DLMF:14.12.E3 (Q4824): Difference between revisions

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Property / constraint
 

0 < θ < π 0 𝜃 𝜋 {\displaystyle{\displaystyle 0<\theta<\pi}}

0<\theta<\pi
Property / constraint: 0 < θ < π 0 𝜃 𝜋 {\displaystyle{\displaystyle 0<\theta<\pi}} / rank
 
Normal rank

Revision as of 16:52, 30 December 2019

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DLMF:14.12.E3
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    Statements

    test
    𝖰 ν μ ⁡ ( cos ⁡ θ ) = π 1 / 2 ⁢ Γ ⁡ ( ν + μ + 1 ) ⁢ ( sin ⁡ θ ) μ 2 μ + 1 ⁢ Γ ⁡ ( μ + 1 2 ) ⁢ Γ ⁡ ( ν - μ + 1 ) ⁢ ( ∫ 0 ∞ ( sinh ⁡ t ) 2 ⁢ μ ( cos ⁡ θ + i ⁢ sin ⁡ θ ⁢ cosh ⁡ t ) ν + μ + 1 ⁢ d t + ∫ 0 ∞ ( sinh ⁡ t ) 2 ⁢ μ ( cos ⁡ θ - i ⁢ sin ⁡ θ ⁢ cosh ⁡ t ) ν + μ + 1 ⁢ d t ) , Ferrers-Legendre-Q-first-kind 𝜇 𝜈 𝜃 superscript 𝜋 1 2 Euler-Gamma 𝜈 𝜇 1 superscript 𝜃 𝜇 superscript 2 𝜇 1 Euler-Gamma 𝜇 1 2 Euler-Gamma 𝜈 𝜇 1 superscript subscript 0 superscript 𝑡 2 𝜇 superscript 𝜃 𝑖 𝜃 𝑡 𝜈 𝜇 1 𝑡 superscript subscript 0 superscript 𝑡 2 𝜇 superscript 𝜃 𝑖 𝜃 𝑡 𝜈 𝜇 1 𝑡 {\displaystyle{\displaystyle\mathsf{Q}^{\mu}_{\nu}\left(\cos\theta\right)=% \frac{\pi^{1/2}\Gamma\left(\nu+\mu+1\right)(\sin\theta)^{\mu}}{2^{\mu+1}\Gamma% \left(\mu+\frac{1}{2}\right)\Gamma\left(\nu-\mu+1\right)}\*\left(\int_{0}^{% \infty}\frac{(\sinh t)^{2\mu}}{(\cos\theta+i\sin\theta\cosh t)^{\nu+\mu+1}}% \mathrm{d}t+\int_{0}^{\infty}\frac{(\sinh t)^{2\mu}}{(\cos\theta-i\sin\theta% \cosh t)^{\nu+\mu+1}}\mathrm{d}t\right),}}
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    DLMF id
    DLMF:14.12.E3
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    constraint
    ℜ ⁡ μ > - 1 2 𝜇 1 2 {\displaystyle{\displaystyle\Re\mu>-\tfrac{1}{2}}}
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    ℜ ⁡ ( ν ± μ ) > - 1 plus-or-minus 𝜈 𝜇 1 {\displaystyle{\displaystyle\Re(\nu\pm\mu)>-1}}
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    0 < θ < π 0 𝜃 𝜋 {\displaystyle{\displaystyle 0<\theta<\pi}}
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