Abstract
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.
Highlights
The fractional calculus is an area of intensive research and development that can be historically divided into old and new parts
The conformable fractional derivative of order α is defined in complex plane
The properties relating to complex conformable fractional derivative of certain functions in complex plane have been considered
Summary
The fractional calculus is an area of intensive research and development that can be historically divided into old and new parts. Discussing the inversion of the integral equation by Grünwald in 1867, and proposing the sum of orders in the product of fractional derivatives by Letnikov in 1868 opened new ways. In 1872, Letnikov clarified the generalization of Cauchy’s integral formula and utilized fractional derivatives to address differential equations. In. 2014, Khalil et al from one side and a few months later Katugampola from the other side proposed two limit based fractional derivatives as conformable derivatives [11,12]. 1. All fractional derivatives do not satisfy the known formula of the derivative of the product of two functions: Daα(f g) = f Daαg + gDaαf. 2. All fractional derivatives do not satisfy the known formula of the derivative α-Differentiable functions in complex plane of the quotient of two functions: Daα(︁.
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