A Course of Modern Analysis

A Course of Modern Analysis: an introduction to the general theory of infinite processes and of analytic functions; with an account of the principal transcendental functions (colloquially known as Whittaker and Watson) is a landmark textbook on mathematical analysis written by Edmund T. Whittaker and George N. Watson, first published by Cambridge University Press in 1915.^[1] The first edition was Whittaker's alone, but later editions were co-authored with Watson.

A Course of Modern Analysis

Cover of a 1996 reissue of the fourth edition of the book.
Author	Edmund T. Whittaker and George N. Watson
Language	English
Subject	Mathematics
Publisher	Cambridge University Press
Publication date	1902

Title page for the third edition of the book.

History

Its first, second, third, and the fourth edition were published in 1902,^[2] 1915,^[3] 1920,^[4] and 1927,^[5] respectively. Since then, it has continuously been reprinted and is still in print today.^[5][6] A revised, expanded and digitally reset fifth edition, edited by Victor H. Moll, was published in 2021.^[7]

The book is notable for being the standard reference and textbook for a generation of Cambridge mathematicians including Littlewood and Godfrey H. Hardy. Mary L. Cartwright studied it as preparation for her final honours on the advice of fellow student Vernon C. Morton, later Professor of Mathematics at Aberystwyth University.^[8] But its reach was much further than just the Cambridge school; André Weil in his obituary of the French mathematician Jean Delsarte noted that Delsarte always had a copy on his desk.^[9] In 1941, the book was included among a "selected list" of mathematical analysis books for use in universities in an article for that purpose published by American Mathematical Monthly.^[10]

Notable features

Some idiosyncratic but interesting problems from an older era of the Cambridge Mathematical Tripos are in the exercises.

The book was one of the earliest to use decimal numbering for its sections, an innovation the authors attribute to Giuseppe Peano.^[11]

Contents

Below are the contents of the fourth edition:

Part I. The Process of Analysis

1. Complex Numbers
2. The Theory of Convergence
3. Continuous Functions and Uniform Convergence
4. The Theory of Riemann Integration
5. The fundamental properties of Analytic Functions; Taylor's, Laurent's, and Liouville's Theorems
6. The Theory of Residues; application to the evaluation of Definite Integrals
7. The expansion of functions in Infinite Series
8. Asymptotic Expansions and Summable Series
9. Fourier Series and Trigonometrical Series
10. Linear Differential Equations
11. Integral Equations

Part II. 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

Reception

Reviews of the first edition

George B. Mathews, in a 1903 review article published in The Mathematical Gazette opens by saying the book is "sure of a favorable reception" because of its "attractive account of some of the most valuable and interesting results of recent analysis".^[12] He notes that Part I deals mainly with infinite series, focusing on power series and Fourier expansions while including the "elements of" complex integration and the theory of residues. Part II, in contrast, has chapters on the gamma function, Legendre functions, the hypergeometric series, Bessel functions, elliptic functions, and mathematical physics.

Arthur S. Hathaway, in another 1903 review published in the Journal of the American Chemical Society, notes that the book centers around complex analysis, but that topics such as infinite series are "considered in all their phases" along with "all those important series and functions" developed by mathematicians such as Joseph Fourier, Friedrich Bessel, Joseph-Louis Lagrange, Adrien-Marie Legendre, Pierre-Simon Laplace, Carl Friedrich Gauss, Niels Henrik Abel, and others in their respective studies of "practice problems".^[13] He goes on to say it "is a useful book for those who wish to make use of the most advanced developments of mathematical analysis in theoretical investigations of physical and chemical questions."^[13]

In a third review of the first edition, Maxime Bôcher, in a 1904 review published in the Bulletin of the American Mathematical Society notes that while the book falls short of the "rigor" of French, German, and Italian writers, it is a "gratifying sign of progress to find in an English book such an attempt at rigorous treatment as is here made".^[1] He notes that important parts of the book were otherwise non-existent in the English language.

