Contenuti:
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.
Le informazioni nella sezione "Su questo libro" possono far riferimento a edizioni diverse di questo titolo.