DLMF:10.22.E56 (Q3430): Difference between revisions

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Property / constraint
 

ℜ ⁡ ( μ + ν + 1 ) > ℜ ⁡ λ > - 1 𝜇 𝜈 1 𝜆 1 {\displaystyle{\displaystyle\Re(\mu+\nu+1)>\Re\lambda>-1}}

\Re(\mu+\nu+1)>\Re\lambda>-1
Property / constraint: ℜ ⁡ ( μ + ν + 1 ) > ℜ ⁡ λ > - 1 𝜇 𝜈 1 𝜆 1 {\displaystyle{\displaystyle\Re(\mu+\nu+1)>\Re\lambda>-1}} / rank
 
Normal rank

Revision as of 16:20, 30 December 2019

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DLMF:10.22.E56
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    Statements

    test
    ∫ 0 ∞ J μ ⁡ ( a ⁢ t ) ⁢ J ν ⁡ ( b ⁢ t ) t λ ⁢ d t = a μ ⁢ Γ ⁡ ( 1 2 ⁢ ν + 1 2 ⁢ μ - 1 2 ⁢ λ + 1 2 ) 2 λ ⁢ b μ - λ + 1 ⁢ Γ ⁡ ( 1 2 ⁢ ν - 1 2 ⁢ μ + 1 2 ⁢ λ + 1 2 ) ⁢ 𝐅 ⁡ ( 1 2 ⁢ ( μ + ν - λ + 1 ) , 1 2 ⁢ ( μ - ν - λ + 1 ) ; μ + 1 ; a 2 b 2 ) , superscript subscript 0 Bessel-J 𝜇 𝑎 𝑡 Bessel-J 𝜈 𝑏 𝑡 superscript 𝑡 𝜆 𝑡 superscript 𝑎 𝜇 Euler-Gamma 1 2 𝜈 1 2 𝜇 1 2 𝜆 1 2 superscript 2 𝜆 superscript 𝑏 𝜇 𝜆 1 Euler-Gamma 1 2 𝜈 1 2 𝜇 1 2 𝜆 1 2 scaled-hypergeometric-bold-F 1 2 𝜇 𝜈 𝜆 1 1 2 𝜇 𝜈 𝜆 1 𝜇 1 superscript 𝑎 2 superscript 𝑏 2 {\displaystyle{\displaystyle\int_{0}^{\infty}\frac{J_{\mu}\left(at\right)J_{% \nu}\left(bt\right)}{t^{\lambda}}\mathrm{d}t=\frac{a^{\mu}\Gamma\left(\frac{1}% {2}\nu+\frac{1}{2}\mu-\frac{1}{2}\lambda+\frac{1}{2}\right)}{2^{\lambda}b^{\mu% -\lambda+1}\Gamma\left(\frac{1}{2}\nu-\frac{1}{2}\mu+\frac{1}{2}\lambda+\frac{% 1}{2}\right)}\*\mathbf{F}\left(\tfrac{1}{2}(\mu+\nu-\lambda+1),\tfrac{1}{2}(% \mu-\nu-\lambda+1);\mu+1;\frac{a^{2}}{b^{2}}\right),}}
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    DLMF id
    DLMF:10.22.E56
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    constraint
    ℜ ⁡ ( μ + ν + 1 ) > ℜ ⁡ λ > - 1 𝜇 𝜈 1 𝜆 1 {\displaystyle{\displaystyle\Re(\mu+\nu+1)>\Re\lambda>-1}}
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    0 < a < b 0 𝑎 𝑏 {\displaystyle{\displaystyle 0<a<b}}
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    ℜ ⁡ ( μ + ν + 1 ) > ℜ ⁡ λ > - 1 𝜇 𝜈 1 𝜆 1 {\displaystyle{\displaystyle\Re(\mu+\nu+1)>\Re\lambda>-1}}
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