied to the complementary Incomplete Gamma Function Γ( α , α x ), furnishing an asymptotic approximation which is uniformly valid for x lying in a complex domain which properly contains all x satisfying |arg(x)| oo.
Gergő Nemes - One of the best experts on this subject based on the ideXlab platform.
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Asymptotic expansions for the Incomplete Gamma Function in the transition regions
Mathematics of Computation, 2018Co-Authors: Gergő Nemes, Adri B. Olde DaalhuisAbstract:We construct, for the first time, asymptotic expansions for the normalised Incomplete Gamma Function $Q(a,z)=\Gamma(a,z)/\Gamma(a)$ that are valid in the transition regions, including the case $z\approx a$, and have simple polynomial coefficients. For Bessel Functions, these type of expansions are well known, but for the Incomplete Gamma Function they were missing from the literature. A detailed historical overview is included. We also derive an asymptotic expansion for the corresponding inverse problem, which has importance in probability theory and mathematical statistics. The coefficients in this expansion are again simple polynomials, and therefore its implementation is straightforward. As a byproduct, we give the first complete asymptotic expansion as $a\to-\infty$ of the unique negative zero of the regularised Incomplete Gamma Function $\Gamma^*(a,x)$.
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The resurgence properties of the Incomplete Gamma Function, I
Analysis and Applications, 2016Co-Authors: Gergő NemesAbstract:In this paper, we derive new representations for the Incomplete Gamma Function, exploiting the reformulation of the method of steepest descents by C. J. Howls [Hyperasymptotics for integrals with finite endpoints, Proc. Roy. Soc. London Ser. A 439 (1992) 373–396]. Using these representations, we obtain a number of properties of the asymptotic expansions of the Incomplete Gamma Function with large arguments, including explicit and realistic error bounds, asymptotics for the late coefficients, exponentially improved asymptotic expansions, and the smooth transition of the Stokes discontinuities.
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The Resurgence Properties of the Incomplete Gamma Function II
Studies in Applied Mathematics, 2015Co-Authors: Gergő NemesAbstract:In this paper, we derive a new representation for the Incomplete Gamma Function, exploiting the reformulation of the method of steepest descents by Howls 1992. Using this representation, we obtain numerically computable bounds for the remainder term of the asymptotic expansion of the Incomplete Gamma Function Γ−a,λa with large a and fixed positive λ, and an asymptotic expansion for its late coefficients. We also give a rigorous proof of Dingle's formal result regarding the exponentially improved version of the asymptotic series of Γ−a,λa.
Richard B. Paris - One of the best experts on this subject based on the ideXlab platform.
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The asymptotic expansion of a generalised Incomplete Gamma Function
Journal of Computational and Applied Mathematics, 2003Co-Authors: Richard B. ParisAbstract:We define a generalised Incomplete Gamma Function Lp(a,z), which coincides with the familiar normalised Function Q(a,z) = Γ(a,z)/Γ(a) when the integer parameter p = 1. It is shown that the large-z asymptotics of Lp(a,z) in the sector |argz| 0, all these expansions are recessive at infinity and form a sequence of increasingly subdominant exponential contributions.
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A uniform asymptotic expansion for the Incomplete Gamma Function
Journal of Computational and Applied Mathematics, 2002Co-Authors: Richard B. ParisAbstract:We describe a new uniform asymptotic expansion for the Incomplete Gamma Function Γ(a,z) valid for large values of z. This expansion contains a complementary error Function of an argument measuring transition across the point z = a (which is different from that in the well-known uniform expansion for large a of Temme), with easily computable coefficients that do not involve a removable singularity at z = a. Our expansion is, however, valid in a smaller domain of the parameters than that of Temme. Numerical examples are given to illustrate the accuracy of the expansion.
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Error bounds for the uniform asymptotic expansion of the Incomplete Gamma Function
Journal of Computational and Applied Mathematics, 2002Co-Authors: Richard B. ParisAbstract:We ∉ derive simple, explicit error bounds for the uniform asymptotic expansion of the Incomplete Gamma Function Γ(a,z) valid for complex values of a and z as |a| → ∞. Their evaluation depends on numerically pre-computed bounds for the coefficients ck(η) in the expansion of Γ(a,z) taken along rays in the complex η plane, where η is a variable related to z/a. The bounds are compared with numerical computations of the remainder in the truncated expansion.
