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  • The Mittag-Leffler Theorem: The Origin, Evolution, and Reception of a Mathematical

    Result, 1876- 1884

    Laura E . Turner

    B.Sc., Acadia University, 2005

    A THESIS SUBMITTED IN PARTIAL FULFILLMENT

    OF T H E REQUIREMENTS FOR T H E DEGREE OF

    in the Department

    of

    Mathematics

    @ Laura E. Turner 2007

    SIMON FRASER UNIVERSITY

    2007

    All rights reserved. This work may not be

    reproduced in whole or in part, by photocopy

    or other means, without the permission of the author.

  • APPROVAL

    Name: Laura E. Turner

    Degree: Master of Science

    Title of Thesis: The Mittag-Leffler Theorem:

    The Origin, Evolution, and Reception of a Mathematical

    Result, 1876-1884

    Examining Committee: Dr. David Muraki

    Chair

    Dr. Tom Archibald

    Senior Supervisor

    Dr. Len Berggren

    Supervisor

    Dr. Peter Borwein

    Internal Examiner

    Date of Defense: August 1, 2007

    11

  • S I M O N FRASER U N I V E R S I T Y L I B R A R Y

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    The original Partial Copyright Licence attesting to these terms, and signed by this author, may be found in the original bound copy of this work, retained in the Simon Fraser University Archive.

    Simon Fraser University Library Burnaby, BC, Canada

    Revised: Summer 2007

  • Abstract

    The Swedish mathematician Gosta Mittag-Leffler (1846-1927) is well-known for founding

    Acta Mathematics, the first international mathematical journal. A "post-doctoral" student

    in Paris and Berlin (1873-76), Mittag-Leffler built on Karl Weierstrass' work by proving

    the Mittag-Leffler theorem, roughly: a meromorphic function is specified by its poles, their

    multiplicities, and the coefficients in the principal part of its Laurent expansion.

    In this thesis, I explore the evolution of the Mittag-Leffler theorem, from its initial (1876)

    state to its final (1884) version. Aspects of the details of Mittag-Leffler's work at various

    stages are analyzed to demonstrate the evolution of Mittag-Leffler's technique. A key finding

    of the thesis is that Mittag-Leffler's research on infinite sets of singular points attracted him

    to Georg Cantor's set-theoretic work. The incorporation of Cantor's theory was controversial,

    but demonstrates Mittag-Leffler's important role in the promotion of abstract mathematics

    over the more concrete mathematics of the previous era.

  • Acknowledgments

    The research behind this thesis was supported financially by a Natural Sciences and Engi-

    neering Research Council of Canada (NSERC) Canada Graduate Scholarship (2005-2007).

    I would like to thank my supervisor Dr. T . Archibald for all of his guidance and en-

    couragement, and for helping me to find a topic that, though we stumbled upon it almost

    accidentally, turned out to be so rich and interesting.

    I would also like to thank the Institut Mittag-Leffler in Djursholm, Sweden, for their

    assistance with the archival research during my stay there.

    Regarding translations: I would like to thank Dr. T . Archibald for his translations of the

    writings of Weierstrass, Cantor, etc., from the German. I am grateful to both Dr. Archibald

    and K. Hegelund for their assistance with the translations of Mittag-Leffler's early work from

    the Swedish.

    Finally, I would like to thank all of the individuals who supplied me with good questions

    and helpful comments. In particular: Dr. Archibald, Dr. 3. Barrow-Green, K. Andersen

    and H. K. Sorensen, and the members of the History of hllathematical Sciences Research

    Colloquium of the California Institute of Technology.

    Personal thanks are extended to my family, friends, and K. Hegelund for their support.

  • Contents

    . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Approval ii ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Abstract 111

    . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments iv

    . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Contents v

    . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . List of Figures vii ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . List of Tables vui

    . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Introduction 1

    . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Weierstrass's Influence 6

    . . . . . . . . . . . . . . . . . . 3 The first versions of Mittag-Leffler's theorem 14

    . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Contact with Cantor's ideas 24

    4.1 Mittag-Leffler's "pre-Cantorian" Comptes Rendus publications . . . . . 26 . . . . . . . . . . 4.1.1 First Letter: Theorem statement and proof 26

    . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2 Second Letter 30 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Georg Cantor 36

    4.3 Mittag-Leffler's Comptes Rendus publications using Cantorian termi-

    . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . nology 38

    . . . . . . . . . . . . . . . . . . . . 5 The CantorIMittag-Leffler correspondence 49

    6 The final version of Mittag-Leffler's theorem . . . . . . . . . . . . . . . . . . . 54

    . . . . . . . . . . . . . . . . . . . . . . . 7 Reception and Concluding Remarks 64 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Reception 64

    7.1.1 The reactions of Hermite and Weierstrass prior to the intro-

    . . . . . . . . . . . . . . . . . duction of Cantor's set theory 65 . . . . . . . . . . . . . . . 7.1.2 Reception of the Cantorian version 67

    . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Concluding Remarks 73

    C Note for "The Final Version of Mittag-Leffler's Theorem" . . . . . . . . 75

    D Modern Statement and Proof . . . . . . . . . . . . . . . . . . . . . . . 77

  • E Table of important publications . . . . . . . . . . . . . . . . . . . . . . 79

    Bibliography . . . . . . . . . . . . . . . . . , . . . . . . . . . . . . . . . . . . . . . 82

  • List of Figures

    Fig . 4.1. An example of singular points arranged in a "double series" . . . . . . . 46

    Fig . 6.1. A set of singular points for a non-monogenic function . . . . . . . . . . . 61

    vii

  • List of Tables

    Table 7.1: Important publications pertaining to the Mittag-Leffler theorem, 1876

    - 1877. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

    Table 7.2: Important publications pertaining to the Mittag-Leffler theorem, 1878

    - 1 8 8 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

  • Chapter 1

    Introduction

    Magnus Gijsta Mittag-Leffler was born March 16, 1846 in Stockholm, Sweden. His childhood

    education was greatly enriched by close contact with his father and frequent visits from family

    friends, and this addition to his traditional schooling proved to be fruitful in developing a

    multi-faceted set of talents. However, when he entered the Gymnasium in Stockholm his

    teachers recognized in particular his aptitude for higher mathematics, which he studied after

    training as an actuary. (Gaa941, pp. 73-74

    His post-secondary mathe