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ET I 68(39)

3.12

${\int }_{0}^{\infty }\frac{{x}^{2}{{m}^{}}^{+1}sin\left(ax\right)\text{d}x}{{\left({b}^{2}+{x}^{2}\right)}^{n+\frac{1}{2}}}\text{?}=\frac{{\left(-\text{1}\right)}^{\text{m}+\text{1}}\sqrt{\pi }{d}^{2}{m+1}^{}}{{\text{2}}^{n}{b}^{n}\Gamma \left(n+\frac{1}{2}\right)d{a}^{2m+1}}\left[{a}^{n}{K}_{n}\left(ab\right)\right]$

ET I 67(37)

4.

$\begin{array}{ll}{\int }_{0}^{\infty }\frac{{x}^{2}{\nu }^{}cos\left(ax\right)\text{d}x}{{\left({x}^{2}+{b}^{2}\right)}^{\mu +1}}\hfill & =\frac{1}{2}{b}^{2\nu }{{-2\mu }^{}}^{-1}\text{B}\left(\nu +\frac{1}{2},\mu -\nu +\frac{1}{2}\right)F_{1}{2}_{}\left(\nu +\frac{1}{2};\nu -\mu +\frac{1}{2},\frac{1}{2};\frac{{b}^{2}{a}^{2}}{4}\right)\hfill \\ +\frac{\sqrt{\pi }{a}^{2}{{\mu -}^{}}^{2}{{\nu }^{}}^{+1}}{{4}^{\mu -\nu +1}}\frac{\Gamma \left(\nu -\mu -\frac{1}{2}\right)}{\Gamma \left(\mu -\nu +\frac{1}{2}\right)}F_{1}{2}_{}\left(\mu +1;\mu -\nu +1,\mu -\nu +\frac{3}{2};\frac{{b}^{2}{a}^{2}}{4}\hfill \\ =\frac{\sqrt{\pi }}{\text{2}\Gamma \left(\mu +\text{1}\right)}{\text{b}}^{\text{2}}{{\nu }^{}}^{-\text{2}\mu }{-\text{1}}^{}\text{?}{\text{G}}_{\text{13}}^{\text{21}}\left(\frac{{\text{a}}^{\text{2}}{\text{b}}^{\text{2}}}{\text{4}}|{\mu -\nu +\frac{1}{2},0,\frac{1}{2}}_{-\nu +\frac{1}{2}}^{}\right)\hfill \end{array}$

ET I 14(29)a, ET II 235(19)

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