EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 4, Article Number 6557 ISSN 1307-5543 – ejpam.com Published by New York Business Global A New Definition of (α, β)-Fractional Derivatives for Complex-Valued Functions and Their Analytic Properties Waseem Ghazi Alshanti1,∗, Ma ′ mon Abu Hammad1, Roshdi Khalil2 1 Department of Mathematics, Al Zaytoonah University of Jordan, Amman, Jordan 2 Department of Mathematics, The University of Jordan, Amman, Jordan Abstract. In this paper, we present a new definition of fractional derivatives of complex-valued functions, defined via two parameters (α, β). We introduce the concepts of both (α, β)-Cauchy- Riemann equations and (α, β)-fractional analytic complex valued function. 2020 Mathematics Subject Classifications: 32Q02, 30K99, 30J15 Key Words and Phrases: (α, β)-derivative, (α, β)-Cauchy-Riemann equations, (α, β)-analytic complex valued function 1. Introduction In mathematics, the positive integer ordered derivative dy dx = f ′(x) of a function y = f(x) is a concept by which we can find the instantaneous rate of change of a vertical variable y with respect to a horizontal variable x, specifically, we write f ′(x) = lim △x→0 △y △x = lim △x→0 f (x+△x)− f (x) △x , where △x and △y are, respectively, the corresponding increments of the variable x and the variable y [1]. Dynamical systems whose state evolves over time, such as population growth and the flowing of a fluid through a pipe, can be described and studied by differential equations (ordinary or partial) that involve derivative of a function. However, some of these dynamical systems that represent phenomena like electromagnetism, economy and finance, and signal processing cannot be handled accurately using standard differential equations that include positive integer ordered derivatives. In such cases, differential equations in which the included derivatives possess fractional orders come into play. It is known that, the most widely used definitions for fractional derivatives are the old ones Riemann–Liouville and Caputo definitions and the latest one [1–4], the so called conformable fractional derivative [5]. These definitions can be stated as follows: ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.6557 Email addresses: w.alshanti@zuj.edu.jo (W. G. Alshanti), m.abuhammad@zuj.edu.jo (M. A. Hammad), roshdi@ju.edu.jo (R. Khalil) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) W. G. Alshanti, M. A. Hammad, R. Khalil / Eur. J. Pure Appl. Math, 18 (4) (2025), 6557 2 of 8 (i) Riemann–Liouville definition [1]. For α ∈ [n− 1, n), the α derivative for f is Dα a (f) (t) = 1 Γ (n− α) dn dtn t∫ a f (x) (t− x)α−n+1dx. (ii) Caputo definition [2]. For α ∈ [n− 1, n), the α derivative of f is Dα a (f) (t) = 1 Γ (n− α) t∫ a f (n) (x) (t− x)α−n+1dx. (iii) Khalil definition [5]. For α ∈ (0, 1), the α-conformable derivative of f is Dαu(x) = lim ϵ→0 u(x+ ϵx1−α)− u(x) ϵ . There is still no general consensus on the concept of complex-order derivatives. The past century has seen few contributions related to this concept. For example in [6], the author assumed, together with certain assumption, that the derivative of complex order w of a complex function f (z), to be defined by the generalized Cauchy integral D(w)f (z) = Γ (1 + w) 2πi ∫ ∂D f (η) (η − z)−w−1 dη, where D is a closed region within the complex plane C. Moreover, as an application in solving hypergeometric integral equations, the concept of derivatives of purely imaginary orders were also studied [7]. Recently, the concept of α-fractional analytic function was carried out with some nice results [8]. In this paper, we present the definition of (α, β)-fractional derivative of a complex function f (z) defined on a region G ⊆ C. 2. The Definition / Proof of the basic result Throughout this paper, let E = {(x, y) : x, y > 0} and f : G ⊆ E → C be a complex valued function defined on a region G ⊆ C. Let z◦ = (x◦, y◦) ∈ G◦, the