DLMF:10.22.E49 (Q3423)

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Language	Label	Description	Also known as
English	DLMF:10.22.E49	No description defined

Statements

Defining formula

{\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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{\displaystyle{\displaystyle\Re(\mu+\nu)>0,\Re(a\pm ib)>0}}

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Bessel function of the first kind

Defining formula

{\displaystyle{\displaystyle J_{\NVar{\nu}}\left(\NVar{z}\right)}}

C10.S2.E2.m2aakdec

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Defining formula

{\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}

C5.S2.E1.m2andec

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Defining formula

{\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}

C1.S4.SS4.m1aavdec

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base of natural logarithm

Defining formula

{\displaystyle{\displaystyle\mathrm{e}}}

C4.S2.E11.m2aadec

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Defining formula

{\displaystyle{\displaystyle\mathrm{i}}}

C1.S9.E1.m2aadec

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Defining formula

{\displaystyle{\displaystyle\int}}

C1.S4.SS4.m3aavdec

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Defining formula

{\displaystyle{\displaystyle\Re}}

C1.S9.E2.m1aaddec

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$$={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$$ Olver’s hypergeometric function

Defining formula

{\displaystyle{\displaystyle\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z% }\right)}}

C15.S2.E2.m2adec

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Defining formula

{\displaystyle{\displaystyle\nu}}

C10.S1.XMD7.m1ahdec

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Sitelinks

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