Journal of Samara State Technical University, Ser. Physical and Mathematical SciencesJournal of Samara State Technical University, Ser. Physical and Mathematical SciencesВестник Самарского государственного технического университета. Серия «Физико-математические науки»1991-86152310-7081Samara State Technical University4199510.14498/vsgtu1734ArticlesСтатьиResearch Article$\alpha$-Differentiable functions in complex plane$\alpha$-Дифференцируемые функции в комплексной плоскостиPashaeiRonakPishkooAmirapishkoo@gmail.comAsgariMohammad Sadeghmsasgari@yahoo.comEbrahimi BaghaDavood

PhD, Professor

PhD, профессор

Islamic Azad University Central Tehran BranchNuclear Science and Technology Research Institute31072020242VOL 24, NO2 (2020)ТОМ 24, №2 (2020)37938904082020Copyright ©; 2020, Samara State Technical UniversityCopyright ©; 2020, Самарский государственный технический университет2020Samara State Technical UniversityСамарский государственный технический университетhttps://creativecommons.org/licenses/by/4.0https://journals.eco-vector.com/1991-8615/article/view/41995

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.

В комплексной плоскости вводится взвешенная дробная производная порядка $\alpha$.Относительно многозначной функции $ z ^ {1- \alpha} $ получены дробные уравнения Коши–Римана, которые при $ \alpha = 1 $ совпадают с классическими уравнениями Коши–Римана.Для некоторых функций в комплексной плоскости рассмотрены свойства, относящиеся к комплексной взвешенной дробной производной.Обсуждаются два комплексных дифференциальных уравнения специальной формы. Для некоторых значений $\alpha$ приводятся римановы поверхности их решений и сравниваются их графики.

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