See also

• Bateman Manuscript Project

References

1. Bôcher, Maxime (1904). "Review: A Course of Modern Analysis, by E. T. Whittaker". Bulletin of the American Mathematical Society (review). 10 (7): 351–354. doi:10.1090/s0002-9904-1904-01123-4. (4 pages)
2. Whittaker, Edmund Taylor (1902). A Course Of Modern Analysis: An Introduction to the General Theory of Infinite Processes and of Analytic Functions; with an Account of the Principal Transcendental Functions (1st ed.). Cambridge, UK: at the University Press. OCLC 1072208628. (xvi+378 pages)
3. Whittaker, Edmund Taylor; Watson, George Neville (1915). A Course Of Modern Analysis: An Introduction to the General Theory of Infinite Processes and of Analytic Functions; with an Account of the Principal Transcendental Functions (2nd ed.). Cambridge, UK: at the University Press. OCLC 474155529. (viii+560 pages)
4. Whittaker, Edmund Taylor; Watson, George Neville (1920). A Course Of Modern Analysis: An Introduction to the General Theory of Infinite Processes and of Analytic Functions; with an Account of the Principal Transcendental Functions (3rd ed.). Cambridge, UK: at the University Press. OCLC 1170617940.
5. Whittaker, Edmund Taylor; Watson, George Neville (1927-01-02). A Course Of Modern Analysis: An Introduction to the General Theory of Infinite Processes and of Analytic Functions; with an Account of the Principal Transcendental Functions (4th ed.). Cambridge, UK: at the University Press. ISBN 0-521-06794-4. ISBN 978-0-521-06794-2. (vi+608 pages) (reprinted: 1935, 1940, 1946, 1950, 1952, 1958, 1962, 1963, 1992)
6. Whittaker, Edmund Taylor; Watson, George Neville (1996) [1927]. A Course of Modern Analysis. Cambridge Mathematical Library (4th reissued ed.). Cambridge, UK: Cambridge University Press. doi:10.1017/cbo9780511608759. ISBN 978-0-521-58807-2. OCLC 802476524. ISBN 0-521-58807-3. (reprinted: 1999, 2000, 2002, 2010)
7. Whittaker, Edmund Taylor; Watson, George Neville (2021-08-26) [2021-08-07]. Moll, Victor Hugo (ed.). A Course of Modern Analysis (5th revised ed.). Cambridge, UK: Cambridge University Press. doi:10.1017/9781009004091. ISBN 978-1-31651893-9. ISBN 1-31651893-0. Archived from the original on 2021-08-10. Retrieved 2021-12-26. (700 pages)
8. O'Connor, John J.; Robertson, Edmund Frederick (October 2003). "Dame Mary Lucy Cartwright". MacTutor. St. Andrews, UK: St. Andrews University. Archived from the original on 2021-03-21. Retrieved 2021-03-21.
9. O'Connor, John J.; Robertson, Edmund Frederick (December 2005). "Jean Frédéric Auguste Delsarte". MacTutor. St. Andrews, UK: St. Andrews University. Archived from the original on 2021-03-21. Retrieved 2021-03-21.
10. "A Selected List of Mathematics Books for Colleges". The American Mathematical Monthly. 48 (9): 600–609. 1941. doi:10.1080/00029890.1941.11991146. ISSN 0002-9890. JSTOR 2303868. (10 pages)
11. Kowalski, Emmanuel [in German] (2008-06-03). "Peano paragraphing". E. Kowalski's blog - Comments on mathematics, mostly. Archived from the original on 2021-02-25. Retrieved 2021-03-21.
12. Mathews, George Ballard (1903). "Review of A Course of Modern Analysis". The Mathematical Gazette (review). 2 (39): 290–292. doi:10.2307/3603560. ISSN 0025-5572. JSTOR 3603560. S2CID 221486387. (3 pages)
13. Hathaway, Arthur Stafford (February 1903). "A Course in Modern Analysis". Journal of the American Chemical Society (review). 25 (2): 220. doi:10.1021/ja02004a022. ISSN 0002-7863.

