6.52 Bessel functions combined with x and x2

6.521

1.

$\begin{array}{cc}{\int }_{0}^{\infty }x{J}_{\nu }\left(ax\right){J}_{\nu }\left(\beta x\right)\text{d}x=\frac{\beta {J}_{\nu -1}\left(\beta \right){J}_{\nu }\left(\alpha \right)-\alpha {J}_{\nu -1}\left(\alpha \right){J}_{\nu }\left(\beta \right)}{{\alpha }^{2}-{\beta }^{2}}& \left[\alpha \ne \beta ,\nu >-1\right]\\ =\frac{\alpha {J}_{\nu -1}\left(\beta \right){{J}^{\prime }}_{\nu }\left(\alpha \right)-\beta {J}_{\nu }\left(\alpha \right){{J}^{\prime }}_{\nu }\left(\beta \right)}{{\beta }^{2}-{\alpha }^{2}}& \left[\alpha \ne \beta ,\nu >-1\right]\end{array}$ WH

2.10

$\underset{0}{\overset{\infty }{\int }}x{\mathbit{K}}_{v}\left(ax\right){J}_{v}\left(bx\right)\text{d}x=\frac{{b}^{v}}{{a}^{v}\left({b}^{2}+{a}^{2}\right)}\left[Rea>0,b>0,Rev>-1\right]$ ET II 63(2)

3.

$\underset{0}{\overset{\infty }{\int }}x{\mathbit{K}}_{v}\left(ax\right){\mathbit{K}}_{v}\left(bx\right)\text{d}x=\frac{\pi {\left(ab\right)}^{-v}\left({a}^{2v}-{b}^{2v}\right)}{2sin\left(v\pi \right)\left({a}^{2}-{b}^{2}\right)}\left[|Rev|>1,Re\left(a+b\right)>0\right]$ ET II 45(48)

4.

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

ET II 367(26)

$[Reν>−1]$

5.

$\underset{0}{\overset{\infty }{\int }}x{\mathbit{K}}_{1}\left(ax\right)=\frac{\pi }{2{a}^{2}}\left[$

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