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on the high order coefficients in the uniform asymptotic expansion for the Incomplete Gamma Function
Methods and applications of analysis, 1998Co-Authors: T M Dunster, Richard B. Paris, S CangAbstract:We examine the asymptotic nature as k → ∞ of the coefficientsck(�) appearing in the uniform asymptotic expansion of the Incomplete Gamma Function ( a,z) whereis a variable that depends on the ratio z/a. It is shown that this expansion diverges like the familiar "factorial divided by a power" dependence multiplied by a Function fk(�). For values ofnear the real axis, fk(�) is a slowly varying Function, but in the left half-plane, there are two lobes situated symmetrically about the negative realaxis in which fk(�) becomes large. The asymptotic expansion of fk(�) as k → ∞ is found to reveal a resurgence-type structure in which the high-order coefficients are related tothe low-order coeffi- cients. Numerical examples are given to illustrate the growth of the coefficients ck(�).
Emin Özçag - One of the best experts on this subject based on the ideXlab platform.
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Defining Incomplete Gamma Type Function with Negative Arguments and PolyGamma Functions $\psi^{(n)}(-m)$
arXiv: Classical Analysis and ODEs, 2014Co-Authors: Emin ÖzçagAbstract:In this paper the Incomplete Gamma Function $\Gamma(\alpha,x)$ and its derivative is considered for negative values of $\alpha $ and the Incomplete Gamma type Function $\Gamma_*(\alpha,x_-)$ is introduced. Further the polyGamma Functions $\psi^{(n)}(x)$ are defined for negative integers via the neutrix setting.
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defining Incomplete Gamma type Function with negative arguments and polyGamma Functions psi n m
arXiv: Classical Analysis and ODEs, 2014Co-Authors: Emin ÖzçagAbstract:In this paper the Incomplete Gamma Function $\Gamma(\alpha,x)$ and its derivative is considered for negative values of $\alpha $ and the Incomplete Gamma type Function $\Gamma_*(\alpha,x_-)$ is introduced. Further the polyGamma Functions $\psi^{(n)}(x)$ are defined for negative integers via the neutrix setting.
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Some remarks on the Incomplete Gamma Function
Mathematical Methods in Engineering, 1Co-Authors: Emin Özçag, İnci Ege, Haşmet Gürçay, Biljana Jolevska-tuneskaAbstract:The Incomplete Gamma Function γ(α, x) is defined for α > 0 and x ≥ 0 by $$ \Gamma \left( {\alpha ,x} \right) = \int_0^x {u^{\alpha - 1} e^{ - u} du} $$ and by using the recurrence formula $$ \Gamma \left( {\alpha + 1,x} \right) = \alpha \Gamma \left( {\alpha ,x} \right) - x^\alpha e^{ - x} $$ the definition of γ(α, x) can be extended to negative, non integer value of α. Recently Fisher et al. [FJK03] defined γ(−m, x) for m = 0, 1, 2, . . . . In this paper we consider the derivatives of the Incomplete Gamma Function γ(α, x) and the derivatives of locally summable Function γ(α, x +) = H(x)γ(α, x) for negative integers, where H(x) denotes the Heaviside Function.
Jonathan M. Borwein - One of the best experts on this subject based on the ideXlab platform.
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Crandall's computation of the Incomplete Gamma Function and the Hurwitz zeta Function, with applications to Dirichlet L-series
Applied Mathematics and Computation, 2015Co-Authors: David H. Bailey, Jonathan M. BorweinAbstract:This paper extends tools developed by Crandall (2012) 16 to provide robust, high-precision methods for computation of the Incomplete Gamma Function and the Lerch transcendent. We then apply these to the corresponding computation of the Hurwitz zeta Function and so of Dirichlet L-series and character polylogarithms.
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Uniform Bounds for the Complementary Incomplete Gamma Function
Mathematical Inequalities & Applications, 2009Co-Authors: Jonathan M. Borwein, O-yeat ChanAbstract:We prove upper and lower bounds for the complementary Incomplete Gamma Function $\G(a,z)$ with complex parameters $a$ and $z$. Our bounds are refined within the circular hyperboloid of one sheet $\{(a,z):|z|>c|a-1|\}$ with $a$ real and $z$ complex. Our results show that within the hyperboloid, $|\G(a,z)|$ is of order $|z|^{a-1}e^{-\Re(z)}$, and extends an upper estimate of Natalini and Palumbo to complex values of $z$.
