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Elliptic functions and applications. (English) Zbl 0689.33001

Applied Mathematical Sciences, 80. New York, NY etc.: Springer-Verlag. xiv, 334 p. DM 124.00 (1989).
With this book the author has set himself the task of providing “An introductory book which can be recommended for self-study by the undergraduate student or ordinary working mathematician...”. His underlying educational philosophy is very clearly set forth in the preface, which also reveals an obvious delight in the subject matter.
The text develops the theory from theta functions, and discusses Jacobi’s elliptic functions, elliptic inetrals, Weierstrass elliptic functions and modular transformations. In accordance with the author’s general plan, complex function theory is given a supporting role, and appears late in the book. A large number of carefully worked out examples of applications are included, sometimes in separate chapters, which should make the book especially attractive to anyone teaching prospective mathematical practitioners. Appropriately, the book ends with a section of tables, including BASIC programs to produce them.
Reviewer: H.Martens

MSC:

33E05 Elliptic functions and integrals
33-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to special functions
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Digital Library of Mathematical Functions:

§19.2(ii) Legendre’s Integrals ‣ §19.2 Definitions ‣ Legendre’s Integrals ‣ Chapter 19 Elliptic Integrals
§19.36(iii) Via Theta Functions ‣ §19.36 Methods of Computation ‣ Computation ‣ Chapter 19 Elliptic Integrals
§20.13 Physical Applications ‣ Applications ‣ Chapter 20 Theta Functions
§20.15 Tables ‣ Computation ‣ 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(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
§20.5(i) Single Products ‣ §20.5 Infinite Products and Related Results ‣ Properties ‣ Chapter 20 Theta Functions
§20.7(i) Sums of Squares ‣ §20.7 Identities ‣ 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
§20.7(vii) Derivatives of Ratios of Theta Functions ‣ §20.7 Identities ‣ Properties ‣ Chapter 20 Theta Functions
§20.7(vi) Landen Transformations ‣ §20.7 Identities ‣ Properties ‣ Chapter 20 Theta Functions
§20.7(vii) Derivatives of Ratios of Theta Functions ‣ §20.7 Identities ‣ Properties ‣ Chapter 20 Theta Functions
Chapter 20 Theta Functions
§22.10(ii) Maclaurin Series in 𝑘 and 𝑘’ ‣ §22.10 Maclaurin Series ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions
§22.10(i) Maclaurin Series in 𝑧 ‣ §22.10 Maclaurin Series ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions
§22.12 Expansions in Other Trigonometric Series and Doubly-Infinite Partial Fractions: Eisenstein Series ‣ 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(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(ii) Indefinite Integrals of Powers of Jacobian Elliptic Functions ‣ §22.14 Integrals ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions
§22.15(ii) Representations as Elliptic Integrals ‣ §22.15 Inverse Functions ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions
§22.15(i) Definitions ‣ §22.15 Inverse Functions ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions
§22.15(ii) Representations as Elliptic Integrals ‣ §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
§22.17(i) Real or Purely Imaginary Moduli ‣ §22.17 Moduli Outside the Interval [0,1] ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions
§22.18(iii) Uniformization and Other Parametrizations ‣ §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
§22.19(ii) Classical Dynamics: The Quartic Oscillator ‣ §22.19 Physical Applications ‣ Applications ‣ Chapter 22 Jacobian Elliptic Functions
