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

6.61 Combinations of Bessel functions and exponentials

6.611

1. 

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

  EH II 49(18), WA 422(8)

2. 

$\begin{array}{ll}\underset{0}{\overset{\infty }{{\displaystyle \int }}}{e}^{-\alpha x}{Y}_v(bx)\mathrm{d}x=\hfill & {({\alpha }^2+b^2)}^{-\frac{1}{2}}cosec(v\pi )\hfill \\ \times \{b^v{[{({\alpha }^2+b^2)}^{\frac{1}{2}}+\alpha ]}^{-v}cos(v\pi )-b^{-v}{[{({\alpha }^2+b^2)}^{\frac{1}{2}}+\alpha ]}^v\}\hfill \end{array}$

  MO 179, ET II 105(1)

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

3. 

$\underset{0}{\overset{\infty }{{\displaystyle \int }}}{e}^{-\alpha x}{\mathit{K}}_v(bx)\mathrm{d}x=\frac{\pi }{bsin(v\pi )}\frac{sin(v\theta )}{sin\theta }$

  ET II 131(22)

$[cos\theta =\frac{\alpha }{b};\text{?}\text{with ...}$