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Ferrers functions

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21: 29.8 Integral Equations
29.8.2 μ w ( z 1 ) w ( z 2 ) w ( z 3 ) = - 2 K 2 K P ν ( x ) w ( z ) d z ,
where P ν ( x ) is the Ferrers function of the first kind (§14.3(i)), …
29.8.5 Ec ν 2 m ( z 1 , k 2 ) w 2 ( K ) - w 2 ( - K ) d w 2 ( z ) / d z | z = 0 = - K K P ν ( y ) Ec ν 2 m ( z , k 2 ) d z ,
29.8.7 Ec ν 2 m + 1 ( z 1 , k 2 ) w 2 ( K ) + w 2 ( - K ) w 2 ( 0 ) = - k 2 sn ( z 1 , k ) - K K sn ( z , k ) d P ν ( y ) d y Ec ν 2 m + 1 ( z , k 2 ) d z ,
29.8.9 Es ν 2 m + 2 ( z 1 , k 2 ) d w 2 ( z ) / d z | z = K - d w 2 ( z ) / d z | z = - K w 2 ( 0 ) = - k 4 k sn ( z 1 , k ) cn ( z 1 , k ) - K K sn ( z , k ) cn ( z , k ) d 2 P ν ( y ) d y 2 Es ν 2 m + 2 ( z , k 2 ) d z .
22: 16.18 Special Cases
As a corollary, special cases of the F 1 1 and F 1 2 functions, including Airy functions, Bessel functions, parabolic cylinder functions, Ferrers functions, associated Legendre functions, and many orthogonal polynomials, are all special cases of the Meijer G -function. …
23: 30.17 Tables
§30.17 Tables
24: 14.15 Uniform Asymptotic Approximations
§14.15 Uniform Asymptotic Approximations
§14.15(i) Large μ , Fixed ν
For asymptotic expansions and explicit error bounds, see Dunster (2003b).
§14.15(iii) Large ν , Fixed μ
25: 14.30 Spherical and Spheroidal Harmonics
§14.30 Spherical and Spheroidal Harmonics
14.30.2 Y l m ( θ , ϕ ) = cos ( m ϕ ) P l m ( cos θ )  or  sin ( m ϕ ) P l m ( cos θ ) .
Most mathematical properties of Y l , m ( θ , ϕ ) can be derived directly from (14.30.1) and the properties of the Ferrers function of the first kind given earlier in this chapter. …
14.30.9 P l ( cos θ 1 cos θ 2 + sin θ 1 sin θ 2 cos ( ϕ 1 - ϕ 2 ) ) = 4 π 2 l + 1 m = - l l Y l , m ( θ 1 , ϕ 1 ) ¯ Y l , m ( θ 2 , ϕ 2 ) .
26: 14.31 Other Applications
The conical functions P - 1 2 + i τ m ( x ) appear in boundary-value problems for the Laplace equation in toroidal coordinates (§14.19(i)) for regions bounded by cones, by two intersecting spheres, or by one or two confocal hyperboloids of revolution (Kölbig (1981)). …
27: 15.9 Relations to Other Functions
§15.9(iv) Associated Legendre Functions; Ferrers Functions
Any hypergeometric function for which a quadratic transformation exists can be expressed in terms of associated Legendre functions or Ferrers functions. …
28: 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 ) ) ,
29: 14.16 Zeros
§14.16(ii) Interval - 1 < x < 1
30: 30.5 Functions of the Second Kind