DLMF:14.20.E22 (Q4943)

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DLMF:14.20.E22
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    Statements

    Defining formula
    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 ) ) , Ferrers-Legendre-P-first-kind 𝜇 1 2 𝑖 𝜏 𝑥 𝛽 𝜇 𝛽 𝛽 Euler-Gamma 𝜇 1 superscript 1 superscript 𝛽 2 𝜇 2 superscript 𝑒 𝜇 𝜌 superscript 1 superscript 𝛽 2 superscript 𝑥 2 superscript 𝛽 2 1 4 1 Big-O 1 𝜇 {\displaystyle{\displaystyle\mathsf{P}^{-\mu}_{-\frac{1}{2}+i\tau}\left(x% \right)=\frac{\beta\exp\left(\mu\beta\operatorname{arctan}\beta\right)}{\Gamma% \left(\mu+1\right)\left(1+\beta^{2}\right)^{\mu/2}}\frac{e^{-\mu\rho}}{\left(1% +\beta^{2}-x^{2}\beta^{2}\right)^{1/4}}\left(1+O\left(\frac{1}{\mu}\right)% \right),}}
    0 references
    DLMF id
    DLMF:14.20.E22
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    Symbols used
    order not exceeding
    Defining formula
    O ⁡ ( x ) Big-O 𝑥 {\displaystyle{\displaystyle O\left(\NVar{x}\right)}}
    xml-id
    C2.S1.E3.m2addec
    0 references
    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2afdec
    0 references
    Ferrers function of the first kind
    Defining formula
    𝖯 ν μ ⁡ ( x ) Ferrers-Legendre-P-first-kind 𝜇 𝜈 𝑥 {\displaystyle{\displaystyle\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}% \right)}}
    xml-id
    C14.S3.E1.m2ajdec
    0 references
    exponential function
    Defining formula
    exp ⁡ z 𝑧 {\displaystyle{\displaystyle\exp\NVar{z}}}
    xml-id
    C4.S2.E19.m2aadec
    0 references
     
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