Journal Menu

- About this Journal ·
- Abstracting and Indexing ·
- Aims and Scope ·
- Article Processing Charges ·
- Articles in Press ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents

Advances in Mathematical Physics

Volume 2013 (2013), Article ID 421685, 13 pages

http://dx.doi.org/10.1155/2013/421685

Research Article

## An Alpha-Beta Phase Diagram Representation of the Zeros and Properties of the Mittag-Leffler Function

Department of Physics, University of Memphis, Memphis, TN 38152, USA

Received 8 May 2013; Accepted 25 June 2013

Academic Editor: Dumitru Baleanu

Copyright © 2013 John W. Hanneken et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Linked References

- I. Podlubny,
*Fractional Differential Equations*, vol. 198 of*Mathematics in Science and Engineering*, Academic Press, San Diego, Calif, USA, 1999. View at Zentralblatt MATH · View at MathSciNet - R. L. Magin,
*Fractional Calculus in Bioengineering*, Begell House Publishers, Redding, Conn, USA, 2006. - A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo,
*Theory and Applications of Fractional Differential Equations*, vol. 204 of*North-Holland Mathematics Studies*, Elsevier Science B.V., Amsterdam, The Netherlands, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - F. Mainardi,
*Fractional Calculus and Waves in Linear Viscoelasticity*, Imperial College Press, London, UK, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - L. V. Ahlfors,
*Complex Analysis*, McGraw-Hill, New York, NY, USA, 2nd edition, 1966. View at Zentralblatt MATH · View at MathSciNet - R. Gorenflo and F. Mainardi, “Fractional calculus: integral and differential equations of fractional order,” in
*Fractals and Fractional Calculus in Continuum Mechanics*, A. Carpenteri and F. Mainardi, Eds., vol. 378 of*CISM Courses and Lectures*, pp. 223–276, Springer, Vienna, Austria, 1997. View at MathSciNet - A. M. Sedletski, “Asymptotic formulas for zeros of functions of Mittag-Leffler type,”
*Analysis Mathematica*, vol. 20, no. 2, pp. 117–132, 1994. View at Publisher · View at Google Scholar · View at MathSciNet - A. M. Sedletski, “Nonasymptotic properties of the roots of a function of Mittag-Leffler type,”
*Matematicheskie Zametki*, vol. 75, no. 3, pp. 405–420, 2004. View at Publisher · View at Google Scholar · View at MathSciNet - R. Gorenflo, Yu. Luchko, and S. V. Rogozin, “Mittag-Leffler type functions: notes on growth properties and distribution of zeros,” A97-04, Fachbereich Mathematik und Informatik, Freie Universitat, Berlin, Germany, 1997.
- A. Yu. Popov and A. M. Sedletski, “Distribution of the zeros of the Mittag-Leffler function,”
*Doklady Mathematics*, vol. 67, pp. 336–339, 2003. - W. R. Schneider, “Completely monotone generalized Mittag-Leffler functions,”
*Expositiones Mathematicae*, vol. 14, no. 1, pp. 3–16, 1996. View at Zentralblatt MATH · View at MathSciNet - K. S. Miller and S. G. Samko, “A note on the complete monotonicity of the generalized Mittag-Leffler function,”
*Real Analysis Exchange*, vol. 23, no. 2, pp. 753–755, 1997/98. View at MathSciNet - K. S. Miller and S. G. Samko, “Completely monotonic functions,”
*Integral Transforms and Special Functions*, vol. 12, no. 4, pp. 389–402, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - T. S. Aleroev and H. T. Aleroeva, “A problem on the zeros of the Mittag-Leffler function and the spectrum of a fractional-order differential operator,”
*Electronic Journal of Qualitative Theory of Differential Equations*, vol. 25, pp. 1–18, 2009. View at Zentralblatt MATH · View at MathSciNet - R. Gorenflo and F. Mainardi, “Fractional oscillations and Mittag-Leffler functions,” in
*Proceedings of the International Workshop on the Recent Advances in Applied Mathematics (RAAM '96)*, Kuwait University, Kuwait City, Kuwait, 1996. - A. Wiman, “Über die Nullstellen der Funktionen E
_{α}(x),”*Acta Mathematica*, vol. 29, no. 1, pp. 217–234, 1905. View at Publisher · View at Google Scholar · View at MathSciNet - J. W. Hanneken, D. M. Vaught, and B. N. N. Achar, “Enumeration of the real zeros of the Mittag-Leffler function E
_{α}(x), $1<\alpha <2$,” in*Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering*, J. Sabatier, O. P. Agrawal, and J. A. T. Machado, Eds., pp. 15–26, Springer, Dordrecht, The Netherlands, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. W. Hanneken, B. N. N. Achar, D. M. Vaught, S. T. Spencer, and T. R. Ensley, “An alpha-beta phase diagram representation of the zeros and properties of the Mittag-Leffler function,” in
*Proceedings of the International Workshop on Fractional Differentiation and its Applications (FDA '10)*, University of Extremadura, Badajoz, Spain, October 2010. - J. S. Duan, Z. Wang, Y. L. Liu, and X. Qiu, “Eigenvalue problems for fractional ordinary differential equations,”
*Chaos, Solitons & Fractals*, vol. 46, pp. 46–53, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. M. Dzhrbashyan, “Interpolation and spectral expansions associated with differential operators of fractional order,”
*Soviet Journal of Contemporary Mathematical Analysis*, vol. 19, no. 2, pp. 1–116, 1984. - M. M. Djrbashian,
*Harmonic Analysis and Boundary Value Problems in the Complex Domain*, vol. 65 of*Operator Theory: Advances and Applications*, Birkhäuser, Basel, Switzerland, 1993. View at Publisher · View at Google Scholar · View at MathSciNet - I. V. Ostrovskii and I. N. Peresyolkova, “Nonasymptotic results on distribution of zeros of the function E
_{ρ}(z,*μ*),”*Analysis Mathematica*, vol. 23, no. 4, pp. 283–296, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A. Y. Popov, “On the Ostrovskii-Peresyolkova conjecture about zeros of the Mittag-Leffler functions,”
*Proceedings of the Steklov Institute of Mathematics*, supplement 1, pp. S167–S182, 2001. View at Zentralblatt MATH · View at MathSciNet - A. Y. Popov and A. M. Sedletski, “Distribution of the zeros of the Mittag-Leffler function,”
*Doklady Mathematics*, vol. 67, pp. 336–339, 2003. - A. Y. Popov, “On zeros of Mittag-Leffler functions with parameter $\rho LTHEXA1/2$,”
*Analysis Mathematica*, vol. 32, no. 3, pp. 207–246, 2006. View at Publisher · View at Google Scholar · View at MathSciNet - A. Y. Popov, “On zeros of a certain family of Mittag-Leffler functions,”
*Journal of Mathematical Sciences*, vol. 144, no. 4, pp. 4228–4231, 2005. View at Publisher · View at Google Scholar · View at MathSciNet - G. Pólya, “Bemerkung über die Mittag-Lefflerschen Funktionen,”
*Tôhoku Mathematical Journal*, vol. 19, pp. 241–248, 1921. - R. Hilfer and H. J. Seybold, “Computation of the generalized Mittag-Leffler function and its inverse in the complex plane,”
*Integral Transforms and Special Functions*, vol. 17, no. 9, pp. 637–652, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet