Legendre relation
(0.003 seconds)
41—50 of 74 matching pages
41: 19.14 Reduction of General Elliptic Integrals
§19.14 Reduction of General Elliptic Integrals
… ►Legendre (1825–1832) showed that every elliptic integral can be expressed in terms of the three integrals in (19.1.2) supplemented by algebraic, logarithmic, and trigonometric functions. The classical method of reducing (19.2.3) to Legendre’s integrals is described in many places, especially Erdélyi et al. (1953b, §13.5), Abramowitz and Stegun (1964, Chapter 17), and Labahn and Mutrie (1997, §3). The last reference gives a clear summary of the various steps involving linear fractional transformations, partial-fraction decomposition, and recurrence relations. …42: 29.14 Orthogonality
…
►First, the orthogonality relations (29.3.19) apply; see §29.12(i).
…
►
29.14.2
…
►
29.14.3
…
►
29.14.4
…
►
29.14.11
…
43: 22.11 Fourier and Hyperbolic Series
44: Bibliography S
…
►
A method of generating integral relations by the simultaneous separability of generalized Schrödinger equations.
SIAM J. Math. Anal. 10 (4), pp. 823–838.
…
►
Chebyshev expansions for the error and related functions.
Math. Comp. 32 (144), pp. 1232–1240.
…
►
Bounds on differences of adjacent zeros of Bessel functions and iterative relations between consecutive zeros.
Math. Comp. 70 (235), pp. 1205–1220.
…
►
Structure of avoided crossings for eigenvalues related to equations of Heun’s class.
J. Phys. A 30 (2), pp. 673–687.
…
►
Integral equations and relations for Lamé functions and ellipsoidal wave functions.
Proc. Cambridge Philos. Soc. 64, pp. 113–126.
…
45: 14.32 Methods of Computation
§14.32 Methods of Computation
… ►In other cases recurrence relations (§14.10) provide a powerful method when applied in a stable direction (§3.6); see Olver and Smith (1983) and Gautschi (1967). …46: Bille C. Carlson
…
►This symmetry led to the development of symmetric elliptic integrals, which are free from the transformations of modulus and amplitude that complicate the Legendre theory.
…
►In Symmetry in c, d, n of Jacobian elliptic functions (2004) he found a previously hidden symmetry in relations between Jacobian elliptic functions, which can now take a form that remains valid when the letters c, d, and n are permuted.
This invariance usually replaces sets of twelve equations by sets of three equations and applies also to the relation between the first symmetric elliptic integral and the Jacobian functions.
…
47: 20.9 Relations to Other Functions
§20.9 Relations to Other Functions
►§20.9(i) Elliptic Integrals
… ►§20.9(ii) Elliptic Functions and Modular Functions
►See §§22.2 and 23.6(i) for the relations of Jacobian and Weierstrass elliptic functions to theta functions. … ►§20.9(iii) Riemann Zeta Function
…48: 18.5 Explicit Representations
…
►Related formula:
…
►
§18.5(iii) Finite Power Series, the Hypergeometric Function, and Generalized Hypergeometric Functions
… ►Laguerre
… ►Hermite
… ►Legendre
…49: 18.30 Associated OP’s
…
►Associated polynomials and the related corecursive polynomials appear in Ismail (2009, §§2.3, 2.6, and 2.10), where the relationship of OP’s to continued fractions is made evident.
…
►