Weber's Discontinuous Integrals

$$\int_0^\infty J_0(ax)\cos (cx)\,dx =\cases{ 0 & $a < c$\cr {1\over \sqrt{a^2-c^2}} & $ a>c$\cr}$$

$$\int_0^\infty J_0(ax)\sin (cx)\,dx=\cases{ {1\over \sqrt{c^2-a^2}} & $ a<c$\cr \strut 0 & $a > c$,\cr}$$

where $J_0(z)$ is a zeroth order Bessel Function of the First Kind.

References

Bowman, F. Introduction to Bessel Functions. New York: Dover, pp. 59-60, 1958.