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Tables of Mellin Transforms
29 Nov 1974-
TL;DR: Inverse Mellin Transform Analysis as discussed by the authors, the Gamma Function and Gamma Function are used to transform the Mellin transform into a generalized function, which is then used to obtain a generalized version of Bessel Function.
Abstract: I. Mellin Transforms.- Some Applications of the Mellin Transform Analysis.- 1.1 General Formulas.- 1.2 Algebraic Functions and Powers of Arbitrary Order.- 1.3 Exponential Functions.- 1.4 Logarithmic Functions.- 1.5 Trigonometric Functions.- 1.6 Hyperbolic Functions.- 1.7 The Gamma Function and Related Functions.- 1.8 Legendre Functions.- 1.9 Orthogonal Polynomials.- 1.10 Bessel Functions.- 1.11 Modified Bessel Function.- 1.12 Functions Related to Bessel Function.- 1.13 Whittaker Functions and Special Cases.- 1.14 Elliptic Integrals and Elliptic Functions.- 1.15 Hyper geometric Functions.- II. Inverse Mellin Transforms.- 2.1 General Formulas.- 2.2 Algebraic Functions and Powers of Arbitrary Order.- 2.3 Exponential and Logarithmic Functions.- 2.4 Trigonometric and Hyperbolic Functions.- 2.5 The Gamma Function and Related Functions.- 2.6 Orthogonal Polynomials and Legendre Functions.- 2.7 Bessel Functions and Related Functions.- 2.8 Whittaker Functions and Special Cases.
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01 Jan 2010
TL;DR: In this article, a cubic Hermite collocation scheme for the solution of the coupled integro-partial differential equations governing the propagation of a hydraulic fracture in a state of plane strain is described.
Abstract: article i nfo We describe a novel cubic Hermite collocation scheme for the solution of the coupled integro-partial differential equations governing the propagation of a hydraulic fracture in a state of plane strain. Special blended cubic Hermite-power-law basis functions, with arbitrary index 0b αb1, are developed to treat the singular behavior of the solution that typically occurs at the tips of a hydraulic fracture. The implementation of blended infinite elements to model semi-infinite crack problems is also described. Explicit formulae for the integrated kernels associated with the cubic Hermite and blended basis functions are provided. The cubic Hermite collocation algorithm is used to solve a number of different test problems with two distinct propagation regimes and the results are shown to converge to published similarity and asymptotic solutions. The convergence rate of the cubic Hermite scheme is determined by the order of accuracy of the tip asymptotic expansion as well as the O(h 4 ) error due to the Hermite cubic interpolation. The errors due to these two approximations need to be matched in order to achieve optimal convergence. Backward Euler time-stepping yields a robust algorithm that, along with geometric increments in the time-step, can be used to explore the transition between propagation regimes over many orders of magnitude in time.
923 citations
01 Jan 2014
TL;DR: In this paper, Dzherbashian [Dzh60] defined a function with positive α 1 > 0, α 2 > 0 and real α 1, β 2, β 3, β 4, β 5, β 6, β 7, β 8, β 9, β 10, β 11, β 12, β 13, β 14, β 15, β 16, β 17, β 18, β 20, β 21, β 22, β 24
Abstract: Consider the function defined for \(\alpha _{1},\ \alpha _{2} \in \mathbb{R}\) (α 1 2 +α 2 2 ≠ 0) and \(\beta _{1},\beta _{2} \in \mathbb{C}\) by the series
$$\displaystyle{ E_{\alpha _{1},\beta _{1};\alpha _{2},\beta _{2}}(z) \equiv \sum _{k=0}^{\infty } \frac{z^{k}} {\varGamma (\alpha _{1}k +\beta _{1})\varGamma (\alpha _{2}k +\beta _{2})}\ \ (z \in \mathbb{C}). }$$
(6.1.1)
Such a function with positive α 1 > 0, α 2 > 0 and real \(\beta _{1},\beta _{2} \in \mathbb{R}\) was introduced by Dzherbashian [Dzh60].
919 citations
23 Feb 2000
TL;DR: Poularikas et al. as discussed by the authors proposed the Hartley Transform, Kraig J. Olejniczak, University of Arkansas, Fayetteville Laplace Transforms, Samuel Seely (deceased), Westbrook, Connecticut The Z-Transform, Alexander D. Poularis et al., University of Alabama in Huntsville Hilbert Transform, Stefan L. Hahn, Warsaw University of Technology Radon and Abel Transforms.
Abstract: Signals and Systems, Alexander D. Poularikas, University of Alabama in Huntsville Fourier Transforms, Kenneth B. Howell, University of Alabama in Huntsville Sine and Cosine Transforms, Pat Yip, McMaster University, Ontario The Hartley Transform, Kraig J. Olejniczak, University of Arkansas, Fayetteville Laplace Transforms, Samuel Seely (deceased), Westbrook, Connecticut The Z-Transform, Alexander D. Poularikas, University of Alabama in Huntsville Hilbert Transforms, Stefan L. Hahn, Warsaw University of Technology Radon and Abel Transforms, Stanley Deans, University of South Florida The Hankel Transform, Robert Piessens , Katholieke Universieit Leuven, Belgium Wavelet Transform, Yunlong Sheng, Laval University, Quebec The Mellin Transform, Jacqueline Bertrand, Pierre Bertrand, Universite de Paris VII, and Jean-Philippe Ovarlez, ONERA/DES, France Mixed Time-Frequency Signal Transformations, G. Faye Boudreaux-Bartels, University of Rhode Island Fractional Fourier Transforms, Dr. Mustafa Abushagur, University of Alabama in Huntsvill, Dr. Ahmed M. Almanasrah, Photronix, Malaysia The Lapped Transforms, Ricardo L. de Queiroz, Xerox Corporation The Discrete Time and the Discrete Transforms, Alexander D. Poularikas, University of Alabama in Huntsville Discrete Time and the Discrete Transforms, Alexander D. Poularikas, University of Alabama in Huntsville Appendices, Alexander D. Poularikas, University of Alabama in Huntsville Index
776 citations
TL;DR: In this paper, a detailed account of relativistic quantum field theory in the grand canonical ensemble is given, where three approaches are discussed: traditional Euclidean Matsubara, and two recently developed real-time methods, namely, Minkowskian time-path and thermo field dynamics.
765 citations
TL;DR: Applying the idea of self-similar dynamics, a fractal scaling model is derived that results in an equation in which the time derivative is replaced by a differentiation (d/dt)beta of non-integer order beta ofNon- integer order beta.
656 citations
Cites methods from "Tables of Mellin Transforms"
...is found from tables of Mellin transforms and inverse Mellin transforms (Oberhettinger, 1974)....
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...Further we use the relation (Oberhettinger, 1974) (41)...
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