Table of Integrals, Series, and Products, 8th Edition

  ET II 106(4)

$\left[Re\alpha >0,\text{?}Re\mu >\left|Rev\right|-1,\text{?}b>0\right]$

3.¹² 

${\displaystyle {\int }_0^{\infty }{x}^{{\mu }^{}}{e}^{-\alpha x^2}{\mathit{K}}_v\left(bx\right)\text{d}x}=\frac{1}{2}{\alpha }^{-\frac{1}{2}\mu }{b}^{-1}\Gamma \left(\frac{1+v+\mu }{2}\right)\Gamma \left(\frac{1-v+\mu }{2}\right)\exp \left(\frac{b^2}{8\alpha }\right)W_{-\frac{1}{2}\mu ,\frac{1}{2}v}\left(\frac{b^2}{4\alpha }\right)$

  ET II 132(25)

$\left[Re\alpha >0,\text{?}Re\mu >\left|Rev\right|-1\right]$

4.¹¹ 

$\begin{array}{cc}{\displaystyle {\int }_0^{\infty }{x}^{{v+1}^{}}{e}^{-\alpha x^2}{J}_v\left(bx\right)\text{d}x}=\frac{b^v}{{\left(2\alpha \right)}^{v+1}}\exp \left(-\frac{b^2}{4\alpha }\right)& \left[Re\alpha >0,\text{?}Rev>-1\right]\end{array}$

  WA 431(4), ET II 29(10)