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21: 14.8 Behavior at Singularities
14.8.7 P ν μ ( x ) 1 Γ ( 1 μ ) ( 2 x 1 ) μ / 2 , μ 1 , 2 , 3 , ,
14.8.8 P ν m ( x ) Γ ( ν + m + 1 ) m ! Γ ( ν m + 1 ) ( x 1 2 ) m / 2 , m = 1 , 2 , 3 , , ν ± m 1 , 2 , 3 , ,
14.8.11 𝑸 ν μ ( x ) Γ ( μ ) 2 Γ ( ν + μ + 1 ) ( 2 x 1 ) μ / 2 , μ > 0 , ν + μ 1 , 2 , 3 , .
22: 14.4 Graphics
§14.4(iii) Associated Legendre Functions: 2D Graphs
§14.4(iv) Associated Legendre Functions: 3D Surfaces
See accompanying text
Figure 14.4.32: 𝑸 0 μ ( x ) , 0 μ 10 , 1 < x < 10 . Magnify 3D Help
23: 14 Legendre and Related Functions
Chapter 14 Legendre and Related Functions
24: 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. …
25: 14.14 Continued Fractions
§14.14 Continued Fractions
14.14.1 1 2 ( x 2 1 ) 1 / 2 P ν μ ( x ) P ν μ 1 ( x ) = x 0 y 0 + x 1 y 1 + x 2 y 2 + ,
14.14.3 ( ν μ ) Q ν μ ( x ) Q ν 1 μ ( x ) = x 0 y 0 x 1 y 1 x 2 y 2 , ν μ ,
26: 29 Lamé Functions
Chapter 29 Lamé Functions
27: Howard S. Cohl
Cohl has published papers in orthogonal polynomials and special functions, and is particularly interested in fundamental solutions of linear partial differential equations on Riemannian manifolds, associated Legendre functions, generalized and basic hypergeometric functions, eigenfunction expansions of fundamental solutions in separable coordinate systems for linear partial differential equations, orthogonal polynomial generating function and generalized expansions, and q -series. …
28: 14.23 Values on the Cut
§14.23 Values on the Cut
14.23.5 𝖰 ν μ ( x ) = 1 2 Γ ( ν + μ + 1 ) ( e μ π i / 2 𝑸 ν μ ( x + i 0 ) + e μ π i / 2 𝑸 ν μ ( x i 0 ) ) ,
14.23.6 𝖰 ν μ ( x ) = e μ π i / 2 Γ ( ν + μ + 1 ) 𝑸 ν μ ( x ± i 0 ) ± 1 2 π i e ± μ π i / 2 P ν μ ( x ± i 0 ) .
14.23.7 𝖰 ^ 1 2 + i τ μ ( x ) = 1 2 e 3 μ π i / 2 Q 1 2 + i τ μ ( x i 0 ) + 1 2 e 3 μ π i / 2 Q 1 2 i τ μ ( x + i 0 ) .
29: 14.19 Toroidal (or Ring) Functions
Most required properties of toroidal functions come directly from the results for P ν μ ( x ) and 𝑸 ν μ ( x ) . …
14.19.4 P n 1 2 m ( cosh ξ ) = Γ ( n + m + 1 2 ) ( sinh ξ ) m 2 m π 1 / 2 Γ ( n m + 1 2 ) Γ ( m + 1 2 ) 0 π ( sin ϕ ) 2 m ( cosh ξ + cos ϕ sinh ξ ) n + m + ( 1 / 2 ) d ϕ ,
14.19.5 𝑸 n 1 2 m ( cosh ξ ) = Γ ( n + 1 2 ) Γ ( n + m + 1 2 ) Γ ( n m + 1 2 ) 0 cosh ( m t ) ( cosh ξ + cosh t sinh ξ ) n + ( 1 / 2 ) d t , m < n + 1 2 .
30: 14.10 Recurrence Relations and Derivatives
§14.10 Recurrence Relations and Derivatives
14.10.1 𝖯 ν μ + 2 ( x ) + 2 ( μ + 1 ) x ( 1 x 2 ) 1 / 2 𝖯 ν μ + 1 ( x ) + ( ν μ ) ( ν + μ + 1 ) 𝖯 ν μ ( x ) = 0 ,
14.10.3 ( ν μ + 2 ) 𝖯 ν + 2 μ ( x ) ( 2 ν + 3 ) x 𝖯 ν + 1 μ ( x ) + ( ν + μ + 1 ) 𝖯 ν μ ( x ) = 0 ,
14.10.6 P ν μ + 2 ( x ) + 2 ( μ + 1 ) x ( x 2 1 ) 1 / 2 P ν μ + 1 ( x ) ( ν μ ) ( ν + μ + 1 ) P ν μ ( x ) = 0 ,
14.10.7 ( x 2 1 ) 1 / 2 P ν μ + 1 ( x ) ( ν μ + 1 ) P ν + 1 μ ( x ) + ( ν + μ + 1 ) x P ν μ ( x ) = 0 .