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

6.52 Bessel functions combined with x and x²

6.521

1. 

$\begin{array}{cc}{\displaystyle {\int }_0^{\infty }x{J}_{\nu }\left(ax\right)J_{\nu }\left(\beta x\right)}\text{d}x=\frac{\beta J_{\nu -1}\left(\beta \right)J_{\nu }\left(\alpha \right)-\alpha J_{\nu -1}\left(\alpha \right)J_{\nu }\left(\beta \right)}{{\alpha }^2-{\beta }^2}& \left[\alpha \ne \beta ,\nu >-1\right]\\ =\frac{\alpha J_{\nu -1}\left(\beta \right){J^{\prime }}_{\nu }\left(\alpha \right)-\beta J_{\nu }\left(\alpha \right){J^{\prime }}_{\nu }\left(\beta \right)}{{\beta }^2-{\alpha }^2}& \left[\alpha \ne \beta ,\nu >-1\right]\end{array}$

  WH

2.¹⁰ 

${\displaystyle \underset{0}{\overset{\infty }{\int }}x{\mathit{K}}_v(ax)J_v(bx)\text{d}x=\frac{b^v}{a^v(b^2+a^2)}[Rea>0,b>0,Rev>-1]}$

  ET II 63(2)

3. 

${\displaystyle \underset{0}{\overset{\infty }{\int }}x{\mathit{K}}_v(ax){\mathit{K}}_v(bx)\text{d}x=\frac{\pi {(ab)}^{-v}(a^{2v}-b^{2v})}{2sin(v\pi )(a^2-b^2)}[|Rev|>1,Re(a+b)>0]}$

  ET II 45(48)

4. 

$\underset{0}{\overset{\infty }{\int }}x{J}_v(\lambda x){\mathit{K}}_v(\mu x)\text{d}x={({\mu }^2+{\lambda }^2)}^{-1}[{(\frac{\lambda }{\mu })}^v+\lambda a{J}_{v+1}(\lambda a){\mathit{K}}_v(\mu a)-\mu a{J}_v(\lambda a){\mathit{K}}_{v+1}(\mu a)]$

  ET II 367(26)

$[Re\nu >-1]$

5. 

${\displaystyle \underset{0}{\overset{\infty }{\int }}x{\mathit{K}}_1(ax)=\frac{\pi }{2a^2}[}$