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Ferrers functions

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1: 14.1 Special Notation
§14.1 Special Notation
The main functions treated in this chapter are the Legendre functions P ν ( x ) , Q ν ( x ) , P ν ( z ) , Q ν ( z ) ; Ferrers functions P ν μ ( x ) , Q ν μ ( x ) (also known as the Legendre functions on the cut); associated Legendre functions P ν μ ( z ) , Q ν μ ( z ) , Q ν μ ( z ) ; conical functions P - 1 2 + i τ μ ( x ) , Q - 1 2 + i τ μ ( x ) , Q ^ - 1 2 + i τ μ ( x ) , P - 1 2 + i τ μ ( x ) , Q - 1 2 + i τ μ ( x ) (also known as Mehler functions). …
2: 14.3 Definitions and Hypergeometric Representations
§14.3 Definitions and Hypergeometric Representations
Ferrers Function of the First Kind
14.3.1 P ν μ ( x ) = ( 1 + x 1 - x ) μ / 2 F ( ν + 1 , - ν ; 1 - μ ; 1 2 - 1 2 x ) .
Ferrers Function of the Second Kind
14.3.2 Q ν μ ( x ) = π 2 sin ( μ π ) ( cos ( μ π ) ( 1 + x 1 - x ) μ / 2 F ( ν + 1 , - ν ; 1 - μ ; 1 2 - 1 2 x ) - Γ ( ν + μ + 1 ) Γ ( ν - μ + 1 ) ( 1 - x 1 + x ) μ / 2 F ( ν + 1 , - ν ; 1 + μ ; 1 2 - 1 2 x ) ) .
3: 14.20 Conical (or Mehler) Functions
For - 1 < x < 1 and τ > 0 , a numerically satisfactory pair of real conical functions is P - 1 2 + i τ - μ ( x ) and P - 1 2 + i τ - μ ( - x ) . …
14.20.2 Q ^ - 1 2 + i τ - μ ( x ) = ( e μ π i Q - 1 2 + i τ - μ ( x ) ) - 1 2 π sin ( μ π ) P - 1 2 + i τ - μ ( x ) .
14.20.4 𝒲 { P - 1 2 + i τ - μ ( x ) , P - 1 2 + i τ - μ ( - x ) } = 2 | Γ ( μ + 1 2 + i τ ) | 2 ( 1 - x 2 ) .
14.20.7 Q ^ - 1 2 + i τ μ ( x ) 1 2 Γ ( μ ) ( 2 1 - x ) μ / 2 ,
14.20.22 P - 1 2 + i τ - μ ( x ) = β exp ( μ β arctan β ) Γ ( μ + 1 ) ( 1 + β 2 ) μ / 2 e - μ ρ ( 1 + β 2 - x 2 β 2 ) 1 / 4 ( 1 + O ( 1 μ ) ) ,
4: 14.2 Differential Equations
§14.2(ii) Associated Legendre Equation
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 ) . … Unless stated otherwise in §§14.214.20 it is assumed that the arguments of the functions P ν μ ( x ) and Q ν μ ( x ) lie in the interval ( - 1 , 1 ) , and the arguments of the functions P ν μ ( x ) , Q ν μ ( x ) , and Q ν μ ( x ) lie in the interval ( 1 , ) . …
§14.2(iv) Wronskians and Cross-Products
14.2.5 P ν + 1 μ ( x ) Q ν μ ( x ) - P ν μ ( x ) Q ν + 1 μ ( x ) = Γ ( ν + μ + 1 ) Γ ( ν - μ + 2 ) ,
5: 14.32 Methods of Computation
§14.32 Methods of Computation
6: 14.7 Integer Degree and Order
§14.7 Integer Degree and Order
§14.7(ii) Rodrigues-Type Formulas
§14.7(iii) Reflection Formulas
§14.7(iv) Generating Functions
7: 14.17 Integrals
§14.17(i) Indefinite Integrals
§14.17(ii) Barnes’ Integral
§14.17(iii) Orthogonality Properties
§14.17(v) Laplace Transforms
§14.17(vi) Mellin Transforms
8: 14.5 Special Values
§14.5 Special Values
14.5.7 Q 0 ( x ) = 1 2 ln ( 1 + x 1 - x ) ,
14.5.8 Q 1 ( x ) = x 2 ln ( 1 + x 1 - x ) - 1 .
§14.5(v) μ = 0 , ν = ± 1 2
9: 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 ,
14.10.2 ( 1 - x 2 ) 1 / 2 P ν μ + 1 ( x ) - ( ν - μ + 1 ) P ν + 1 μ ( x ) + ( ν + μ + 1 ) x 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 ,
14.10.4 ( 1 - x 2 ) d P ν μ ( x ) d x = ( μ - ν - 1 ) P ν + 1 μ ( x ) + ( ν + 1 ) x P ν μ ( x ) ,
10: 14.4 Graphics
§14.4(i) Ferrers Functions: 2D Graphs
§14.4(ii) Ferrers Functions: 3D Surfaces
See accompanying text
Figure 14.4.15: P 0 - μ ( x ) , 0 μ 10 , - 1 < x < 1 . Magnify 3D Help
See accompanying text
Figure 14.4.16: Q 0 μ ( x ) , 0 μ 6.2 , - 1 < x < 1 . Magnify 3D Help