About the Project
NIST

relation to associated Legendre functions

AdvancedHelp

(0.006 seconds)

1—10 of 28 matching pages

1: 14.3 Definitions and Hypergeometric Representations
§14.3 Definitions and Hypergeometric Representations
§14.3(ii) Interval 1 < x <
§14.3(iii) Alternative Hypergeometric Representations
14.3.14 w 2 ( ν , μ , x ) = 2 μ Γ ( 1 2 ν + 1 2 μ + 1 ) Γ ( 1 2 ν - 1 2 μ + 1 2 ) x ( 1 - x 2 ) - μ / 2 F ( 1 2 - 1 2 ν - 1 2 μ , 1 2 ν - 1 2 μ + 1 ; 3 2 ; x 2 ) .
§14.3(iv) Relations to Other Functions
2: 14.7 Integer Degree and Order
§14.7(i) μ = 0
3: 14.5 Special Values
§14.5(v) μ = 0 , ν = ± 1 2
4: 15.9 Relations to Other Functions
§15.9(iv) Associated Legendre Functions; Ferrers Functions
5: 30.6 Functions of Complex Argument
Relations to Associated Legendre Functions
6: 14.2 Differential Equations
Ferrers functions and the associated Legendre functions are related to the Legendre functions by the equations P ν 0 ( x ) = P ν ( x ) , Q ν 0 ( x ) = Q ν ( x ) , P ν 0 ( x ) = P ν ( x ) , Q ν 0 ( x ) = Q ν ( x ) , Q ν 0 ( x ) = Q ν ( x ) = Q ν ( x ) / Γ ( ν + 1 ) . …
7: 14.21 Definitions and Basic Properties
§14.21(i) Associated Legendre Equation
Standard solutions: the associated Legendre functions P ν μ ( z ) , P ν - μ ( z ) , Q ν μ ( z ) , and Q - ν - 1 μ ( z ) . …
§14.21(ii) Numerically Satisfactory Solutions
§14.21(iii) Properties
8: 14 Legendre and Related Functions
Chapter 14 Legendre and Related Functions
9: 16.18 Special Cases
§16.18 Special Cases
This is a consequence of the following relations: …As a corollary, special cases of the F 1 1 and F 1 2 functions, including Airy functions, Bessel functions, parabolic cylinder functions, Ferrers functions, associated Legendre functions, and many orthogonal polynomials, are all special cases of the Meijer G -function. Representations of special functions in terms of the Meijer G -function are given in Erdélyi et al. (1953a, §5.6), Luke (1969a, §§6.4–6.5), and Mathai (1993, §3.10).
10: 34.3 Basic Properties: 3 j Symbol
§34.3(iii) Recursion Relations
§34.3(vii) Relations to Legendre Polynomials and Spherical Harmonics
For the polynomials P l see §18.3, and for the function Y l , m see §14.30. …Equations (34.3.19)–(34.3.22) are particular cases of more general results that relate rotation matrices to 3 j symbols, for which see Edmonds (1974, Chapter 4). …