1. q-analytic functions, fractals and generalized analytic functions
- Author
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Oktay K. Pashaev, Sengul Nalci, TR57865, TR57807, Pashaev, Oktay, Nalcı, Şengül, and Izmir Institute of Technology. Mathematics
- Subjects
Statistics and Probability ,Pure mathematics ,Complex-valued function ,Calculus ,Mathematical analysis ,General Physics and Astronomy ,Statistical and Nonlinear Physics ,01 natural sciences ,Quantum ,Fock-Bargman representation ,010305 fluids & plasmas ,Complex analysis ,Complex dynamics ,Fractals ,Quasi-analytic function ,Special functions ,Generalized analytic functions ,Modeling and Simulation ,0103 physical sciences ,Non-analytic smooth function ,010306 general physics ,Mathematical Physics ,Wirtinger derivatives ,Analytic function ,Mathematics - Abstract
We introduce a new class of complex functions of complex argument which we call q-analytic functions. These functions satisfy q-Cauchy-Riemann equations and have real and imaginary parts as q-harmonic functions. We show that q-analytic functions are not the analytic functions. However some of these complex functions fall in the class of generalized analytic functions. As a main example we study the complex q-binomial functions and their integral representation as a solution of the D-bar problem. In terms of these functions the complex q-analytic fractal, satisfying the self-similar q-difference equation is derived. A new type of quantum states as q-analytic coherent states and corresponding q-analytic Fock-Bargmann representation are constructed. As an application, we solve quantum q-oscillator problem in this representation, and show that the wave functions of quantum states are given by complex q-binomials., The Scientific and Technological Research Council of Turkey (110T679); Izmir Institute of Technology
- Published
- 2014
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