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

6.55 Combinations of Bessel functions and algebraic functions

6.551¹⁰

1. 

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

  ET || 21(1)

$\left[y>0,\text{?}Rev>-\frac{3}{2}\right]$

2. 

${\displaystyle {\int }_1^{\infty }x1/2J_v(xy)\text{d}x=y^{-1/2}\left[J_{v-1}\left(y\right)S_{1/2,v}(y)+\left(\frac{1}{2}-v\right)J_v\left(y\right)S_{-1/2,v-1}(y)\right]}$

  ET || 22(2)

[y > 0]

6.552

1. 

${\displaystyle {\int }_0^{\infty }{J}_v(xy)\frac{\text{d}x}{{\left(x^2+a^2\right)}^{1/2}}}=I_{v/2}\left(\frac{1}{2}ay\right){\mathit{K}}_{v/2}\left(\frac{1}{2}ay\right)$

  ET ||23(11), ...