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**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 |

### Citations:

JFM 33.0390.01
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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)

### Digital Library of Mathematical Functions:

§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions§12.11(i) Distribution of Real Zeros ‣ §12.11 Zeros ‣ Properties ‣ Chapter 12 Parabolic Cylinder Functions

§12.9(i) Poincaré-Type Expansions ‣ §12.9 Asymptotic Expansions for Large Variable ‣ Properties ‣ Chapter 12 Parabolic Cylinder Functions

§1.3(iii) Infinite Determinants ‣ §1.3 Determinants ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods

Chapter 13 Confluent Hypergeometric Functions

§15.6 Integral Representations ‣ Properties ‣ Chapter 15 Hypergeometric Function

Chapter 15 Hypergeometric Function

Chapter 19 Elliptic Integrals

§20.13 Physical Applications ‣ Applications ‣ Chapter 20 Theta Functions

Other Notations ‣ §20.1 Special Notation ‣ Notation ‣ Chapter 20 Theta Functions

Other Notations ‣ §20.1 Special Notation ‣ Notation ‣ Chapter 20 Theta Functions

§20.2(iii) Translation of the Argument by Half-Periods ‣ §20.2 Definitions and Periodic Properties ‣ Properties ‣ Chapter 20 Theta Functions

§20.2(ii) Periodicity and Quasi-Periodicity ‣ §20.2 Definitions and Periodic Properties ‣ Properties ‣ Chapter 20 Theta Functions

§20.2(i) Fourier Series ‣ §20.2 Definitions and Periodic Properties ‣ Properties ‣ Chapter 20 Theta Functions

§20.2(iv) 𝑧-Zeros ‣ §20.2 Definitions and Periodic Properties ‣ Properties ‣ Chapter 20 Theta Functions

§20.4(ii) Higher Derivatives ‣ §20.4 Values at 𝑧 = 0 ‣ Properties ‣ Chapter 20 Theta Functions

§20.4(i) Functions and First Derivatives ‣ §20.4 Values at 𝑧 = 0 ‣ Properties ‣ Chapter 20 Theta Functions

§20.5(ii) Logarithmic Derivatives ‣ §20.5 Infinite Products and Related Results ‣ Properties ‣ Chapter 20 Theta Functions

§20.7(ii) Addition Formulas ‣ §20.7 Identities ‣ Properties ‣ Chapter 20 Theta Functions

§20.7(iv) Reduction Formulas for Products ‣ §20.7 Identities ‣ Properties ‣ Chapter 20 Theta Functions

§20.7(viii) Transformations of Lattice Parameter ‣ §20.7 Identities ‣ Properties ‣ Chapter 20 Theta Functions

§20.9(ii) Elliptic Functions and Modular Functions ‣ §20.9 Relations to Other Functions ‣ Properties ‣ Chapter 20 Theta Functions

§20.9(i) Elliptic Integrals ‣ §20.9 Relations to Other Functions ‣ Properties ‣ Chapter 20 Theta Functions

§20.9(ii) Elliptic Functions and Modular Functions ‣ §20.9 Relations to Other Functions ‣ Properties ‣ Chapter 20 Theta Functions

Chapter 20 Theta Functions

§22.11 Fourier and Hyperbolic Series ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions

§22.13(ii) First-Order Differential Equations ‣ §22.13 Derivatives and Differential Equations ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions

§22.13(i) Derivatives ‣ §22.13 Derivatives and Differential Equations ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions

§22.14(iii) Other Indefinite Integrals ‣ §22.14 Integrals ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions

§22.14(i) Indefinite Integrals of Jacobian Elliptic Functions ‣ §22.14 Integrals ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions

§22.14(iv) Definite Integrals ‣ §22.14 Integrals ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions

§22.15(i) Definitions ‣ §22.15 Inverse Functions ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions

§22.16(iii) Jacobi’s Zeta Function ‣ §22.16 Related Functions ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions

§22.16(ii) Jacobi’s Epsilon Function ‣ §22.16 Related Functions ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions

§22.18(iv) Elliptic Curves and the Jacobi–Abel Addition Theorem ‣ §22.18 Mathematical Applications ‣ Applications ‣ Chapter 22 Jacobian Elliptic Functions

Lemniscate ‣ §22.18(i) Lengths and Parametrization of Plane Curves ‣ §22.18 Mathematical Applications ‣ Applications ‣ Chapter 22 Jacobian Elliptic Functions

§22.2 Definitions ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions

§22.4(iii) Translation by Half or Quarter Periods ‣ §22.4 Periods, Poles, and Zeros ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions

§22.4(ii) Graphical Interpretation via Glaisher’s Notation ‣ §22.4 Periods, Poles, and Zeros ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions

