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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: 14.32 Methods of Computation
§14.32 Methods of Computation
3: 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
4: 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
5: 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
6: 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 ) ,
7: 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
8: 14.18 Sums
§14.18 Sums
§14.18(ii) Addition Theorems
14.18.3 Q ν ( cos θ 1 cos θ 2 + sin θ 1 sin θ 2 cos ϕ ) = P ν ( cos θ 1 ) Q ν ( cos θ 2 ) + 2 m = 1 ( - 1 ) m P ν - m ( cos θ 1 ) Q ν m ( cos θ 2 ) cos ( m ϕ ) .
The formulas are also valid with the Ferrers functions as in §14.3(i) with μ = 0 . …
9: 14.13 Trigonometric Expansions
§14.13 Trigonometric Expansions
14.13.1 P ν μ ( cos θ ) = 2 μ + 1 ( sin θ ) μ π 1 / 2 k = 0 Γ ( ν + μ + k + 1 ) Γ ( ν + k + 3 2 ) ( μ + 1 2 ) k k ! sin ( ( ν + μ + 2 k + 1 ) θ ) ,
14.13.2 Q ν μ ( cos θ ) = π 1 / 2 2 μ ( sin θ ) μ k = 0 Γ ( ν + μ + k + 1 ) Γ ( ν + k + 3 2 ) ( μ + 1 2 ) k k ! cos ( ( ν + μ + 2 k + 1 ) θ ) .
14.13.3 P n ( cos θ ) = 2 2 n + 2 ( n ! ) 2 π ( 2 n + 1 ) ! k = 0 1 3 ( 2 k - 1 ) k ! ( n + 1 ) ( n + 2 ) ( n + k ) ( 2 n + 3 ) ( 2 n + 5 ) ( 2 n + 2 k + 1 ) sin ( ( n + 2 k + 1 ) θ ) ,
14.13.4 Q n ( cos θ ) = 2 2 n + 1 ( n ! ) 2 ( 2 n + 1 ) ! k = 0 1 3 ( 2 k - 1 ) k ! ( n + 1 ) ( n + 2 ) ( n + k ) ( 2 n + 3 ) ( 2 n + 5 ) ( 2 n + 2 k + 1 ) cos ( ( n + 2 k + 1 ) θ ) ,
10: 14.11 Derivatives with Respect to Degree or Order
§14.11 Derivatives with Respect to Degree or Order
14.11.2 ν Q ν μ ( x ) = - 1 2 π 2 P ν μ ( x ) + π sin ( μ π ) sin ( ν π ) sin ( ( ν + μ ) π ) Q ν μ ( x ) - 1 2 cot ( ( ν + μ ) π ) A ν μ ( x ) + 1 2 csc ( ( ν + μ ) π ) A ν μ ( - x ) ,