The publisher of this book utilises modern printing technologies as well as photocopying processes for reprinting and preserving rare works of literature that are out-of-print or on the verge of becoming lost. This book is one such reprint.

INTRODUCTION TO THE THEORY OF ANALYTIC FUNCTIONS BY PROFESSOR JAMES HARKNESS AND PROFESSOR FRANK MORLRY RFPRINT 1924. G. E. STECHERT Co. NEW-YORK PREFACE. THE present book is not to be regarded a. s an abridged and more elementary version of our treatise dh the theory of functions, but as an independent work. It has been composed with different ends in mind, deals in many places with distinct orders of ideas, and presents from an independent point of view such portions of the subject-matter as are common to both volumes. In the treatise our desire was to cover as fully as possible within the limits alt our disposal a very extensive field of analysis, and the execution of this plan precluded the possibility of allotting much space to preliminary notions. At the same time we recognized that readers approaching the subject for the first time could not fail to be hampered by the non-existence in English of any text-book giving a consecutive and elementary account of the fundamental concepts and processes employed in the theory of functions subsequent experience and inquiry has only strengthened our belief that if English and American students are not to be placed at a disadvantage as compared with their foreign brethren they should have ready access to text- ooks which discuss topics of the kind indicated above. It was to an attempt to meet these requirements to the best of our ability that the present volum owes its genesis, and this is its bond of connexion with the treatise. The theory of functions, by virtue of its immense range and vitality and its innumerable points of contact with other branches VI PREFACE, of mathematics, has taken a central position in modern analysis, and has madeits influence felt in all parts of the mathematical domain. It i not surprising accordingly to find that such of the current text-books as have been composed in the modern spirit show numerous traces of the inrush of new ideas due to a wider acquaintance with the theory of functions, and that they are, both as regards structure and aim, poles apart from those of the preceding generation. There is, however, much still to be done in the direction of recasting elementary mathe matics in the light of recent knowledge in particular works in English that treat of the scientific parts of arithmetic show little if any trace of recent discoveries with respect to the number system. As we felt that it would be unsafe to assume any acquaintance with the various modern views on the nature of ordinal and cardinal numbers, and as it was indispensable for the proper comprehension of the succeeding chapters that the meaning of the term ordinal number should be clearly appre ciated, we have devoted the first chapter to the discussion of what-is meant by an ordinal number. This chapter is not and lays no claim to being a scientifically complete account of the matter it will serve its object if it conveys to the reader a distinct image of a number divorced from measurement. In places we have gone afresh over old ground this has been done either for the sake of organic unity, or in order to emphasize by means of simple examples ideas which appear later in more difficult and complicated forms. It has in fact been our desire to keep the difficulties of the subject apart from those which are merely difficulties of technique. In carrying out this plan we have consistently chosen the simplest available examples.As regards the theory of functions proper we had to make a choice between the methods of Cauchy and those of Weierstrass. While fully alive to the wonderful beauty and power of Cauchys theory, we decided eventually in favour of Weierstrasss system. Weierstrass has himself stated with his usual lucidity and force the reasons which have led him to PREFACE. Vil prefer his own scheme to that of Cauchy and Riemann for the purposes of a systematic construction of a theory of functions. In a letter to Prof. Schwarz Weierstrass, Ges, Werke, vol. ii. p...