§22.4(i) Distribution ‣ §22.4 Periods, Poles, and Zeros ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions

§22.5(i) Special Values of 𝑧 ‣ §22.5 Special Values ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions

§22.6(i) Sums of Squares ‣ §22.6 Elementary Identities ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions

§22.6(iv) Rotation of Argument (Jacobi’s Imaginary Transformation) ‣ §22.6 Elementary Identities ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions

§22.7(ii) Ascending Landen Transformation ‣ §22.7 Landen Transformations ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions

§22.7(i) Descending Landen Transformation ‣ §22.7 Landen Transformations ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions

§22.8(iii) Special Relations Between Arguments ‣ §22.8 Addition Theorems ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions

§22.8(ii) Alternative Forms for Sum of Two Arguments ‣ §22.8 Addition Theorems ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions

§22.8(iii) Special Relations Between Arguments ‣ §22.8 Addition Theorems ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions

§22.8(i) Sum of Two Arguments ‣ §22.8 Addition Theorems ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions

Chapter 22 Jacobian Elliptic Functions

§23.10(i) Addition Theorems ‣ §23.10 Addition Theorems and Other Identities ‣ Weierstrass Elliptic Functions ‣ Chapter 23 Weierstrass Elliptic and Modular Functions

Other Notations ‣ §23.1 Special Notation ‣ Notation ‣ Chapter 23 Weierstrass Elliptic and Modular Functions

Starting from Invariants ‣ §23.22(ii) Lattice Calculations ‣ §23.22 Methods of Computation ‣ Computation ‣ Chapter 23 Weierstrass Elliptic and Modular Functions

§23.2(iii) Periodicity ‣ §23.2 Definitions and Periodic Properties ‣ Weierstrass Elliptic Functions ‣ Chapter 23 Weierstrass Elliptic and Modular Functions

§23.2(ii) Weierstrass Elliptic Functions ‣ §23.2 Definitions and Periodic Properties ‣ Weierstrass Elliptic Functions ‣ Chapter 23 Weierstrass Elliptic and Modular Functions

§23.3(ii) Differential Equations and Derivatives ‣ §23.3 Differential Equations ‣ Weierstrass Elliptic Functions ‣ Chapter 23 Weierstrass Elliptic and Modular Functions

§23.3(i) Invariants, Roots, and Discriminant ‣ §23.3 Differential Equations ‣ Weierstrass Elliptic Functions ‣ Chapter 23 Weierstrass Elliptic and Modular Functions

§23.6(i) Theta Functions ‣ §23.6 Relations to Other Functions ‣ Weierstrass Elliptic Functions ‣ Chapter 23 Weierstrass Elliptic and Modular Functions

Chapter 23 Weierstrass Elliptic and Modular Functions

Bernoulli Numbers and Polynomials ‣ §24.1 Special Notation ‣ Notation ‣ Chapter 24 Bernoulli and Euler Polynomials

Table 28.1.1 ‣ §28.1 Special Notation ‣ Notation ‣ Chapter 28 Mathieu Functions and Hill’s Equation

§29.12(iii) Zeros ‣ §29.12 Definitions ‣ Lamé Polynomials ‣ Chapter 29 Lamé Functions

Chapter 29 Lamé Functions

§31.8 Solutions via Quadratures ‣ Properties ‣ Chapter 31 Heun Functions

§31.8 Solutions via Quadratures ‣ Properties ‣ Chapter 31 Heun Functions

Chapter 4 Elementary Functions

§5.11(ii) Error Bounds and Exponential Improvement ‣ §5.11 Asymptotic Expansions ‣ Properties ‣ Chapter 5 Gamma Function

Terminology ‣ §5.11(i) Poincaré-Type Expansions ‣ §5.11 Asymptotic Expansions ‣ Properties ‣ Chapter 5 Gamma Function

§5.17 Barnes’ 𝐺-Function (Double Gamma Function) ‣ Properties ‣ Chapter 5 Gamma Function

§5.8 Infinite Products ‣ Properties ‣ Chapter 5 Gamma Function

(5.9.10_2) ‣ Binet’s Formula ‣ §5.9(i) Gamma Function ‣ §5.9 Integral Representations ‣ Properties ‣ Chapter 5 Gamma Function

§5.9(ii) Psi Function, Euler’s Constant, and Derivatives ‣ §5.9 Integral Representations ‣ Properties ‣ Chapter 5 Gamma Function

§5.9(i) Gamma Function ‣ §5.9 Integral Representations ‣ Properties ‣ Chapter 5 Gamma Function

Binet’s Formula ‣ §5.9(i) Gamma Function ‣ §5.9 Integral Representations ‣ Properties ‣ Chapter 5 Gamma Function

Chapter 5 Gamma Function

Notations K ‣ Notations