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

In this paper, we examine singular points of an integral representation of the two-parameter Mittag-Leffler function $E_{\rho,\mu}(z)$. We establish that this integral representation possesses two singular points: the first-order pole $\zeta=1$ and the point $\zeta=0$, which is either a pole, or a branch point, or a regular point depending on the value of the parameters $\rho$ and $\mu$. For some values of the parameters $\rho$ and $\mu$, the integral in the representation considered can be calculated by methods of the theory of residues and hence the function $E_{\rho, \mu}(z)$ can be expressed through elementary functions.