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General Theory Of Dirichlet's Series

Marcel Riesz

G. Hardy

CreateSpace Independent Publishing Platform

From the Preface. My object in writing this Tract was to collect into a single volume those propositions which are employed in the course of a rigorous proof of Cauchy's theorem, together with a brief account of some of the applications of the theorem to the evaluation of definite integrals.  My endeavour has been to place the whole theory on a definitely arithmetical basis without appealing to geometrical intuitions. With that end in view, it seemed necessary to include an account of various propositions of Analysis Situs, on which depends the proof of the theorem in its most general form. In proving these propositions, I have followed the general course of a memoir by Ames; my indebtedness to it and to the textbooks on Analysis by Goursat and by de la Vallée Poussin will be obvious to those who are acquainted with those works.

Complex Integration And Cauchys Theorem

G. Watson

J. Leathem

From the PREFACE. THE publication of this tract has been delayed by a variety of causes, and I am now compelled to issue it without Dr Riesz's help in the final correction of the proofs. This has at any rate, one advantage, that it gives me the opportunity of saying how conscious I am that whatever value it possesses is due mainly to his contributions to it, and in particular to the fact, that it contains the first systematic, account of his beautiful theory of the summation of series by 'typical means'.  The task of condensing any account of so extensive a theory into the compass of one of these tracts has proved an exceedingly difficult one. Many important theorems are stated without proof, and many details are left to the reader. I believe, however, that our account is full enough to serve as a guide to other mathematicians researching in this and allied subjects. Such readers will be familiar with Landau's Handbuch der Lehre con der Verteilung der Primzahlen, and will hardly need to be told how much we, in common with all other investigators in this field, owe to the writings and to the personal encouragement of its author.  G. H. H.  * * * * * CONTENTS I. Introduction II. Elementary Theory of the Convergence of Dirichlet's Series III. The Formula for the Sum of the Coefficients of a Dirichlet's Series: The Order of the Function Represented by the Series IV. The Summation of Series by Typical Means V. General Arithmetic Theorems Concerning Typical Means VI. Abelian and Tauberian Theorems VII. Further Developments of the Theory of Functions Represented By Dirichlet's Series  VIII. The Multiplication of Dirichlet's Series Bibliography

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Books by G. Hardy

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General Theory Of Dirichlet's Series

Complex Integration And Cauchys Theorem

General Theory Of Dirichlet's Series