Chapter 7.13, Problem 17MP

Show that the Fourier sine transform of $x^{-1/2}$ is ${\alpha }^{-1/2}$.

Hint: Make the change of variable $z=\alpha x.$ The integral ${\displaystyle {\int }_0^{\infty }{z}^{-1/2}}\sin zdz$ can be found by computer or in tables.