DLMF:10.22.E54 (Q3428)

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Language	Label	Description	Also known as
English	DLMF:10.22.E54	No description defined

Statements

Defining formula

{\displaystyle{\displaystyle\int_{0}^{\infty}J_{\nu}\left(bt\right)\exp\left(-% p^{2}t^{2}\right)t^{\mu-1}\mathrm{d}t=\frac{(\tfrac{1}{2}b/p)^{\nu}\Gamma\left% (\tfrac{1}{2}\nu+\tfrac{1}{2}\mu\right)}{2p^{\mu}}\exp\left(-\frac{b^{2}}{4p^{% 2}}\right)\*{\mathbf{M}}\left(\tfrac{1}{2}\nu-\tfrac{1}{2}\mu+1,\nu+1,\frac{b^% {2}}{4p^{2}}\right),}}

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DLMF:10.22.E54

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{\displaystyle{\displaystyle\Re(\mu+\nu)>0}}

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{\displaystyle{\displaystyle\Re(p^{2})>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.m2aandec

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Defining formula

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

C5.S2.E1.m2apdec

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Olver’s confluent hypergeometric function

Defining formula

{\displaystyle{\displaystyle{\mathbf{M}}\left(\NVar{a},\NVar{b},\NVar{z}\right% )}}

C13.S2.E3.m2adec

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Defining formula

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

C1.S4.SS4.m1abadec

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exponential function

Defining formula

{\displaystyle{\displaystyle\exp\NVar{z}}}

C4.S2.E19.m2acdec

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Defining formula

{\displaystyle{\displaystyle\int}}

C1.S4.SS4.m3abadec

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Defining formula

{\displaystyle{\displaystyle\Re}}

C1.S9.E2.m1aaidec

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Defining formula

{\displaystyle{\displaystyle\nu}}

C10.S1.XMD7.m1amdec

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