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. Repr. of the 4th ed. 1927.

*(English)*Zbl 0951.30002
Cambridge: Cambridge University Press. 608 p. £29.95; $ 49.95 (1996).

It is quite an exceptional event that a mathematical text is reprinted almost 100 years after its first edition. Looking up the reviews of the previous editions the reviewer found out that this great classic never was honoured by a full review in the Zbl simply because the Zbl exists only since 1931, and even the forerunner JFM gave a full review only of the first edition of this work (E. T. Whittaker: A course of modern analysis: An introduction of infinite series and of analytic functions; with an account of the principal transcendental functions. Cambridge: At the University Press. XVI+378 pp. (1902; JFM 33.0390.01).

Compared to the first edition, the fourth edition under review is enlarged by more than 200 pages. As before, the work splits into two parts. Part I deals with the processes of analysis and covers the classical theory of real and complex analysis (theory of convergence, Riemann integration, fundamental properties of analytic functions, theory of residues and application to the evaluation of definite integrals, asymptotic expansion, and methods of summation, Fourier series and trigonometrical series, linear differential equations of the second order in the complex domain, integral equations). This part develops the general theory clearly and compactly on just 231 pages.

The larger part of the book is devoted to part II dealing with the classical transcendental functions. Many generations of students (like this reviewer) were introduced to the special functions of mathematical physics by reading the pertinent chapters of this part II, and it is a pleasure to say that the work under review ever was and still is one of the best texts on this subject. The authors deal with the following special functions: the Gamma function, the Riemann zeta function, the hypergeometric function, Legendre functions, the confluent hypergeometric function, Whittaker functions, Bessel functions, Mathieu functions, elliptic functions of Weierstraß and Jacobi, theta functions, ellipsoidal harmonics and Lamé’s equation. The style is very lucid and never lengthy. There is a large number of fully worked out examples from which the reader will grasp the relevant techniques and there is a large collection of miscellaneous examples which form an excellent supply of exercises.

The reader interested in the historical development of classical analysis will appreciate the many apt references to the sources of important classical results from the eighteenth century to the beginning of this century. Of course, from today’s point of view this work is no longer a course on modern analysis but rather a course on classical analysis. For example, the Lebesgue integral is missing, but this is a minor drawback since the reader equipped with this tool will easily supply the simplifications which are possible here and there by means of this concept. The work well represents the state of the art at the time of writing; e.g. Fejér’s theorem (of 1904) is included and the Riemann-Lebesgue lemma also (with a quotation of Lebesgue’s Leçons sur les séries trigonométriques of 1906), and even the first edition of Courant and Hilbert’s Methoden der mathematischen Physik (of 1924) appears among the references.

May this great classic continue to serve its purpose for the next one hundred years – and beyond that modest bound!

Compared to the first edition, the fourth edition under review is enlarged by more than 200 pages. As before, the work splits into two parts. Part I deals with the processes of analysis and covers the classical theory of real and complex analysis (theory of convergence, Riemann integration, fundamental properties of analytic functions, theory of residues and application to the evaluation of definite integrals, asymptotic expansion, and methods of summation, Fourier series and trigonometrical series, linear differential equations of the second order in the complex domain, integral equations). This part develops the general theory clearly and compactly on just 231 pages.

The larger part of the book is devoted to part II dealing with the classical transcendental functions. Many generations of students (like this reviewer) were introduced to the special functions of mathematical physics by reading the pertinent chapters of this part II, and it is a pleasure to say that the work under review ever was and still is one of the best texts on this subject. The authors deal with the following special functions: the Gamma function, the Riemann zeta function, the hypergeometric function, Legendre functions, the confluent hypergeometric function, Whittaker functions, Bessel functions, Mathieu functions, elliptic functions of Weierstraß and Jacobi, theta functions, ellipsoidal harmonics and Lamé’s equation. The style is very lucid and never lengthy. There is a large number of fully worked out examples from which the reader will grasp the relevant techniques and there is a large collection of miscellaneous examples which form an excellent supply of exercises.

The reader interested in the historical development of classical analysis will appreciate the many apt references to the sources of important classical results from the eighteenth century to the beginning of this century. Of course, from today’s point of view this work is no longer a course on modern analysis but rather a course on classical analysis. For example, the Lebesgue integral is missing, but this is a minor drawback since the reader equipped with this tool will easily supply the simplifications which are possible here and there by means of this concept. The work well represents the state of the art at the time of writing; e.g. Fejér’s theorem (of 1904) is included and the Riemann-Lebesgue lemma also (with a quotation of Lebesgue’s Leçons sur les séries trigonométriques of 1906), and even the first edition of Courant and Hilbert’s Methoden der mathematischen Physik (of 1924) appears among the references.

May this great classic continue to serve its purpose for the next one hundred years – and beyond that modest bound!

Reviewer: J.Elstrodt (Münster)

##### MSC:

30-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to functions of a complex variable |

26-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to real functions |

01A75 | Collected or selected works; reprintings or translations of classics |

33-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to special functions |

34-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to ordinary differential equations |

42-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to harmonic analysis on Euclidean spaces |

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\textit{E. T. Whittaker} and \textit{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. Repr. of the 4th ed. 1927. Cambridge: Cambridge University Press (1996; Zbl 0951.30002)