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Enumeration of the real zeros of the Mittag-Leffler function E α (z), 1<α<2. (English) Zbl 1124.33020
Sabatier, J. (ed.) et al., Advances in fractional calculus. Theoretical developments and applications in physics and engineering. Dordrecht: Springer (ISBN 978-1-4020-6041-0/hbk; 978-1-4020-6042-7/e-book). 15-26 (2007).
Summary: The Mittag-Leffler function E α (z), which is a generalization of the exponential function, arises frequently in the solutions of physical problems described by differential and/or integral equations of fractional order. Consequently, the zeros of E α (z) and their distribution are of fundamental importance and play a significant role in the dynamic solutions. The Mittag-Leffler function E α (z) is known to have a finite number of real zeros in the range 1<α<2 which is applicable for many physical problems. What has not been known is the exact number of real zeros of E α (z) for a given value of α in this range. An iteration formula is derived for calculating the number of real zeros of E α (z) for any value of α in the range 1<α<2 and some specific results are tabulated.
MSC:
33E12Mittag-Leffler functions and generalizations
26A33Fractional derivatives and integrals (real functions)