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6.61 Combinations of Bessel functions and exponentials

6.611

1.

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

EH II 49(18), WA 422(8)

2.

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

MO 179, ET II 105(1)

$[Reα>0,?b>0,?|Rev|<1]$

3.

$\underset{0}{\overset{\infty }{\int }}{e}^{-\alpha x}{\mathbit{K}}_{v}\left(bx\right)\mathrm{d}x=\frac{\pi }{bsin\left(v\pi \right)}\frac{sin\left(v\theta \right)}{sin\theta }$

ET II 131(22)

$[cosθ=αb;?with ...$

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