Legendre%0Aelliptic%20integrals
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11: 36.2 Catastrophes and Canonical Integrals
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Canonical Integrals
… ►with the contour passing to the lower right of . …with the contour passing to the upper right of . … ► … ►§36.2(iii) Symmetries
…12: 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 ). …13: 14.21 Definitions and Basic Properties
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§14.21(i) Associated Legendre Equation
… ► ►§14.21(ii) Numerically Satisfactory Solutions
►When and , a numerically satisfactory pair of solutions of (14.21.1) in the half-plane is given by and . ►§14.21(iii) Properties
…14: 14.6 Integer Order
15: 14.4 Graphics
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§14.4(iii) Associated Legendre Functions: 2D Graphs
… ► … ► ►§14.4(iv) Associated Legendre Functions: 3D Surfaces
… ►16: 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
… ►17: 14.5 Special Values
18: 14.18 Sums
§14.18 Sums
… ► … ►In these formulas the Legendre functions are as in §14.3(ii) with . … ►Dougall’s Expansion
… ►19: 14.2 Differential Equations
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§14.2(i) Legendre’s Equation
… ►§14.2(ii) Associated Legendre Equation
… ►Ferrers functions and the associated Legendre functions are related to the Legendre functions by the equations , , , , . … ►§14.2(iii) Numerically Satisfactory Solutions
… ►When and , and are linearly independent, and recessive at and , respectively. …20: 14.33 Tables
§14.33 Tables
►Abramowitz and Stegun (1964, Chapter 8) tabulates for , , 5–8D; for , , 5–7D; and for , , 6–8D; and for , , 6S; and for , , 6S. (Here primes denote derivatives with respect to .)
Zhang and Jin (1996, Chapter 4) tabulates for , , 7D; for , , 8D; for , , 8S; for , , 8D; for , , , , 8S; for , , 8S; for , , , 5D; for , , 7S; for , , 8S. Corresponding values of the derivative of each function are also included, as are 6D values of the first 5 -zeros of and of its derivative for , .
Belousov (1962) tabulates (normalized) for , , , 6D.