Tables of Mellin Transforms

ISBN-13: 9783540069423 / Angielski / Miękka / 1974 / 278 str.

F. Oberhettinger

Opis

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 cents(z) corresponding to a given q, (x) by (a) is known, then cents(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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ."

Spis treści

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.