interior of G, and x◦, y◦ > 0. Definition 1. A function f : G ⊆ C → C is said to be (α, β)-differentiable at z◦ = x◦ + iy◦ ∈ G◦ and denoted by f (α,β)(z◦), where |α| , |β| < 1, if f (α,β)(z◦) = D(α,β)f (z◦) = lim (ϵ1,ϵ2)→(0,0) f(x◦ + ϵ1x 1−α ◦ , y◦ + ϵ2y 1−β ◦ )− f(x◦, y◦) ϵ1 + iϵ2 , (1) exists . W. G. Alshanti, M. A. Hammad, R. Khalil / Eur. J. Pure Appl. Math, 18 (4) (2025), 6557 3 of 8 Accordingly, let us assume f (z) = u (x, y)+iv (x, y) such that it is (α, β)-differentiable at z◦ = (x◦, y◦) . Thus, the above limit (1) exists along any bath toward z◦ and all are equal. Hence, if we move along (x, y◦) towards z◦ = (x◦, y◦), we get D(α,β)f (z◦) = lim ϵ1→0 [ u(x◦ + ϵ1x 1−α ◦ , y◦) + iv(x◦ + ϵ1x 1−α ◦ , y◦) ] − [u (x◦, y◦) + iv (x◦, y◦)] ϵ1 = uαx(x◦, y◦) + ivαx (x◦, y◦), (2) where uαx(x◦, y◦) is the α-fractional partial derivative of u with respect to x at (x◦, y◦) and vαx (x◦, y◦) is the α-fractional partial derivative of v with respect to x at (x◦, y◦). Similarly, if we move along (x◦, y) towards z◦ = (x◦, y◦), we get D(α,β)f (z◦) = lim ϵ2→0 [ u(x◦, y◦ + ϵ2y 1−β ◦ ) + iv(x◦, y◦ + ϵ2y 1−β ◦ ) ] − [u (x◦, y◦) + iv (x◦, y◦)] iϵ2 = vβy (x◦, y◦)− iuβy (x◦, y◦), (3) where vβy (x◦, y◦) is the β-fractional partial derivative of v with respect to y at (x◦, y◦) and uβy (x◦, y◦) is the β-fractional partial derivative of v with respect to y at (x◦, y◦). Therefore, for (α, β)-differentiability of f at z◦ = (x◦, y◦), we must have uαx(z◦) = vβy (z◦), uβy (z◦) = −vαx (z◦). (4) Equations (4) will be called the (α, β)-Cauchy-Riemann equtions. Noting that if α = β, then the two equations (4) represent the α-Cauchy-Riemann equations that are reported in [8]. Example 1. Consider f (z) = f(x, y) = β2x2α−α2y2β α2β2 + i2xαyβ αβ , where |α| , |β| < 1. Then we have uαx = vβy = 2 αx α, and uβy = −vαx = − 2 βy β [5]. Hence, uαx , uβy , vαx , and vβy are all exist and continuous at z◦ = x◦ + iy◦ ∈ G◦. Moreover, they satisfy the (α, β)-Cauchy-Riemann equations (4). Example 2. For f (z) = f(x, y) = e ( xα α +i y β β ) = e xα α cos yβ β + ie xα α sin yβ β , where |α| , |β| < 1. We have uαx = vβy = e xα α cos yβ β , and uβy = −vαx = −e xα α sin yβ β [5]. Hence, uαx , u β y , vαx , and vβy are all exist and continuous at z◦ = x◦+ iy◦ ∈ G◦. Moreover, they satisfy the (α, β)-Cauchy-Riemann equations (4). Clearly, as in the classical case, (α, β)-Cauchy-Riemann equations are necessary con- dition for (α, β)-differentiability of f (z) at z◦ but not a sufficient one. This leads to the following result. W. G. Alshanti, M. A. Hammad, R. Khalil / Eur. J. Pure Appl. Math, 18 (4) (2025), 6557 4 of 8 Theorem 1. (Sufficient conditions for (α, β)-differentiability) Let f : G ⊆ C → C be a complex valued function defined on a region G ⊆ C. Let z◦ = (x◦, y◦) ∈ G◦ the interior of G. If i) uαx , uβy , vαx , and vβy all exist and are continuous at z◦, ii) (α, β) -Cauchy-Riemann equations (4) are satisfied at z◦, then, f is (α, β)-differentiable at z◦. Proof. Suppose that the two conditions of the Theorem 1 hold. Now, f(x◦ + x−x◦ x1−α ◦ x1−α ◦ + i [ y◦ + y−y◦ y1−β ◦ y1−β ◦ ] )− f(x◦, y◦) x−x◦ x1−α ◦ + iy−y◦ y1−β ◦ = u(x◦ + x−x◦ x1−α ◦ x1−α ◦ , y◦ + y−y◦ y1−β ◦ y1−β ◦ )− u(x◦, y◦) x−x◦ x1−α ◦ + iy−y◦ y1−β ◦ +i v(x◦ + x−x◦ x1−α ◦ x1−α ◦ , y◦ + y−y◦ y1−β ◦ y1−β ◦ )− v(x◦, y◦) x−x◦ x1−α ◦ + iy−y◦ y1−β ◦ . But, u(x◦ + x− x◦ x1−α ◦ x1−α ◦ , y◦ + y − y◦ y1−β ◦ y1−β ◦ )− u(x◦, y◦) = u (x, y)− u(x◦, y◦) = x− x◦ x1−α ◦ uαx(x◦, y◦) + y − y◦ y1−β ◦ uβy (x◦, y◦) + √ (x− x◦) 2 + (y − y◦) 2δ1 (z) , and v(x◦ + x− x◦ x1−α ◦ x1−α ◦ , y◦ + y − y◦ y1−β ◦ y1−β ◦ )− v(x◦, y◦) = v (x, y)− v(x◦, y◦) = x− x◦ x1−α ◦ vαx (x◦, y◦) + y − y◦ y1−β ◦ vβy (x◦, y◦) + √ (x− x◦) 2 + (y − y◦) 2δ2 (z) , where z = (x, y) . By the first condition (i) of Theorem 1, that is uαx , u β y , vαx , and vβy exist and con- tinuous at z◦, we deduce that both lim z→z◦ δ1 (z) = lim z→z◦ δ2 (z) = 0. In other words, as( x−x◦ x1−α ◦ + iy−y◦ y1−β ◦ ) → 0, we have δ1 (z) → 0 and δ2 (z) → 0. Moreover, by the second condition (ii) of Theorem 1, we have u(x◦ + x− x◦ x1−α ◦ x1−α ◦ , y◦ + y − y◦ y1−β ◦ y1−β ◦ )− u(x◦, y◦) W. G. Alshanti, M. A. Hammad, R. Khalil / Eur. J. Pure Appl. Math, 18 (4) (2025), 6557 5 of 8 +i [ v(x◦ + x− x◦ x1−α ◦ x1−α ◦ , y◦ + y − y◦ y1−β ◦ y1−β ◦ )− v(x◦, y◦) ] = x− x◦ x1−α ◦ (uαx(x◦, y◦) + ivαx (x◦, y◦)) + y − y◦ y1−β ◦ ( uβy (x◦, y◦) + ivβy (x◦, y◦) ) + √ (x− x◦) 2 + (y − y◦) 2 (δ1 (z) + iδ2 (z)) = x− x◦ x1−α ◦ (uαx(x◦, y◦) + ivαx (x◦, y◦)) + y − y◦ y1−β ◦ (−vαx (x◦, y◦) + iuαx(x◦, y◦)) + √ (x− x◦) 2 + (y − y◦) 2 (δ1 (z) + iδ2 (z)) = ( x− x◦ x1−α ◦ + i y − y◦ y1−β ◦ ) (uαx(x◦, y◦) + ivαx (x◦, y◦)) + √ (x− x◦) 2 + (y − y◦) 2 (δ1 (z) + iδ2 (z)) . Therefore, uαx(x◦, y◦) + ivαx (x◦, y◦) + √ (x− x◦) 2 + (y − y◦) 2 (δ1 (z) + iδ2 (z)) x−x◦ x1−α ◦ + iy−y◦ y1−β ◦ . Which, as ( x−x◦ x1−α ◦ + iy−y◦ y1−β ◦ ) → 0, implies D(α,β)f (z◦) = uαx(x◦, y◦) + ivαx (x◦, y◦) = vβy (x◦, y◦)− iuβy (x◦, y◦). Example 3. Let f (z) = f(x, y) = β2x2α−α2y2β α2β2 , where |α| , |β| < 1. Then (α, β)-Cauchy- Riemann equations are not satisfied for all z = x + iy such that x, y ̸= 0 since uαx = 2 αx α ̸= 0 = vβy , and uβy = − 2 β y β ̸= 0 = −vαx . Consequently, by Theorem 1, f is not (α, β)-differentiable at any non-zero point. However, at z = 0 the two conditions of Theorem 1 are satisfied. So, f (α,β)(0) exists such that f (α,β)(0) = uαx(0, 0) + ivαx (0, 0) = vβy (0, 0)− iuβy (0, 0) = 0. Definition 2. A function f : G ⊆ E → C is said to be (α, β)-differentiable at on a domain G if f is (α, β)-differentiable at every z ∈ G. Clearly, that both functions in examples 1, 2 are (α, β)-differentiable over C. Now, to restate the sufficient conditions theorem for (α, β)-differentiability in polar coordinates, we proceed as follows: Let x = r cos θ and y = r sin θ. Then z = x+ iy = r (cos θ + i sin θ) = reiθ and f (z) = f(reiθ) = u (r, θ) + iv (r, θ) . (5) The (α, β)-derivative of (5), with respect to r and with respect to θ are, respectively, f (α,β)(reiθ)r1−αeiθ = uαr (r, θ) + ivαr (r, θ) , W. G. Alshanti, M. A. Hammad, R. Khalil / Eur. J. Pure Appl. Math, 18 (4) (2025), 6557 6 of 8 f (α,β)(reiθ)irθ1−βeiθ = uβθ (r, θ) + ivβθ (r, θ) . (6) Hence, by equating both equations in (6), we get the polar version of the (α, β)-Cauchy- Riemann equations, namely, uαr = θβ−1 rα vβθ , uβθ = − rα θβ−1 vαr . (7) Definition 3. A function f : G ⊆ E → C is said to be (α, β)-analytic at z◦ = x◦+iy◦ ∈ G◦ where |α| , |β| < 1, if i) f is (α, β) -differentiable at z◦, ii) there exists ϵ > 0 such that f is (α, β) -differentiable for all z ∈ D (ϵ, z◦) . Clearly, both functions in examples 1 and 2 are (α, β)-analytic. Definition 4. A function f : G ⊆ C → C is said to be (α, β)-analytic the domain G where |α| , |β| < 1, if it is (α, β)-analytic at every z ∈ G. 3. Applications, Discussion, and Implications The concept of the (α, β)-fractional derivative offers promising potential in a variety of scientific and engineering domains. In the area of signal and image processing, classical derivatives often fail to capture the memory and hereditary properties inherent in certain signals and textures. By introducing two parameters that independently influence the phase and magnitude in two dimensions, the (α, β)-fractional derivative provides finer control over 2D signal representations. This approach can be leveraged for more precise techniques in edge detection, texture segmentation, and in adaptations of the fractional Fourier transform. In fractional control theory, modern controllers such as fractional PID systems al- ready benefit from generalized derivatives to describe systems with memory (See [9], [10] ). Extending these ideas to the complex domain, the (α, β)-fractional derivative could be incorporated into the design of complex transfer functions and impulse responses for systems with asymmetrical time scales—where, for example, position and velocity evolve under different fractional dynamics. In