Singular points of the integral representation of the Mittag-Leffler function

Abstract

The paper presents an integral representation of the two-parameter Mittag-Leffler function $E_{\rho ,\mu }(z)$ and singular points of this representation have been studied. It has been found that there are two singular points for this integral representation: $\zeta =1$ and $\zeta =0$. The point $\zeta =1$ is a pole of the first order and the point $\zeta =0$, depending on the values of parameters $\rho ,\mu$ is either a pole or a branch point, or a regular point. The subsequent study showed that at some values of parameters $\rho ,\mu$ with the help of the residue theory one can calculate the integral included in the studied integral representation and express the function $E_{\rho ,\mu }(z)$ through elementary functions.