DLMF:14.3.E23 (Q4712)

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DLMF:14.3.E23
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    Statements

    Defining formula
    P ν μ ⁡ ( x ) = 1 Γ ⁡ ( 1 - μ ) ⁢ ( x + 1 x - 1 ) μ / 2 ⁢ ϕ - i ⁢ ( 2 ⁢ ν + 1 ) ( - μ , μ ) ⁡ ( arcsinh ⁡ ( ( 1 2 ⁢ x - 1 2 ) 1 / 2 ) ) . Legendre-P-first-kind 𝜇 𝜈 𝑥 1 Euler-Gamma 1 𝜇 superscript 𝑥 1 𝑥 1 𝜇 2 Jacobi-hypergeometric-phi 𝜇 𝜇 imaginary-unit 2 𝜈 1 hyperbolic-inverse-sine superscript 1 2 𝑥 1 2 1 2 {\displaystyle{\displaystyle P^{\mu}_{\nu}\left(x\right)=\frac{1}{\Gamma\left(% 1-\mu\right)}\left(\frac{x+1}{x-1}\right)^{\mu/2}\phi^{(-\mu,\mu)}_{-\mathrm{i% }(2\nu+1)}\left(\operatorname{arcsinh}\left((\tfrac{1}{2}x-\tfrac{1}{2})^{% \ifrac{1}{2}}\right)\right).}}
    0 references
    DLMF id
    DLMF:14.3.E23
    0 references
    Symbols used
    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2aodec
    0 references
    Jacobi function
    Defining formula
    ϕ λ ( α , β ) ⁡ ( t ) Jacobi-hypergeometric-phi 𝛼 𝛽 𝜆 𝑡 {\displaystyle{\displaystyle\phi^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{\lambda% }}\left(\NVar{t}\right)}}
    xml-id
    C15.S9.E11.m2adec
    0 references
    Q11481
    Defining formula
    P ν μ ⁡ ( z ) Legendre-P-first-kind 𝜇 𝜈 𝑧 {\displaystyle{\displaystyle P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C14.S21.SS1.p1.m1ahdec
    0 references
    Q11247
    Defining formula
    arcsinh ⁡ z hyperbolic-inverse-sine 𝑧 {\displaystyle{\displaystyle\operatorname{arcsinh}\NVar{z}}}
    xml-id
    C4.S37.SS2.p1.m8adec
    0 references
     
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