Chapter 12.23, Problem 29MP

Show that the Fourier cosine transform (Chapter 7, Section 12) of $J_0(x)$ is

Hence show that ${\displaystyle {\int }_0^{\infty }{J}_0}(x)dx=1.$ Hints: Show that the integral in Problem 20 gives $J_0(x)=(2/\pi ){\displaystyle {\int }_0^{\pi /2}\cos }(x\sin \theta )d\theta .$ (Replace $\theta$ by $\pi -\theta$ in the $\pi /2$ to $\pi$ integral.) Let $\sin \theta =\alpha$ to find $J_0$ as a cosine transform; write the inverse transform. Now let $\alpha =0$.