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1: 14 Legendre and Related Functions
Chapter 14 Legendre and Related Functions
2: 15 Hypergeometric Function
3: T. Mark Dunster
4: 19.35 Other Applications
Generalizations of elliptic integrals appear in analysis of modular theorems of Ramanujan (Anderson et al. (2000)); analysis of Selberg integrals (Van Diejen and Spiridonov (2001)); use of Legendre’s relation (19.7.1) to compute π to high precision (Borwein and Borwein (1987, p. 26)). …
5: 15.16 Products
Generalized Legendre’s Relation
6: 19.10 Relations to Other Functions
§19.10(i) Theta and Elliptic Functions
For relations of Legendre’s integrals to theta functions, Jacobian functions, and Weierstrass functions, see §§20.9(i), 22.15(ii), and 23.6(iv), respectively. …
7: 19.7 Connection Formulas
Legendre’s Relation
§19.7(iii) Change of Parameter of Π ( ϕ , α 2 , k )
The third relation (missing from the literature of Legendre’s integrals) maps each circular region onto the other and each hyperbolic region onto the other: …
8: 14.7 Integer Degree and Order
§14.7(i) μ = 0
9: 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
10: 34.3 Basic Properties: 3 j Symbol
§34.3(vii) Relations to Legendre Polynomials and Spherical Harmonics