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Olver associated Legendre function

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1: 14.21 Definitions and Basic Properties
14.21.1 ( 1 - z 2 ) d 2 w d z 2 - 2 z d w d z + ( ν ( ν + 1 ) - μ 2 1 - z 2 ) w = 0 .
2: 14.24 Analytic Continuation
14.24.1 P ν - μ ( z e s π i ) = e s ν π i P ν - μ ( z ) + 2 i sin ( ( ν + 1 2 ) s π ) e - s π i / 2 cos ( ν π ) Γ ( μ - ν ) Q ν μ ( z ) ,
14.24.2 Q ν μ ( z e s π i ) = ( - 1 ) s e - s ν π i Q ν μ ( z ) ,
14.24.4 Q ν , s μ ( z ) = e - s μ π i Q ν μ ( z ) - π i sin ( s μ π ) sin ( μ π ) Γ ( ν - μ + 1 ) P ν - μ ( z ) ,
3: 14.12 Integral Representations
14.12.9 Q n m ( x ) = 1 n ! 0 u ( x - ( x 2 - 1 ) 1 / 2 cosh t ) n cosh ( m t ) d t ,
14.12.11 Q n m ( x ) = ( x 2 - 1 ) m / 2 2 n + 1 n ! - 1 1 ( 1 - t 2 ) n ( x - t ) n + m + 1 d t ,
14.12.12 Q n m ( x ) = 1 ( n - m ) ! P n m ( x ) x d t ( t 2 - 1 ) ( P n m ( t ) ) 2 , n m .
14.12.13 Q n ( x ) = 1 2 ( n ! ) - 1 1 P n ( t ) x - t d t .
4: 14.4 Graphics
§14.4(i) Ferrers Functions: 2D Graphs
§14.4(ii) Ferrers Functions: 3D Surfaces
§14.4(iii) Associated Legendre Functions: 2D Graphs
§14.4(iv) Associated Legendre Functions: 3D Surfaces
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Figure 14.4.32: Q 0 μ ( x ) , 0 μ 10 , 1 < x < 10 . Magnify 3D Help
5: 14.19 Toroidal (or Ring) Functions
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 .
14.19.6 Q - 1 2 μ ( cosh ξ ) + 2 n = 1 Γ ( μ + n + 1 2 ) Γ ( μ + 1 2 ) Q n - 1 2 μ ( cosh ξ ) cos ( n ϕ ) = ( 1 2 π ) 1 / 2 ( sinh ξ ) μ ( cosh ξ - cos ϕ ) μ + ( 1 / 2 ) , μ > - 1 2 .
6: 14.25 Integral Representations
14.25.2 Q ν μ ( z ) = π 1 / 2 ( z 2 - 1 ) μ / 2 2 μ Γ ( μ + 1 2 ) Γ ( ν - μ + 1 ) 0 ( sinh t ) 2 μ ( z + ( z 2 - 1 ) 1 / 2 cosh t ) ν + μ + 1 d t , ( ν + 1 ) > μ > - 1 2 ,
7: 14.3 Definitions and Hypergeometric Representations
14.3.8 P ν m ( x ) = Γ ( ν + m + 1 ) 2 m Γ ( ν - m + 1 ) ( x 2 - 1 ) m / 2 F ( ν + m + 1 , m - ν ; m + 1 ; 1 2 - 1 2 x ) .
14.3.19 Q ν μ ( x ) = 2 ν Γ ( ν + 1 ) ( x + 1 ) μ / 2 ( x - 1 ) ( μ / 2 ) + ν + 1 F ( ν + 1 , ν + μ + 1 ; 2 ν + 2 ; 2 1 - x ) ,
14.3.20 2 sin ( μ π ) π Q ν μ ( x ) = ( x + 1 ) μ / 2 Γ ( ν + μ + 1 ) ( x - 1 ) μ / 2 F ( ν + 1 , - ν ; 1 - μ ; 1 2 - 1 2 x ) - ( x - 1 ) μ / 2 Γ ( ν - μ + 1 ) ( x + 1 ) μ / 2 F ( ν + 1 , - ν ; μ + 1 ; 1 2 - 1 2 x ) .
8: 14.23 Values on the Cut
14.23.3 Q ν μ ( x ± i 0 ) = e ν π i / 2 π 3 / 2 ( 1 - x 2 ) μ / 2 2 ν + 1 ( x F ( 1 2 μ - 1 2 ν + 1 2 , 1 2 ν + 1 2 μ + 1 ; 3 2 ; x 2 ) Γ ( 1 2 ν - 1 2 μ + 1 2 ) Γ ( 1 2 ν + 1 2 μ + 1 2 ) i F ( 1 2 μ - 1 2 ν , 1 2 ν + 1 2 μ + 1 2 ; 1 2 ; x 2 ) Γ ( 1 2 ν - 1 2 μ + 1 ) Γ ( 1 2 ν + 1 2 μ + 1 ) ) .
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 ) .
9: 14.8 Behavior at Singularities
14.8.9 Q ν ( x ) = - ln ( x - 1 ) 2 Γ ( ν + 1 ) + 1 2 ln 2 - γ - ψ ( ν + 1 ) Γ ( ν + 1 ) + O ( x - 1 ) , ν - 1 , - 2 , - 3 , ,
14.8.10 Q - n ( x ) ( - 1 ) n + 1 ( n - 1 ) ! , n = 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 , .
14.8.15 Q ν μ ( x ) π 1 / 2 Γ ( ν + 3 2 ) ( 2 x ) ν + 1 , ν - 3 2 , - 5 2 , - 7 2 , ,
14.8.16 Q - n - ( 1 / 2 ) μ ( x ) π 1 / 2 Γ ( μ + n + 1 2 ) n ! Γ ( μ - n + 1 2 ) ( 2 x ) n + ( 1 / 2 ) , n = 1 , 2 , 3 , , μ - n + 1 2 0 , - 1 , - 2 , .
10: 14.22 Graphics
§14.22 Graphics
In the graphics shown in this section, height corresponds to the absolute value of the function and color to the phase. …
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Figure 14.22.1: P 1 / 2 0 ( x + i y ) , - 5 x 5 , - 5 y 5 . … Magnify 3D Help
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Figure 14.22.4: Q 0 0 ( x + i y ) , - 5 x 5 , - 5 y 5 . … Magnify 3D Help