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

V. V. Saenko

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.

Résumé

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.

Détail

Keywords: Mittag-Leffler function, integral representation.

Bibliographie

[1] Beitmen G., Erdeii A., Vysshie transtseidentnye funktsii. Ellipticheskie i avtomorfnye funktsii. Funktsii Lame i Mate, Nauka, M., 1967

[2] Dzhrbashyan M. M., “Ob integralnom predstavlenii funktsii, nepreryvnykh na neskolkikh luchakh (obobschenie integrala fure)”, Izv. AN SSSR. Ser. mat., 18:5 (1954), 427–448 | Zbl

[3] Dzhrbashyan M. M., Integralnye preobrazovaniya i predstavleniya funktsii v kompleksnoi oblasti, Nauka, M., 1966

[4] Agarwal R. P., “A propos d'une note de M. Pierre Humbert”, C. R. Acad. Sci. Paris., 236 (1953), 2031–2032 | MR | Zbl

[5] Djrbashian M. M., Harmonic Analysis and Boundary-Value Problems in the Complex Domain, Birkhäuser, Basel, 1993 | MR | Zbl

[6] Gorenflo R., Kilbas A. A., Mainardi F., Rogosin S. V., Mittag-Leffler Functions, Related Topics and Applications, Springer-Verlag, Berlin–Heidelberg, 2014 | MR | Zbl

[7] Gorenflo R., Loutchko J., Luchko Yu., “Computation of the Mittag-Leffler function $E_{\alpha,\beta}(z)$ and its derivative”, Fract. Calc. Appl. Anal., 5:4 (2002), 491–518 | MR | Zbl

[8] Gorenflo R., Luchko Yu., Rogosin S., Mittag-Leffler-type functions: notes on growth properties and distribution of zeros, Universit{ä}t Berlin Fachbereich Mathematik und Informatik, 1991

[9] Gorenflo R., Mainardi F., “Fractional Calculus”, Fractals Fract. Calc. Cont. Mech., 378 (1997), 223–276 | DOI | MR | Zbl

[10] Gorenflo R., Mainardi F., Rogosin S., “Mittag-Leffler function: properties and applications”, Handbook of Fractional Calculus with Applications, v. 1. Basic Theory, De Gruyter, Berlin–Boston, 2019, 269–296 | MR

[11] Hilfer R., Seybold H. J., “Computation of the generalized Mittag-Leffler function and its inverse in the complex plane”, Integr. Transforms Spec. Funct., 17:9 (2006), 637–652 | DOI | MR | Zbl

[12] Humbert P., “Quelques résultats rélatifs à la fonction de Mittag-Leffler”, C. R. Acad. Sci. Paris., 236 (1953), 1467–1468 | MR | Zbl

[13] Humbert P., Agarwal R. P., “Sur la fonction de Mittag-Leffler et quelques-unes de ses généralisations”, Bull. Sci. Math., II. Sér., 77 (1953), 180–185 | MR | Zbl

[14] Saenko V. V., “Integral representation of the Mittag-Leffler function”, Proc. 2nd Int. Conf. on Mathematical Modeling in Applied Sciences (August 20–24, 2019, Belgorod, Russia), Belgorod, 2019, 157–158

[15] Seybold H. J., Hilfer R., “Numerical results for the generalized Mittag-Leffler function”, Fract. Calc. Appl. Anal., 8:2 (2005), 129–139 | MR

[16] Wiman A., “Über den Fundamentalsatz in der Teorie der Funktionen $E_a(x)$”, Acta Math., 29:1 (1905), 191–201 | DOI | MR | Zbl

[17] Wiman A., “Über die Nullstellen der Funktionen $E_a(x)$”, Acta Math., 29 (1905), 217–234 | DOI | MR | Zbl