## With Safari, you learn the way you learn best. Get unlimited access to videos, live online training, learning paths, books, tutorials, and more.

No credit card required

### 6.69–6.74 Combinations of Bessel and trigonometric functions and powers

6.691

$\begin{array}{cc}{\int }_{0}^{\infty }xsin\left(bx\right){\mathbit{K}}_{0}\left(ax\right)\text{d}x=\frac{\pi b}{2}{\left({a}^{2}+{b}^{2}\right)}^{-\frac{3}{2}}& \left[\begin{array}{cc}Rea>0,& b>0\end{array}\right]\end{array}$

ET I 105(47)

6.692

1.

$\begin{array}{cc}{\int }_{0}^{\infty }x{\mathbit{K}}_{\nu }\left(ax\right){I}_{\nu }\left(bx\right)sin\left(cx\right)\text{d}x=-\frac{1}{2}{\left(ab\right)}^{-\frac{3}{2}}c{\left({u}^{2}-1\right)}^{-\frac{1}{2}}{Q}_{\nu -\frac{1}{2}}^{1}\left(u\right),& u={\left(2ab\right)}^{-1}\left({a}^{2}+{b}^{2}+{c}^{2}\right)\end{array}$

ET I 106(54)

$[Rea>|Reb|,c>0,Reν>−32]$

2.

${\int }_{0}^{\infty }x{\mathbit{K}}_{v}\left(ax\right){\mathbit{K}}_{\nu }\left(bx\right)sin\left(cx\right)\text{d}x=\frac{\pi }{4}{\left(ab\right)}^{-\frac{3}{2}}c{\left({u}^{2}-1\right)}^{-\frac{1}{2}}\Gamma \left(\frac{3}{2}+\nu \right)\Gamma \left(\frac{3}{2}-\nu \right){P}_{\nu -\frac{1}{2}}^{-1}\left(u\right)$

ET I 107(61) ...

## With Safari, you learn the way you learn best. Get unlimited access to videos, live online training, learning paths, books, interactive tutorials, and more.

No credit card required