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Legendre functions

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11: 14.7 Integer Degree and Order
§14.7 Integer Degree and Order
§14.7(i) μ = 0
§14.7(ii) Rodrigues-Type Formulas
§14.7(iv) Generating Functions
12: 14.18 Sums
§14.18 Sums
§14.18(ii) Addition Theorems
Dougall’s Expansion
13: 14.3 Definitions and Hypergeometric Representations
§14.3(ii) Interval 1 < x <
Associated Legendre Function of the First Kind
Associated Legendre Function of the Second Kind
§14.3(iv) Relations to Other Functions
14: 14.5 Special Values
14.5.9 Q 0 ( x ) = 1 2 ln ( x + 1 x - 1 ) ,
14.5.10 Q 1 ( x ) = x 2 ln ( x + 1 x - 1 ) - 1 .
§14.5(v) μ = 0 , ν = ± 1 2
14.5.30 Q 2 ( x ) = 3 x 2 - 1 8 ln ( x + 1 x - 1 ) - 3 4 x .
15: 14.32 Methods of Computation
§14.32 Methods of Computation
16: 14.25 Integral Representations
§14.25 Integral Representations
14.25.1 P ν - μ ( z ) = ( z 2 - 1 ) μ / 2 2 ν Γ ( μ - ν ) Γ ( ν + 1 ) 0 ( sinh t ) 2 ν + 1 ( z + cosh t ) ν + μ + 1 d t , μ > ν > - 1 ,
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 ,
17: 14.12 Integral Representations
§14.12(ii) 1 < x <
14.12.5 P ν - μ ( x ) = ( x 2 - 1 ) - μ / 2 Γ ( μ ) 1 x P ν ( t ) ( x - t ) μ - 1 d t , μ > 0 .
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 .
Neumann’s Integral
Heine’s Integral
18: 14.24 Analytic Continuation
§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 ) ,
14.24.4 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 ) ,
For fixed z , other than ± 1 or , each branch of P ν - μ ( z ) and Q ν μ ( z ) is an entire function of each parameter ν and μ . …
19: 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 , ) . …
14.2.11 P ν + 1 μ ( x ) Q ν μ ( x ) - P ν μ ( x ) Q ν + 1 μ ( x ) = e μ π i Γ ( ν + μ + 1 ) Γ ( ν - μ + 2 ) .
20: 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 , .