Double and single integrals of the Mittag-Leffler Function: Derivation and Evaluation

One-dimensional and two-dimensional integrals containing $E_b(-u)$ and $E_{\alpha ,\beta }\left(\delta x^{\gamma }\right)$ are considered. $E_b(-u)$ is the Mittag-Leffler function and the integral is taken over the rectangle $0 \leq x < \infty, 0 \leq u < \infty$ and $E_{\alpha ,\beta }\left(\delta x^{\gamma }\right)$ is the generalized Mittag-Leffler function and the integral is over $0\leq x \leq b$ with infinite intervals explored. A representation in terms of the Hurwitz-Lerch zeta function and other special functions are derived for the double and single integrals, from which special cases can be evaluated in terms of special function and fundamental constants.