DLMF:14.12.E6 (Q4827)

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DLMF:14.12.E6
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    Statements

    Defining formula
    𝑸 ν μ ⁡ ( x ) = π 1 / 2 ⁢ ( x 2 - 1 ) μ / 2 2 μ ⁢ Γ ⁡ ( μ + 1 2 ) ⁢ Γ ⁡ ( ν - μ + 1 ) ⁢ ∫ 0 ∞ ( sinh ⁡ t ) 2 ⁢ μ ( x + ( x 2 - 1 ) 1 / 2 ⁢ cosh ⁡ t ) ν + μ + 1 ⁢ d t , associated-Legendre-black-Q 𝜇 𝜈 𝑥 superscript 𝜋 1 2 superscript superscript 𝑥 2 1 𝜇 2 superscript 2 𝜇 Euler-Gamma 𝜇 1 2 Euler-Gamma 𝜈 𝜇 1 superscript subscript 0 superscript 𝑡 2 𝜇 superscript 𝑥 superscript superscript 𝑥 2 1 1 2 𝑡 𝜈 𝜇 1 𝑡 {\displaystyle{\displaystyle\boldsymbol{Q}^{\mu}_{\nu}\left(x\right)=\frac{\pi% ^{1/2}\left(x^{2}-1\right)^{\mu/2}}{2^{\mu}\Gamma\left(\mu+\frac{1}{2}\right)% \Gamma\left(\nu-\mu+1\right)}\*\int_{0}^{\infty}\frac{(\sinh t)^{2\mu}}{\left(% x+(x^{2}-1)^{1/2}\cosh t\right)^{\nu+\mu+1}}\mathrm{d}t,}}
    0 references
    DLMF id
    DLMF:14.12.E6
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    constraint
    ℜ ⁡ ( ν + 1 ) > ℜ ⁡ μ > - 1 2 𝜈 1 𝜇 1 2 {\displaystyle{\displaystyle\Re(\nu+1)>\Re\mu>-\tfrac{1}{2}}}
    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.m2aedec
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    Q11592
    Defining formula
    𝑸 ν μ ⁡ ( z ) associated-Legendre-black-Q 𝜇 𝜈 𝑧 {\displaystyle{\displaystyle\boldsymbol{Q}^{\NVar{\mu}}_{\NVar{\nu}}\left(% \NVar{z}\right)}}
    xml-id
    C14.S21.SS1.p1.m3adec
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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.m2acdec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aedec
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    hyperbolic cosine function
    Defining formula
    cosh ⁡ z 𝑧 {\displaystyle{\displaystyle\cosh\NVar{z}}}
    xml-id
    C4.S28.E2.m2abdec
    0 references
    hyperbolic sine function
    Defining formula
    sinh ⁡ z 𝑧 {\displaystyle{\displaystyle\sinh\NVar{z}}}
    xml-id
    C4.S28.E1.m2aadec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3aedec
    0 references
    Q10811
    Defining formula
    ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1aedec
    0 references
    Q11589
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C14.S1.XMD1.m1cdec
    0 references
     
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