Handbook of Elliptic Integrals for Engineers and Scientists

ISBN-13: 9783642651403 / Angielski / Miękka / 2013 / 360 str.

Paul F. Byrd;Morris David Friedman

Opis

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."

Spis treści

Definitions and Fundamental Relations.- 110. Elliptic Integrals.- Definitions, p. 8. — Legendre’s relation, p. 10. — Special values, p. 10. — Limiting values, p. 11. — Extension of the range of ? and k, p. 12. — Addition formulas p. 13. — Special addition formulas, p. 13. — Differential equations, p. 15. — Sketches of E(?, k), F(?, k), E(k) and K(k), p. 16. — Conformal Mappings, p. 17.- 120. Jacobian Elliptic Functions.- Definitions, p. 18. — Fundamental relations, p. 20. — Special values, p. 20. — Addition formulas, p. 23. — Double and half arguments, p. 24. — Complex and imaginary arguments, p. 24. — Relation to Theta functions, p. 24. — Approximation formulas, p. 24. — Differential equations, p. 25. — Identities, p. 25. — Sketches, p. 26. — Conformal Mappings, p. 28. — Applications, p. 28.- 130. Jacobi’s Inverse Elliptic Functions.- Definitions, p. 29. — Identities, p. 31. — Special values, p. 31. — Addition formulas, p. 32. — Special addition formulas, p. 32.- 140. Jacobian Zeta Function.- Definitions, p. 33. — Special values, p. 33. Maximum value, p. 34. — Limiting value, p. 34. — Approximation formula, p. 34. — Addition formulas, p. 34. — Special addition formula, p. 34. — Complex and imaginary arguments, p. 34. — Relation to Theta functions, p. 34. — Sketches, p. 35.- 150. Heuman’s Lambda Function.- Definitions, p. 35. — Special values, p. 36. — Limiting value, p. 36. — Addition formula, p. 36. — Special addition formulas, p. 36. — Relation to Theta functions, p. 37. — Sketches, p. 37.- 160. Transformation Formulas for Elliptic Functions and Elliptic Integrals.- Imaginary modulus transformation, p. 38. — Imaginary argument transformation, p. 38. — Reciprocal modulus transformation, p. 38. — Landen’s transformation, p. 39. — Gauss’ transformation, p. 39. — Other transformations, p. 40.- Reduction of Algebraic Integrands to Jacobian Elliptic Functions.- 200. Introduction.- 210. Integrands Involving Square Roots of Sums and Differences of Squares.- Introduction, p. 43. — Table of Integrals, p. 45.- 230. Integrands Involving the Square root of a Cubic.- p. 65. — Table of Integrals p. 68.- 250. Integrands Involving the Square root of a Quartic.- p. 95. — Table of Integrals p. 98.- 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.- Complete integrals, p. 225. — Incomplete integrals, p. 232.- 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.- Differentiation of the elliptic integrals, p. 282. Differentiation of the Jacobian elliptic functions, p. 283.- 730. With Respect to the Argument.- Differentiation of the elliptic integrals, p. 284. — Differentiation of the Jacobian elliptic functions, p. 284. — Differentiation of the Jacobian inverse functions, p. 285.- 733. With Respect to the Parameter.- Differentiation of the normal elliptic integral of the third kind, p. 286.-Differentiation of other elliptic integrals, p. 287.- Miscellaneous Integrals and Formulas.- Expansions in Series.- 900. Developments of the Elliptic Integrals.- Complete elliptic integrals of the first and second kind, p. 298. — The nome, p. 300. — Incomplete elliptic integrals of the first and second kind, p. 300. — Heuman’s function, p. 301. — Jacobian Zeta function, p. 301. — The elliptic integral of the third kind, p. 302.- 907. Developments of Jacobian Elliptic Functions.- Maclaurin’s series, p. 303. — Fourier series, p. 304. — Infinite products, P. 306. — Other developments, p. 307.- 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.