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




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.

[1]TeX, a typesetting system that Donald Knuth originally designed to typeset mathematics, now forms the core for the LaTeX document preparation system widely used to prepare scientific papers for publication.  ↩
[2]There are software tools for converting LaTeX into .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. ↩
[3]I prepare the text for these pages with Fletcher Penny’s MultiMarkdown, which can generate either web pages or a LaTeX document from the text. The excellent MathJax script renders the mathematics. ↩

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.

[4]Whittaker and Watson, in the British tradition, refer to problems for the reader as examples. Those marked with “Math.Trip.” are from the famous Cambridge Tripos exam and can be difficult. ↩
[5]Contact me at nithardt@uw.edu. ↩

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.


Construction Notes

Contents

    PROCESSES OF ANALYSIS

  1. Complex Numbers    (pp. 3–10)
    1. 1.1 Rational numbers
    2. 1.2 Dedekind’s theory of irrational numbers
    3. 1.3 Complex numbers
    4. 1.4 The modulus of a complex number
    5. 1.5 The Argand diagram
    6. References
    7. Miscellaneous Examples
  2. The Theory of Convergence    (pp. 11–40)
    1. 2.1 The Definition of the Limit of a Sequence
    2. 2.2 The Limit of an Increasing Sequence
    3. 2.3 Convergence of an Infinite Series
    4. 2.4 Changing the Order of the Terms in a Series
    5. 2.5 Double Series
    6. 2.6 Power Series
    7. 2.7 Infinite Products
    8. 2.8 Infinite Determinants
    9. References
    10. Miscellaneous Examples
  3. Continuous Functions and Uniform Convergence    (pp. 41–60)
    1. 3.1 The Dependence of One Complex Number on Another
    2. 3.2 Continuity of Functions of Real Variables
    3. 3.3 Series of Variable Terms. Uniformity of Convergence.
    4. 3.4 A Particular Double Series
    5. 3.5 The Concept of Uniformity
    6. 3.6 The Modified Heine-Borel Theorem
    7. 3.7 Uniformity of Convergence of Power Series
    8. References
    9. Miscellaneous Examples
  4. The Theory of Reimann Integration    (pp. 61–81)
    1. 4.1 The Concept of Integration
    2. 4.2 Differentiation of Integrals containing a Parameter
    3. 4.3 Double Integrals & Repeated Integrals
    4. 4.4 Infinite Integrals
    5. 4.5 Improper Integrals. Principle Values
    6. 4.6 Complex Integration
    7. 4.7 Integration of Infinite Series
    8. References
    9. Miscellaneous Examples
  5. The Fundamental Properties of Analytic Functions    (pp. 82–110)
    1. 5.1 A Property of Elementary Functions
    2. 5.2 Cauchy’s Theorem on Contour Integrals
    3. 5.3 Analytic Functions as Uniformly Convergent Series
    4. 5.4 Taylor’s Theorem
    5. 5.5 The Process of Continuation
    6. 5.6 Laurent’s Theorem
    7. 5.7 Many-valued Functions
    8. References
    9. Miscellaneous Examples
  6. The Theory of Residues    (pp. 111–124)
    1. 6.1 Residues
    2. 6.2 The Evaluation of Definite Integrals
    3. 6.3 Cauchy’s Integral
    4. 6.4 Connecting a Function’s Zeros & its Derivate’s Zeros
    5. References
    6. Miscellaneous Examples
  7. The Expansion of Functions in Infinite Series    (pp. 125–149)
    1. 7.1 A Formula due to Darboux
    2. 7.2 The Bernoullian Numbers and Bernoullian Polynomials
    3. 7.3 Bürmann’s Theorem
    4. 7.4 The Expansion of Functions in Rational Fractions
    5. 7.5 The Expansion of Functions as Infinite Products
    6. 7.6 The Factor Theorem of Weierstrass
    7. 7.7 The Expansion of Periodic Functions in a Series of Cotangents
    8. 7.8 Borel’s Theorem
    9. References
    10. Miscellaneous Examples
  8. Asymptotic Expansions and Summable Series    (pp. 150–159)
    1. 8.1 Simple Example of an Asymptotic Expansion
    2. 8.2 Definition of an Asymptotic Expansion
    3. 8.3 Multiplication of Asymptotic Expansions
    4. 8.4 Methods of ‘Summing’ Series
    5. 8.5 Hardy’s Convergence Theorem
    6. References
    7. Miscellaneous Examples
  9. Fourier Series and Trigonometrical Series    (pp. 160–193)
    1. 9.1 Definition of Fourier Series
    2. 9.2 Dirichlet’s conditions & Fourier’s theorem
    3. 9.3 Coefficients of a Fourier series
    4. 9.4 Fejér’s theorem
    5. 9.5 Hurwitz-Liapounoff theorem concerning Fourier constants
    6. 9.6 Riemann’s theory of trigonometrical series
    7. 9.7 Fourier’s representation of a function by an integral
    8. References
    9. Miscellaneous Examples
  10. Linear Differential Equations
  11. Integral Equations

    12. THE TRANSCENDENTAL FUNCTIONS

  12. The Gamma Function
  13. The Zeta Function of Riemann
  14. The Hypergeometric Function
  15. Legendre Functions
  16. The Confluent Hypergeometric Function
  17. Bessel Functions
  18. The Equations of Mathematical Physics
  19. Mathieu Functions
  20. Elliptic Functions. General theorems and the Weierstrassian Functions
  21. The Theta Functions
  22. The Jacobian Elliptic Functions
  23. Ellipsoidal Harmonics and Lamé’s Equation