An Introduction to Special Functions
The subjects treated in this book have been especially chosen to represent a bridge connecting the content of a first course on the elementary theory of analytic functions with a rigorous treatment of some of the most important special functions: the Eule
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Carlo Viola
An Introduction to Special Functions
UNITEXT - La Matematica per il 3+2 Volume 102
Editor-in-chief A. Quarteroni Series editors L. Ambrosio P. Biscari C. Ciliberto M. Ledoux W.J. Runggaldier
More information about this series at http://www.springer.com/series/5418
Carlo Viola
An Introduction to Special Functions
123
Carlo Viola Department of Mathematics University of Pisa Pisa Italy
ISSN 2038-5722 ISSN 2038-5757 (electronic) UNITEXT - La Matematica per il 3+2 ISBN 978-3-319-41344-0 ISBN 978-3-319-41345-7 (eBook) DOI 10.1007/978-3-319-41345-7 Library of Congress Control Number: 2016944325 Mathematics Subject Classification (2010): 30-01, 30A10, 33B15, 33C05, 33C15, 34M03, 65B15 © Springer International Publishing Switzerland 2016 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer International Publishing AG Switzerland
Preface
These lecture notes stem from a course that I gave at the doctoral school in mathematics at Pisa University during the academic year 2013–2014. Their primary purpose is to expound some material well suited for a second course on analytic functions of one complex variable, after a first elementary course dealing with the basic concepts in this theory, such as the residue theorem, Cauchy’s integral formula, Taylor and Laurent series expansions, poles and essential singularities, branch points, etc. These basic subjects are the only background assumed in this book; I have made a serious attempt to avoid more advanced prerequisites, sometimes at the cost of choosing slightly longer but more elementary proofs of the theorems. As the title suggests, however, the topics included have been especially chosen to provide the reader with the main notions and results in the theory of functions of one complex variable leading to a rigorous treatment of some special functions: in the first place, the Euler gamma function, which can be
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