## A COURSE OF

# MODERN ANALYSIS

#### an introduction to the general theory of infinite processes and analytic functions;

#### with an account of the principal transcendental functions

###### by

### E. T. WHITTAKER, Sc.D., F.R.S.

###### PROFESSOR OF MATHEMATICS IN THE UNIVERSITY OF EDINBURGH

###### and

### G. N. WATSON, Sc.D., F.R.S.

###### PROFESSOR OF MATHEMATICS IN THE UNIVERSITY OF BIRMINGHAM

#### Third Edition

##### CAMBRIDGE

##### AT THE UNIVERSITY PRESS

##### 1920

###### transcribed and edited as a website by

##### Eric Nitardy

##### 2012

#### CAMBRIDGE UNIVERSITY PRESS

#### C. F. CLAY, Manager

#### LONDON: FETTER LANE, E.C. 4

NEW YORK : | THE MACMILLAN CO. |

BOMBAY | |

CALCUTTA | MACMILLAN AND CO., Ltd. |

MADRAS | |

TORONTO : | THE MACMILLAN CO. OF |

CANADA, Ltd. | |

TOKYO : | MARUZEN-KABUSHIKI-KAISHA |

##### ALL RIGHTS RESERVED

*First Edition* 1902

*Second Edition* 1915

*Third Edition* 1920

## Preface

Advantage has been taken of the preparation of the third edition of this work to add a chapter on Ellipsoidal Harmonics and Lame’s Equation, and to rearrange the chapter on Trigonometrical Series so that the parts which are used in Applied Mathematics come at the beginning of the chapter. A number of minor errors have been corrected and we have endeavoured to make the references more complete.

Our thanks are due to Miss Wrinch for reading the greater part of the proofs and to the staff of the University Press for much courtesy and consideration during the progress of the printing.

E.T.W. |

G.N.W. |

July, 1920.

## Editor’s Note

After twenty-five years of failed predictions, the paperless office finally seems to have arrived. The near print quality resolution of the ‘New iPad’ and its competitors will likely further reduce the use of paper. Most mathematicians and physicists have been distributing their work electronically for a long time using files produced by TeX based typesetting software.[1] This software, as excellent as it is, was designed around print, which, I have noted, is quickly vanishing. It might be time to think seriously about publishing mathematics directly on the web or on its cousin the e-book.[2] As an experiment,[3] I am transcribing for the web Whittaker and Watson’s classic work on complex analysis and special functions. Its value as a reference work, extensive use of parenthetical comments, and numerous footnotes and references may combine to make it more pleasant and useful to read on the web than on paper. You may judge for yourself.

`.html`

and `.css`

files from which one might build a website, notably LaTeX2HTML and TeX4ht, but the quality of the result is poor, and the `.css`

files produced are muddled and difficult to customize. ↩
I am transcribing the third edition from 1920, which is in the public domain. According to the preface for the fourth and final edition of 1927, very little had changed from the third edition. I have retained the old-fashioned spelling and heavy use of semi-colons, but I have altered the typography and layout to fit the web and improve readability. The footnotes have been reformed as margin notes which are initially hidden, but become visible when tapped. I have, when possible, added links to on-line versions of the mentioned references and, occasionally, added editorial comments, which I have clearly marked.

This effort has just begun: I have only the first two chapters finished as of this writing. Given my other commitments and my determination to do all the *examples*,[4] I expect to complete a chapter every month, roughly. Thus, I should finish sometime late 2013 or early 2014. Feel free to offer comments or corrections.[5] I will update this note as things move along.

*examples*. Those marked with “Math.Trip.” are from the famous Cambridge Tripos exam and can be difficult. ↩

Eric Nitardy

April, 2012.

Imagining that the this effort would be finished by early 2014 was wildly too optimistic. At the moment, I have completed chapters one through six and the appendix, have begun work on chapter seven, and am proceeding at about a chapter every month and a half. Whether that rate continues, we shall see.

I am grateful to Brian Wignall for the use of portions of his standard Latex transcription of Whittaker and Watson.

Eric Nitardy

April, 2014.

## Contents

- Complex Numbers
**(pp. 3–10)** - The Theory of Convergence
**(pp. 11–40)** - Continuous Functions and Uniform Convergence
**(pp. 41–60)**- 3.1 The Dependence of One Complex Number on Another
- 3.2 Continuity of Functions of Real Variables
- 3.3 Series of Variable Terms. Uniformity of Convergence.
- 3.4 A Particular Double Series
- 3.5 The Concept of Uniformity
- 3.6 The Modified Heine-Borel Theorem
- 3.7 Uniformity of Convergence of Power Series
- References
- Miscellaneous Examples

- The Theory of Reimann Integration
**(pp. 61–81)** - The Fundamental Properties of Analytic Functions
**(pp. 82–110)** - The Theory of Residues
**(pp. 111–124)** - The Expansion of Functions in Infinite Series
**(pp. 125–149)**- 7.1 A Formula due to Darboux
- 7.2 The Bernoullian Numbers and Bernoullian Polynomials
- 7.3 Bürmann’s Theorem
- 7.4 The Expansion of Functions in Rational Fractions
- 7.5 The Expansion of Functions as Infinite Products
- 7.6 The Factor Theorem of Weierstrass
- 7.7 The Expansion of Periodic Functions in a Series of Cotangents
- 7.8 Borel’s Theorem
- References
- Miscellaneous Examples

- Asymptotic Expansions and Summable Series
**(pp. 150–159)** - Fourier Series and Trigonometrical Series
**(pp. 160–193)**- 9.1 Definition of Fourier Series
- 9.2 Dirichlet’s conditions & Fourier’s theorem
- 9.3 Coefficients of a Fourier series
- 9.4 Fejér’s theorem
- 9.5 Hurwitz-Liapounoff theorem concerning Fourier constants
- 9.6 Riemann’s theory of trigonometrical series
- 9.7 Fourier’s representation of a function by an integral
- References
- Miscellaneous Examples

- Linear Differential Equations
- Integral Equations
- The Gamma Function
- The Zeta Function of Riemann
- The Hypergeometric Function
- Legendre Functions
- The Confluent Hypergeometric Function
- Bessel Functions
- The Equations of Mathematical Physics
- Mathieu Functions
- Elliptic Functions. General theorems and the Weierstrassian Functions
- The Theta Functions
- The Jacobian Elliptic Functions
- Ellipsoidal Harmonics and Lamé’s Equation