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. Partiยญ cularly widespread and effective is its application to problems arising in analytic number theory. This is partially due to the fact that if ยข(z) corresponding to a given q,(x) by (a) is known, then ยข(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 conceñing Mellin transยญ 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. . . โข . โข โข โข . โข . . . . . . . . . . . . โข โข โข โข . . . โข . โข . . โข โข โข . โข . 1 Some Applications of the Mellin Transform Analysis. โขโข. โขโขโข. . . โข. โข. . . . โขโข . โข . . . . . . โขโข. . . . . โขโข 6 1. 1 General Formulas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1. 2 Algebraic Functions and Powers of Arbitrary Order . . . 13 1. 3 Exponential Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
CONTENT
I. Mellin Transforms -- Some Applications of the Mellin Transform Analysis -- II. Inverse Mellin Transforms
