International Journal of Engineering Research and Reviews
ISSN 2348-697X (Online) Vol. 10, Issue 4, pp: (9-14), Month: October - December 2022, Available at: www.researchpublish.com
Limits of Some Fractional Power Exponential Functions Chii-Huei Yu School of Mathematics and Statistics, Zhaoqing University, Guangdong, China DOI: https://doi.org/10.5281/zenodo.7351367
Published Date: 23-November-2022
Abstract: In this article, we study the limit problems of some fractional power exponential functions. Jumarie type of Riemann-Liouville (R-L) fractional derivative, fractional L’Hospital’s rule, and a new multiplication of fractional analytic functions play important roles in this paper. In fact, our results are the generalization of these results in classical calculus. Keyword: fractional power exponential functions, Jumarie type of R-L fractional derivative, fractional L’Hospital’s rule, new multiplication, fractional analytic functions.
I. INTRODUCTION Fractional calculus was proposed by Leibniz and L’Hospital in a letter dated September 30, 1695. Leibniz and L’Hospital both know about ordinary calculus and put forward the problem of noninteger differential of simple functions. Leibniz concluded the discussion by asserting that one day, the noninteger differential problem will be solved for the benefit of mankind. This unsatisfactory answer has inspired the further development of Lacroix, Fourier, Abel, Riemann, Riemann, Liouville and others in the past 300 years. In recent years, fractional calculus has become an increasingly popular research area due to its effective applications in different scientific fields such as economics, viscoelasticity, physics, dynamics, biology, control theory, and so on [1-7]. However, the definition of fractional derivative is not unique. The commonly used definitions include Riemann-Liouville (R-L) fractional derivative, Caputo fractional derivative, Grunwald-Letnikov (G-L) fractional derivative, and Jumarie’s modified R-L fractional derivative [8-11]. Since Jumarie type of R-L fractional derivative helps to avoid non-zero fractional derivative of constant function, it is easier to use this definition to connect fractional calculus with traditional calculus. This paper studies the limit problems of some fractional power exponential functions. Jumarie’s modified R-L fractional derivative, fractional L’Hospital’s rule, and a new multiplication of fractional analytic functions play important roles in this article. In fact, our results are the generalization of these results in ordinary calculus.
II. DEFINITIONS AND PROPERTIES Firstly, the fractional derivative used in this paper and its properties are introduced below. Definition 2.1 ([12]): Let 0 < 𝛼 ≤ 1, and 𝑥0 be a real number. The Jumarie type of Riemann-Liouville (R-L) 𝛼-fractional derivative is defined by ( 𝑥0 𝐷𝑥𝛼 )[𝑓(𝑥)] =
1 𝑑 𝑥 𝑓(𝑡)−𝑓(𝑥0 ) 𝑑𝑡 ∫ Γ(1−𝛼) 𝑑𝑥 𝑥0 (𝑥−𝑡)𝛼
.
(1)
where Γ( ) is the gamma function. Proposition 2.2 ([13]): If 𝛼, 𝛽, 𝑥0 , 𝑐 are real numbers and 𝛽 ≥ 𝛼 > 0, then
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