DLMF:14.19.E2 (Q4914)

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DLMF:14.19.E2
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    Statements

    Defining formula
    P ν - 1 2 μ ⁡ ( cosh ⁡ ξ ) = Γ ⁡ ( 1 2 - μ ) π 1 / 2 ⁢ ( 1 - e - 2 ⁢ ξ ) μ ⁢ e ( ν + ( 1 / 2 ) ) ⁢ ξ ⁢ 𝐅 ⁡ ( 1 2 - μ , 1 2 + ν - μ ; 1 - 2 ⁢ μ ; 1 - e - 2 ⁢ ξ ) , Legendre-P-first-kind 𝜇 𝜈 1 2 𝜉 Euler-Gamma 1 2 𝜇 superscript 𝜋 1 2 superscript 1 superscript 𝑒 2 𝜉 𝜇 superscript 𝑒 𝜈 1 2 𝜉 scaled-hypergeometric-bold-F 1 2 𝜇 1 2 𝜈 𝜇 1 2 𝜇 1 superscript 𝑒 2 𝜉 {\displaystyle{\displaystyle P^{\mu}_{\nu-\frac{1}{2}}\left(\cosh\xi\right)=% \frac{\Gamma\left(\frac{1}{2}-\mu\right)}{\pi^{1/2}\left(1-e^{-2\xi}\right)^{% \mu}e^{(\nu+(1/2))\xi}}\*\mathbf{F}\left(\tfrac{1}{2}-\mu,\tfrac{1}{2}+\nu-\mu% ;1-2\mu;1-e^{-2\xi}\right),}}
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    DLMF id
    DLMF:14.19.E2
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    constraint
    μ ≠ 1 2 , 3 2 , 5 2 , … 𝜇 1 2 3 2 5 2 … {\displaystyle{\displaystyle\mu\neq\frac{1}{2},\frac{3}{2},\frac{5}{2},\ldots}}
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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.m2adec
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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.m2adec
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    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2adec
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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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    $$={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$$ Olver’s hypergeometric function
    Defining formula
    𝐅 ⁡ ( a , b ; c ; z ) scaled-hypergeometric-bold-F 𝑎 𝑏 𝑐 𝑧 {\displaystyle{\displaystyle\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z% }\right)}}
    xml-id
    C15.S2.E2.m2adec
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    Q11591
    Defining formula
    μ 𝜇 {\displaystyle{\displaystyle\mu}}
    xml-id
    C14.S1.XMD7.m1dec
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    Q11590
    Defining formula
    ν 𝜈 {\displaystyle{\displaystyle\nu}}
    xml-id
    C14.S1.XMD8.m1dec
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    Q11626
    Defining formula
    ξ > 0 𝜉 0 {\displaystyle{\displaystyle\xi>0}}
    xml-id
    C14.S19.XMD7.m1dec
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