Handbook of Elliptic Integrals for Engineers and Scientists

Engineers and physicists are more and more encountering integrations involving nonelementary integrals and higher transcendental functions. Such integrations frequently involve (not always in immediately re­ cognizable form) elliptic functions and elliptic integrals. The numerous books written on elliptic integrals, while of great value to the student or mathematician, are not especially suitable for the scientist whose primary objective is the ready evaluation of the integrals that occur in his practical problems. As a result, he may entirely avoid problems which lead to elliptic integrals, or is likely to resort to graphical methods or other means of approximation in dealing with all but the simplest of these integrals. It became apparent in the course of my work in theoretical aero­ dynamics that there was a need for a handbook embodying in convenient form a comprehensive table of elliptic integrals together with auxiliary formulas and numerical tables of values. Feeling that such a book would save the engineer and physicist much valuable time, I prepared the present volume.

Definitions and Fundamental Relations.- 110. Elliptic Integrals.- 120. Jacobian Elliptic Functions.- 130. Jacobi’s Inverse Elliptic Functions.- 140. Jacobian Zeta Function.- 150. Heuman’s Lambda Function.- 160. Transformation Formulas for Elliptic Functions and Elliptic Integrals.- Reduction of Algebraic Integrands to Jacobian Elliptic Functions.- 200. Introduction.- 210. Integrands Involving Square Roots of Sums and Differences of Squares.- 230. Integrands Involving the Square root of a Cubic.- 250. Integrands Involving the Square root of a Quartic.- 270. Integrands Involving Miscellaneous Fractional Powers of Polynomials.- Reduction of Trigonometric Integrands to Jacobian Elliptic Functions.- Reduction of Hyperbolic Integrands to Jacobian Elliptic Functions.- Tables of Integrals of Jacobian Elliptic Functions.- 310. Recurrence Formulas for the Integrals of the Twelve Jacobian Elliptic Functions.- 330. Additional Recurrence Formulas.- 360. Integrands Involving Various Combinations of Jacobian Elliptic Functions.- 390. Integrals of Jacobian Inverse Elliptic Functions.- Elliptic Integrals of the Third Kind.- 400. Introduction.- 410. Table of Integrals.- Table of Miscellaneous Elliptic Integrals Involving Trigonometric or Hyperbolic Integrands.- 510. Single Integrals.- 530. Multiple Integrals.- Elliptic Integrals Resulting from Laplace Transformations.- Hyperelliptic Integrals.- 575. Introduction.- 576. Table of Integrals.- Integrals of the Elliptic Integrals.- 610. With Respect to the Modulus.- 630. With Respect to the Argument.- Derivatives.- 710. With Respect to the Modulus.- 730. With Respect to the Argument.- 733. With Respect to the Parameter.- Miscellaneous Integrals and Formulas.- Expansions in Series.- 900. Developments of the Elliptic Integrals.- 907.Developments of Jacobian Elliptic Functions.- 1030. Weierstrassian Elliptic Functions and Elliptic Integrals.- Definition, p. 308. — Relation to Jacobian elliptic functions, p. 309. — Fundamental relations, p. 309. — Derivatives, p. 309. — Special values, p. 310. — Addition formulas, p. 310. — Relation to Theta functions, p. 310. — Weierstrassian normal elliptic integrals, p. 311. — Other integrals, p. 312. — Illustrative example, p. 313..- 1050. Theta Functions.- Definitions, p. 315. — Special values, p. 316. — Quasi-Addition Formulas, p.317. — Differential equation, p. 317. — Relation to Jacobian elliptic functions, p. 318. — Relation to elliptic integrals, p. 318..- 1060. Pseudo-elliptic Integrals.- Definition, p. 320. — Examples, p. 321..- Table of Numerical Values.- Supplementary Bibliography.