Operator of fractional derivative in the complex plane

Abstract

The paper deals with a fractional derivative introduced by means of the Fourier transform. The explicit form of the kernel of general derivative operator acting on the functions analytic on a curve in complex plane is deduced and the correspondence with some well known approaches is shown. In particular, it is shown how the uniqueness of the operation depends on the derivative order type (integer, rational, irrational, complex) and the number of poles of considered function in the complex plane.

30 pages, Tex, 10 PostScript figures. This slightly compressed version will appear in Comm. in Math. Phys

References in corpus (1)

• Non-Local Pseudo-Differential Operators

Cited by in corpus (11)

• The fractional - order controllers: Methods for their synthesis and application
• Phenomenology of local scale invariance: from conformal invariance to dynamical scaling
• Nonlocal Effective Gravitational Field Equations and the Running of Newton's G
• Relativistic wave equations with fractional derivatives and pseudo-differential operators
• Running of Newton's constant and non integer powers of the d'Alembertian
• Stationarity-conservation laws for certain linear fractional differential equations
• Differentiation of the Wright functions with respect to parameters and other results
• Analytic Continuation of Operators -- operators acting complex s-times -- Applications: from Number Theory and Group Theory to Quantum Field and String Theories
• Nonlocal Effective Field Equations for Quantum Cosmology
• Real and complex order integrals and derivatives operators over the set of causal functions
• A New Proof of a Theorem in Analysis by Generating Integrals and Fractional Calculus