DLMF:10.22.E49 (Q3423): Difference between revisions

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( μ + ν ) > 0 , ( a ± i b ) > 0 formulae-sequence 𝜇 𝜈 0 plus-or-minus 𝑎 𝑖 𝑏 0 {\displaystyle{\displaystyle\Re(\mu+\nu)>0,\Re(a\pm ib)>0}}

\realpart@@{(\mu+\nu)}>0,\realpart@@{(a\pm ib)}>0
Property / constraint: ( μ + ν ) > 0 , ( a ± i b ) > 0 formulae-sequence 𝜇 𝜈 0 plus-or-minus 𝑎 𝑖 𝑏 0 {\displaystyle{\displaystyle\Re(\mu+\nu)>0,\Re(a\pm ib)>0}} / rank
 
Normal rank

Revision as of 17:20, 30 December 2019

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DLMF:10.22.E49
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    0 t μ - 1 e - a t J ν ( b t ) d t = ( 1 2 b ) ν a μ + ν Γ ( μ + ν ) 𝐅 ( μ + ν 2 , μ + ν + 1 2 ; ν + 1 ; - b 2 a 2 ) , superscript subscript 0 superscript 𝑡 𝜇 1 superscript 𝑒 𝑎 𝑡 Bessel-J 𝜈 𝑏 𝑡 𝑡 superscript 1 2 𝑏 𝜈 superscript 𝑎 𝜇 𝜈 Euler-Gamma 𝜇 𝜈 scaled-hypergeometric-bold-F 𝜇 𝜈 2 𝜇 𝜈 1 2 𝜈 1 superscript 𝑏 2 superscript 𝑎 2 {\displaystyle{\displaystyle\int_{0}^{\infty}t^{\mu-1}e^{-at}J_{\nu}\left(bt% \right)\mathrm{d}t=\frac{(\tfrac{1}{2}b)^{\nu}}{a^{\mu+\nu}}\Gamma\left(\mu+% \nu\right)\*\mathbf{F}\left(\frac{\mu+\nu}{2},\frac{\mu+\nu+1}{2};\nu+1;-\frac% {b^{2}}{a^{2}}\right),}}
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    DLMF:10.22.E49
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    ( μ + ν ) > 0 , ( a ± i b ) > 0 formulae-sequence 𝜇 𝜈 0 plus-or-minus 𝑎 𝑖 𝑏 0 {\displaystyle{\displaystyle\Re(\mu+\nu)>0,\Re(a\pm ib)>0}}
    0 references