Speaker: Arran Fernandez, Department of Mathematics, Eastern Mediterranean University, Northern Cyprus

Time : 15:00 - 16:00 CEST (Rome/Paris)

Hosted at: SISSA, International School of Advanced Studies, Trieste, Italy

Zoom : A Zoom link will appear here, an hour before the talk

Organizers : Pavan Pranjivan Mehta* (pavan.mehta@sissa.it) and Arran Fernandez** (arran.fernandez@emu.edu.tr)

* SISSA, International School of Advanced Studies, Italy

** Eastern Mediterranean University, Northern Cyprus

Keywords: complex analysis, branch cuts, residue theorem

Abstract: The residue theorem is one of the fundamental results of complex analysis, including various facts such as Cauchy's theorem and Cauchy's integral formula as particular cases. Even though the complex-analysis viewpoint inspired some of the pioneering work on fractional calculus in the 19th century, it is surprising that a fractional residue theorem has (apparently) not been created in the literature up to today. The current work originated in an idea of an amateur mathematician, Egor Zaytsev, who proposed a natural idea for fractional "pseudo-residues'' which always seemed to give correct results when applied to real integral problems that could be verified computationally. Some theoretical work from my side was able to prove a fractional residue theorem, subject to some necessary complications around branch cuts and non-local pseudo-residues. We used this result to prove rigorously several elegant formulae for real integrals involving fractional powers, and it is expected that our result may have various further applications and ramifications in the study of complex analysis and special functions.

This work is dedicated to the memory of Jerry B. Keiper and Thomas J. Osler, 20th-century contributors to fractional calculus from the complex analysis viewpoint, who are underappreciated nowadays and whose approach to fractional calculus has shaped my own.

Biography: Arran Fernandez is a pure mathematician and associate professor at the Eastern Mediterranean University, specialising in fractional calculus. He completed his bachelor's, master's, and PhD at the University of Cambridge, where he began as the youngest student at the university (aged 15) and came top of his year to be the youngest-ever senior wrangler. Since completing his PhD in 2018, he has worked at the Eastern Mediterranean University, firstly as an assistant professor and then as an associate professor. He also spent a year working at Sultan Qaboos University, also as an associate professor. His research interests lie in connecting fractional calculus with other branches of mathematics, such as abstract algebra, analytic number theory, and Clifford analysis. His achievements in these directions include extending Mikusinski’s operational calculus to fractional PDEs, expressing the Riemann zeta function as a fractional differintegral of an elementary function, and defining fractional d-bar derivatives in complex and hypercomplex settings. He is also listed as an editorial board member for several international mathematical journals, including Communications in Nonlinear Science & Numerical Simulation and Fractional Calculus & Applied Analysis.

Bibliography

[1] E. Zaytsev, A. Fernandez, “A fractional residue theorem and its applications in calculating real integrals”, Bulletin of the London Mathematical Society, 58 (2026), 70351.