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### 6.55 Combinations of Bessel functions and algebraic functions

6.55110

1.

$\underset{0}{\overset{1}{\int }}{x}^{1/2}{J}_{v}\left(xy\right)\text{d}x=\sqrt{2}{y}^{-3/2}\frac{\Gamma \left(\frac{3}{4}+\frac{1}{2}v\right)}{\Gamma \left(\frac{1}{4}+\frac{1}{2}v\right)}+{y}^{-1/2}\left[\left(v-\frac{1}{2}\right){J}_{v}\left(y\right){S}_{-1/2,v-1}\left(y\right)-{J}_{v-1}\left(y\right){S}_{1/2,v}\left(y\right)\right]$

ET || 21(1)

$[y>0,?Rev>−32]$

2.

${\int }_{1}^{\infty }x1/2{J}_{v}\left(xy\right)\text{d}x={y}^{-1/2}\left[{J}_{v-1}\left(y\right){S}_{1/2,v}\left(y\right)+\left(\frac{1}{2}-v\right){J}_{v}\left(y\right){S}_{-1/2,v-1}\left(y\right)\right]$

ET || 22(2)

[y > 0]

6.552

1.

${\int }_{0}^{\infty }{J}_{v}\left(xy\right)\frac{\text{d}x}{{\left({x}^{2}+{a}^{2}\right)}^{1/2}}={I}_{v/2}\left(\frac{1}{2}ay\right){\mathbit{K}}_{v/2}\left(\frac{1}{2}ay\right)$

ET ||23(11), ...

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