Geodesic
$\alpha$-Differentiable functions in complex plane
Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, Tome 24 (2020) no. 2, pp. 379-389

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In this paper, the conformable fractional derivative of order $\alpha$ is defined in complex plane. Regarding to multi-valued function $z^{1-\alpha}$, we obtain fractional Cauchy–Riemann equations which in case of $\alpha=1$ give classical Cauchy–Riemann equations. The properties relating to complex conformable fractional derivative of certain functions in complex plane have been considered. Then, we discuss about two complex conformable differential equations and solutions with their Riemann surfaces. For some values of order of derivative, $\alpha$, we compare their plots.
Keywords: conformable fractional derivative, limit based fractional derivative.
Mots-clés : Cauchy–Riemann equations 
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     title = {$\alpha${-Differentiable} functions in complex plane},
     journal = {Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences},
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R. Pashaei; A. Pishkoo; M. S. Asgari; D. Ebrahimi Bagha. $\alpha$-Differentiable functions in complex plane. Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, Tome 24 (2020) no. 2, pp. 379-389. http://geodesic.mathdoc.fr/item/VSGTU_2020_24_2_a8/