the study of complex dynamics and fractals, the generalized derivative opens a pathway to modeling non-local interactions and generating fractional Julia or Mandelbrot sets. The flexibility of the parameters α and β may lead to the creation of entirely new families of fractals whose dimensions and symmetries can be tuned, with possible applications in data compression, encryption, and chaotic modeling. W. G. Alshanti, M. A. Hammad, R. Khalil / Eur. J. Pure Appl. Math, 18 (4) (2025), 6557 7 of 8 Physical modeling in areas such as fluid flow or electromagnetic field theory may also benefit from the (α, β)-Cauchy-Riemann framework. In inhomogeneous or anisotropic media, conventional calculus often cannot account for asymmetric diffusion or non-local interactions. The polar-coordinate form derived in this paper suggests that the new deriva- tive could capture these effects with greater accuracy. Finally, from a purely mathematical perspective, the new definition unifies conformable derivatives with the classical Cauchy-Riemann theory, thereby bridging real and com- plex fractional calculus. It invites further investigation into its relationships with other fractional analytic function classes, such as Hadamard-type or Caputo-type, potentially contributing to a more comprehensive theory of fractional complex analysis. 4. Conclusions This work has introduced and developed the concept of the (α, β)-derivative for complex- valued functions, along with its associated (α, β)-Cauchy-Riemann equations and the def- inition of (α, β)-analytic functions. By generalizing the idea of fractional differentiation to allow independent fractional orders in two orthogonal directions, we have extended both the theoretical framework and the potential applicability of fractional complex analysis. The proposed derivative not only preserves many of the desirable properties of classical complex derivatives but also adds new degrees of freedom for modeling phenomena where asymmetry, non-locality, and memory effects are present. Its polar-coordinate formulation further enriches the theory by connecting it naturally to problems with radial and angular components. Looking ahead, there are several promising directions for future work. Analytically, it would be valuable to explore the connections between (α, β)-analyticity and other gen- eralized analytic classes, as well as to investigate the spectral and mapping properties of operators defined via this derivative. Numerically, developing efficient algorithms for computing (α, β)-derivatives could enable applications in image analysis, control system simulation, and computational physics. From an applied standpoint, the flexibility of the new framework could be harnessed to model asymmetric processes in engineering, physics, and finance, where traditional integer-order calculus fails to capture the observed behavior. In essence, the (α, β)-fractional derivative provides a robust and adaptable mathemat- ical tool that has the potential to deepen our understanding of complex-valued functions and to inspire innovative solutions across multiple disciplines. Acknowledgements The authors would like to thank the editor of EJPAM as well as the anonymous reviewers for their valuable comments and suggestions. W. G. Alshanti, M. A. Hammad, R. Khalil / Eur. J. Pure Appl. Math, 18 (4) (2025), 6557 8 of 8 References [1] K. S. Miller. An Introduction to Fractional Calculus and Fractional Differential Equa- tions. J. Wiley and Sons, New York, USA, 1993. [2] K. Oldham and J. Spanier. The Fractional Calculus: Theory and Applications of Differentiation and Integration of Arbitrary Order. Elsevier, Massachusetts, USA, 1974. [3] A. Kilbas, H. Srivastava, and J. Trujillo. Theory and Applications of Fractional Differential Equations. Elsevier, Massachusetts, USA, 2006. [4] I. Podlubny. Fractional Differential Equations. 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