§22.19(i) Classical Dynamics: The Pendulum ‣ §22.19 Physical Applications ‣ Applications ‣ Chapter 22 Jacobian Elliptic Functions
§22.19(ii) Classical Dynamics: The Quartic Oscillator ‣ §22.19 Physical Applications ‣ Applications ‣ Chapter 22 Jacobian Elliptic Functions
§22.19(i) Classical Dynamics: The Pendulum ‣ §22.19 Physical Applications ‣ Applications ‣ Chapter 22 Jacobian Elliptic Functions
§22.19(iv) Tops ‣ §22.19 Physical Applications ‣ Applications ‣ Chapter 22 Jacobian Elliptic Functions
Case I: 𝑉(𝑥)={1/2}⁢𝑥²+{1/4}⁢𝛽⁢𝑥⁴ ‣ §22.19(ii) Classical Dynamics: The Quartic Oscillator ‣ §22.19 Physical Applications ‣ Applications ‣ Chapter 22 Jacobian Elliptic Functions
§22.19(v) Other Applications ‣ §22.19 Physical Applications ‣ Applications ‣ Chapter 22 Jacobian Elliptic Functions
§22.20(vii) Further References ‣ §22.20 Methods of Computation ‣ Computation ‣ Chapter 22 Jacobian Elliptic Functions
§22.21 Tables ‣ Computation ‣ 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(i) Distribution ‣ §22.4 Periods, Poles, and Zeros ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions
§22.5(ii) Limiting Values of 𝑘 ‣ §22.5 Special Values ‣ 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(iii) Half Argument ‣ §22.6 Elementary Identities ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions
§22.6(ii) Double Argument ‣ §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(ii) Alternative Forms for Sum of Two 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(iii) 𝑛-Tuple Formulas ‣ §23.10 Addition Theorems and Other Identities ‣ Weierstrass Elliptic Functions ‣ Chapter 23 Weierstrass Elliptic and Modular Functions
§23.10(i) Addition Theorems ‣ §23.10 Addition Theorems and Other Identities ‣ Weierstrass Elliptic Functions ‣ Chapter 23 Weierstrass Elliptic and Modular Functions
§23.10(i) Addition Theorems ‣ §23.10 Addition Theorems and Other Identities ‣ Weierstrass Elliptic Functions ‣ Chapter 23 Weierstrass Elliptic and Modular Functions
§23.10(iv) Homogeneity ‣ §23.10 Addition Theorems and Other Identities ‣ Weierstrass Elliptic Functions ‣ Chapter 23 Weierstrass Elliptic and Modular Functions
§23.12 Asymptotic Approximations ‣ Weierstrass Elliptic Functions ‣ Chapter 23 Weierstrass Elliptic and Modular Functions
§23.21(i) Classical Dynamics ‣ §23.21 Physical Applications ‣ Applications ‣ 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
§23.2(iii) Periodicity ‣ §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(ii) Jacobian Elliptic Functions ‣ §23.6 Relations to Other Functions ‣ Weierstrass Elliptic Functions ‣ Chapter 23 Weierstrass Elliptic and Modular Functions
§23.6(iii) General Elliptic Functions ‣ §23.6 Relations to Other Functions ‣ 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
§23.6(ii) Jacobian Elliptic Functions ‣ §23.6 Relations to Other Functions ‣ 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
Rectangular Lattice ‣ §23.6(iv) Elliptic Integrals ‣ §23.6 Relations to Other Functions ‣ Weierstrass Elliptic Functions ‣ Chapter 23 Weierstrass Elliptic and Modular Functions
§23.7 Quarter Periods ‣ Weierstrass Elliptic Functions ‣ Chapter 23 Weierstrass Elliptic and Modular Functions
§23.8(iii) Infinite Products ‣ §23.8 Trigonometric Series and Products ‣ Weierstrass Elliptic Functions ‣ Chapter 23 Weierstrass Elliptic and Modular Functions
§23.8(ii) Series of Cosecants and Cotangents ‣ §23.8 Trigonometric Series and Products ‣ Weierstrass Elliptic Functions ‣ Chapter 23 Weierstrass Elliptic and Modular Functions
§23.8(i) Fourier Series ‣ §23.8 Trigonometric Series and Products ‣ Weierstrass Elliptic Functions ‣ Chapter 23 Weierstrass Elliptic and Modular Functions
Chapter 23 Weierstrass Elliptic and Modular Functions