Geodesic
Singular points of the integral representation of the Mittag-Leffler function
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the International Conference on Mathematical Modelling in Applied Sciences — ICMMAS'19. Belgorod, August 20–24, 2019, Tome 195 (2021), pp. 97-107

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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.
Keywords: Mittag-Leffler function, integral representation. 
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     author = {V. V. Saenko},
     title = {Singular points of the integral representation of the {Mittag-Leffler} function},
     journal = {Itogi nauki i tehniki. Sovremenna\^a matematika i e\"e prilo\v{z}eni\^a. Temati\v{c}eskie obzory},
     pages = {97--107},
     publisher = {mathdoc},
     volume = {195},
     year = {2021},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/INTO_2021_195_a11/}
}
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V. V. Saenko. Singular points of the integral representation of the Mittag-Leffler function. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the International Conference on Mathematical Modelling in Applied Sciences — ICMMAS'19. Belgorod, August 20–24, 2019, Tome 195 (2021), pp. 97-107. http://geodesic.mathdoc.fr/item/INTO_2021_195_a11/