DLMF:14.12.E4 (Q4825)

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DLMF:14.12.E4
No description defined

    Statements

    Defining formula
    P ν - μ ⁡ ( x ) = 2 1 / 2 ⁢ Γ ⁡ ( μ + 1 2 ) ⁢ ( x 2 - 1 ) μ / 2 π 1 / 2 ⁢ Γ ⁡ ( ν + μ + 1 ) ⁢ Γ ⁡ ( μ - ν ) ⁢ ∫ 0 ∞ cosh ⁡ ( ( ν + 1 2 ) ⁢ t ) ( x + cosh ⁡ t ) μ + ( 1 / 2 ) ⁢ d t , Legendre-P-first-kind 𝜇 𝜈 𝑥 superscript 2 1 2 Euler-Gamma 𝜇 1 2 superscript superscript 𝑥 2 1 𝜇 2 superscript 𝜋 1 2 Euler-Gamma 𝜈 𝜇 1 Euler-Gamma 𝜇 𝜈 superscript subscript 0 𝜈 1 2 𝑡 superscript 𝑥 𝑡 𝜇 1 2 𝑡 {\displaystyle{\displaystyle P^{-\mu}_{\nu}\left(x\right)=\frac{2^{1/2}\Gamma% \left(\mu+\frac{1}{2}\right)\left(x^{2}-1\right)^{\mu/2}}{\pi^{1/2}\Gamma\left% (\nu+\mu+1\right)\Gamma\left(\mu-\nu\right)}\*\int_{0}^{\infty}\frac{\cosh% \left(\left(\nu+\frac{1}{2}\right)t\right)}{(x+\cosh t)^{\mu+(1/2)}}\mathrm{d}% t,}}
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    DLMF id
    DLMF:14.12.E4
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    constraint
    ℜ ⁡ ( μ - ν ) > 0 𝜇 𝜈 0 {\displaystyle{\displaystyle\Re(\mu-\nu)>0}}
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    ν + μ ≠ - 1 , - 2 , - 3 , … 𝜈 𝜇 1 2 3 … {\displaystyle{\displaystyle\nu+\mu\neq-1,-2,-3,\dots}}
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    ℜ ⁡ ( μ - ν ) > 0 𝜇 𝜈 0 {\displaystyle{\displaystyle\Re(\mu-\nu)>0}}
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    Symbols used
    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2acdec
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    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.m1adec
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    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2abdec
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    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1acdec
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    hyperbolic cosine function
    Defining formula
    cosh ⁡ z 𝑧 {\displaystyle{\displaystyle\cosh\NVar{z}}}
    xml-id
    C4.S28.E2.m2aadec
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    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3acdec
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    Q10811
    Defining formula
    ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1acdec
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    Q11589
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C14.S1.XMD1.m1adec
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    Q11591
    Defining formula
    μ 𝜇 {\displaystyle{\displaystyle\mu}}
    xml-id
    C14.S1.XMD7.m1cdec
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    Q11590
    Defining formula
    ν 𝜈 {\displaystyle{\displaystyle\nu}}
    xml-id
    C14.S1.XMD8.m1cdec
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