Legendre
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1: 14.1 Special Notation
§14.1 Special Notation
… ►The main functions treated in this chapter are the Legendre functions , , , ; Ferrers functions , (also known as the Legendre functions on the cut); associated Legendre functions , , ; conical functions , , , , (also known as Mehler functions). … ►Magnus et al. (1966) denotes , , , and by , , , and , respectively. Hobson (1931) denotes both and by ; similarly for and .2: 18.3 Definitions
§18.3 Definitions
… ►This table also includes the following special cases of Jacobi polynomials: ultraspherical, Chebyshev, and Legendre. … ►For expressions of ultraspherical, Chebyshev, and Legendre polynomials in terms of Jacobi polynomials, see §18.7(i). … ►Legendre
►Legendre polynomials are special cases of Legendre functions, Ferrers functions, and associated Legendre functions (§14.7(i)). …3: 19.2 Definitions
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§19.2(ii) Legendre’s Integrals
… ►Also, if and are real, then is called a circular or hyperbolic case according as is negative or positive. … ►Legendre’s complementary complete elliptic integrals are defined via … ►Bulirsch’s integrals are linear combinations of Legendre’s integrals that are chosen to facilitate computational application of Bartky’s transformation (Bartky (1938)). … ►Lastly, corresponding to Legendre’s incomplete integral of the third kind we have …4: 27.9 Quadratic Characters
§27.9 Quadratic Characters
►For an odd prime , the Legendre symbol is defined as follows. If divides , then the value of is . If does not divide , then has the value when the quadratic congruence has a solution, and the value when this congruence has no solution. The Legendre symbol , as a function of , is a Dirichlet character (mod ). …5: 14.21 Definitions and Basic Properties
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§14.21(i) Associated Legendre Equation
… ►Standard solutions: the associated Legendre functions , , , and . … ►§14.21(ii) Numerically Satisfactory Solutions
… ►§14.21(iii) Properties
…6: 14.7 Integer Degree and Order
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§14.7(i)
… ►where is the Legendre polynomial of degree . … ►§14.7(ii) Rodrigues-Type Formulas
… ►§14.7(iv) Generating Functions
… ►7: 14.26 Uniform Asymptotic Expansions
§14.26 Uniform Asymptotic Expansions
►The uniform asymptotic approximations given in §14.15 for and for are extended to domains in the complex plane in the following references: §§14.15(i) and 14.15(ii), Dunster (2003b); §14.15(iii), Olver (1997b, Chapter 12); §14.15(iv), Boyd and Dunster (1986). …8: 14.6 Integer Order
9: 14.31 Other Applications
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►The conical functions appear in boundary-value problems for the Laplace equation in toroidal coordinates (§14.19(i)) for regions bounded by cones, by two intersecting spheres, or by one or two confocal hyperboloids of revolution (Kölbig (1981)).
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