Tables of Laplace Transforms

Tables of Laplace Transforms PDF

Author: F. Oberhettinger

Publisher: Springer Science & Business Media

Published: 2012-12-06

Total Pages: 438

ISBN-13: 3642656455

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This material represents a collection of integrals of the Laplace- and inverse Laplace Transform type. The usef- ness of this kind of information as a tool in various branches of Mathematics is firmly established. Previous publications include the contributions by A. Erdelyi and Roberts and Kaufmann (see References). Special consideration is given to results involving higher functions as integrand and it is believed that a substantial amount of them is presented here for the first time. Greek letters denote complex parameters within the given range of validity. Latin letters denote (unless otherwise stated) real positive parameters and a possible extension to complex values by analytic continuation will often pose no serious problem. The authors are indebted to Mrs. Jolan Eross for her tireless effort and patience while typing this manu script. Oregon State University Corvallis, Oregon Eastern Michigan University Ypsilanti, Michigan The Authors Contents Part I. Laplace Transforms In troduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1. 1 General Formulas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1. 2 Algebraic Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1. 3 Powers of Arbitrary Order. . . . . . . . . . . . . . . . . . . . . . . . 21 1. 4 Sectionally Rational- and Rows of Delta Functions 28 1. 5 Exponential Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 1. 6 Logarithmic Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 1. 7 Trigonometric Functions. . . . . . . . . . . . . . . . . . . . . . . . . . 54 1. 8 Inverse Trigonometric Functions. . . . . . . . . . . . . . . . . . 81 1. 9 Hyperbolic Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 1. 10 Inverse Hyperbolic Functions. . . . . . . . . . . . . . . . . . . . . 99 1. 11 Orthogonal Polynomials . . . . . . . •. . . . . . . . . . . . . . . . . . . 103 1. 12 Legendre Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 1. 13 Bessel Functions of Order Zero and Unity . . . . . . . . . 119 1. 14 Bessel Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 1. 15 Modified Bessel Functions . . . . . . . . . . . . . . . . . . . . . . . .

Tables of Mellin Transforms

Tables of Mellin Transforms PDF

Author: F. Oberhettinger

Publisher: Springer Science & Business Media

Published: 2012-12-06

Total Pages: 284

ISBN-13: 3642659756

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This book contains tables of integrals of the Mellin transform type z-l J (a) 1> (z) q,(x)x dx o t Since the substitution x = e- transforms (a) into (b) 1> (z) the Mellin transform is sometimes referred to as the two sided Laplace transform. The use of the Mellin transform in various problems in mathematical analysis is well established. Parti cularly widespread and effective is its application to problems arising in analytic number theory. This is partially due to the fact that if ¢(z) corresponding to a given q,(x) by (a) is known, then ¢(z) belonging to xaq,(x) or more general to P xaq,(x ) (p real) is likewise known. (See particularly the rules in sections 1. 1 and 2. 1 of this book. ) A list of major contributions conce~ning Mellin trans forms is added at the end of the introduction. Latin letters (unless otherwise stated) denote real positive numbers while Greek letters denote complex parameters within the given range of validity. The author is indebted to Mrs. Jolan Eross for her tireless effort and patience while typing this manuscript. Oregon State University Corvallis, Oregon May 1974 Fritz Oberhettinger Contents Part I. Mellin Transforms Introduction. . . • . • • • . • . . . . . . . . . . . . • • • • . . . • . • . . • • • . • . 1 Some Applications of the Mellin Transform Analysis. ••. •••. . . •. •. . . . •• . • . . . . . . ••. . . . . •• 6 1. 1 General Formulas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1. 2 Algebraic Functions and Powers of Arbitrary Order . . . 13 1. 3 Exponential Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Table of Integrals, Series, and Products

Table of Integrals, Series, and Products PDF

Author: I. S. Gradshteyn

Publisher: Academic Press

Published: 2014-05-10

Total Pages: 1207

ISBN-13: 1483265641

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Table of Integrals, Series, and Products provides information pertinent to the fundamental aspects of integrals, series, and products. This book provides a comprehensive table of integrals. Organized into 17 chapters, this book begins with an overview of elementary functions and discusses the power of binomials, the exponential function, the logarithm, the hyperbolic function, and the inverse trigonometric function. This text then presents some basic results on vector operators and coordinate systems that are likely to be useful during the formulation of many problems. Other chapters consider inequalities that range from basic algebraic and functional inequalities to integral inequalities and fundamental oscillation and comparison theorems for ordinary differential equations. This book discusses as well the important part played by integral transforms. The final chapter deals with Fourier and Laplace transforms that provides so much information about other integrals. This book is a valuable resource for mathematicians, engineers, scientists, and research workers.

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