Catalan Integrals

Special cases of general Formulas due to Bessel.

$$J_0(\sqrt{z^2-y^2}\,)={1\over\pi}\int_0^\pi e^{y\cos\theta}\cos(z\sin\theta)\,d\theta,$$

where $J_0$ is a Bessel Function of the First Kind. Now, let $z\equiv 1-z'$ and $y\equiv 1+z'$. Then

$$J_0(2i\sqrt{z}\,) = {1\over\pi} \int_0^\pi e^{(1+z)\cos\theta}\cos[(1-z)\sin\theta]\,d\theta.$$