Further reading

• Jourdain, Philip E. B. (1916-01-01). "(1) A Course of Pure Mathematics. By G. H. Hardy. Cambridge University Press, 1908. Pp. xvi, 428. Cloth, 12s. net. (2) A Course of Pure Mathematics. By G. H. Hardy. Second edition. Cambridge University Press, 1914. Pp. xii, 443. Cloth, 12s. net. (3) A Course of Modern Analysis: An Introduction to the General Theory of Infinite Processes and of Analytic Functions; with an Account of the Principal Transcendental Functions. By E. T. Whittaker. Cambridge University Press, 1902. Pp. xvi, 378. Cloth, 12s. 6d. net. (4) A Course of Modern Analysis: An Introduction to the General Theory of Infinite Processes and of Analytic Functions; with an Account of the Principal Transcendental Functions. Second edition, completely revised. By E. T. Whittaker and G. N. Watson. Cambridge University Press, 1915. Pp. viii, 560. Cloth, 18s. net". VI. Critical Notices. Mind (review). XXV (4): 525–533. doi:10.1093/mind/XXV.4.525. ISSN 0026-4423. JSTOR 2248860. (9 pages)
• Neville, Eric Harold (1921). "Review of A Course of Modern Analysis". The Mathematical Gazette (review). 10 (152): 283. doi:10.2307/3604927. ISSN 0025-5572. JSTOR 3604927. (1 page)
• Wrinch, Dorothy Maud (1921). "Review of A Course of Modern Analysis. Third Edition". Science Progress in the Twentieth Century (1919-1933) (review). 15 (60). Sage Publications, Inc.: 658. ISSN 2059-4941. JSTOR 43769035. (1 page)
• "Review of A Course of Modern Analysis". The Mathematical Gazette (review). 14 (196): 245. 1928. doi:10.2307/3606904. ISSN 0025-5572. JSTOR 3606904. S2CID 3980161. (1 page)
• "Review of A Course of Modern Analysis. An Introduction to the General Theory of Infinite Processes and of Analytic Functions; with an Account of the Principal Transcendental Functions". The American Mathematical Monthly (review). 28 (4): 176. 1921. doi:10.2307/2972291. hdl:2027/coo1.ark:/13960/t17m0tq6p. ISSN 0002-9890. JSTOR 2972291.
• Φ (1916). "Review of A Course of Modern Analysis: An Introduction to the General Theory of Infinite Processes and of Analytic Functions; with an Account of the Principal Transcendental Functions. Second edition, completely revised". The Monist (review). 26 (4): 639–640. ISSN 0026-9662. JSTOR 27900617. (2 pages)
• "Review of A Course of Modern Analysis. An Introduction to the General Theory of Infinite Processes and of Analytical Functions, with an Account of the Principal Transcendental Functions. Second Edition". Science Progress (1916–1919) (review). 11 (41). Sage Publications, Inc.: 160–161. 1916. ISSN 2059-495X. JSTOR 43426733. (2 pages)
• "Review of A Course of Modern Analysis: An introduction to the General Theory of Infinite Processes and of Analytical Functions; With an Account of the Principal Transcendental Functions". The Mathematical Gazette (review). 8 (124): 306–307. 1916. doi:10.2307/3604810. ISSN 0025-5572. JSTOR 3604810. S2CID 40238008. (2 pages)
• Schubert, A. (1963). "E. T. Whittaker and G. N. Watson, A Course of Modern Analysis. An introduction to the general theory of infinite processes and of analytic functions; with an account of the principal transcendental functions. Fourth Edition. 608 S. Cambridge 1962. Cambridge University Press. Preis brosch. 27/6 net". ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik (review). 43 (9): 435. Bibcode:1963ZaMM...43R.435S. doi:10.1002/zamm.19630430916. ISSN 1521-4001. (1 page)
• "Modern Analysis. By E. T. Whittaker and G. N. Watson Pp. 608. 27s. 6d. 1962. (Cambridge University Press)". The Mathematical Gazette (review). 47 (359): 88. February 1963. doi:10.1017/S0025557200049032. ISSN 0025-5572.
• "A Course of Modern Analysis". Nature (review). 97 (2432): 298–299. 1916-06-08. Bibcode:1916Natur..97..298.. doi:10.1038/097298a0. ISSN 1476-4687. S2CID 3980161. (1 page)
• "A Course of Modern Analysis: An Introduction to the General Theory of Infinite Processes and of Analytic Functions; with an Account of the Principal Transcendental Functions". Nature (review). 106 (2669): 531. 1920-12-23. Bibcode:1920Natur.106R.531.. doi:10.1038/106531c0. hdl:2027/coo1.ark:/13960/t17m0tq6p. ISSN 1476-4687. S2CID 40238008. (1 page)
• M.-T., L. M. (1928-03-17). "A Course of Modern Analysis: an Introduction to the General Theory of Infinite Processes and of Analytic Functions; with an Account of the Principal Transcendental Functions". Nature (review). 121 (3046): 417. Bibcode:1928Natur.121..417M. doi:10.1038/121417a0. ISSN 1476-4687. (1 page)
• Stuart, S. N. (1981). "Table errata: A course of modern analysis [fourth edition, Cambridge Univ. Press, Cambridge, 1927; Jbuch 53, 180] by E. T. Whittaker and G. N. Watson". Mathematics of Computation (errata). 36 (153). American Mathematical Society: 315–320 [319]. doi:10.1090/S0025-5718-1981-0595076-1. ISSN 0025-5718. JSTOR 2007758. (1 of 6 pages)