DLMF:14.20.E22 (Q4943)

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DLMF:14.20.E22
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    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),}}
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    DLMF:14.20.E22
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