New results on complex conformable integral

A new theory of analytic functions has been recently introduced in the sense of conformable fractional derivative. In addition, the concept of fractional contour integral has also been developed. In this paper, we propose and prove some new results on complex fractional integration. First, we establish necessary and sufficient conditions for a continuous function to have antiderivative in the conformable sense. Finally, some of the well-known Cauchy’s integral theorems will also be the subject of the extension that we do in this paper.


Introduction
Integrals are very important in studying the functions of a complex variable. The integration theory is notable for its mathematical elegance. The theorems are generally concise and powerful, and their proofs are simple. The theory has also multiple applications in applied mathematics, engineering, and natural sciences, [1].
Fractional Calculus has attracted a great interest from mathematicians, scientists, and engineers due to the fact that many systems can be modeled in the context of fractional-order derivatives better than modeling them using the classical integer-order derivatives because many systems and phenomena exhibit memory effect and have internal damping [25]. The main idea of Fractional Calculus was initiated at the end of the 17 th century when the French mathematician L'Hôpital proposed an interesting research question about the half-order derivative [25]. This was the point where the discussion started going into discovering this new mathematical idea in more details, and recently this topic of research created a revolution in applied mathematics and sciences. Various fractional derivative definitions have been formulated, and the most common ones in research studies are the Caputo fractional derivative and Riemann-Liouville fractional derivative.
In this current paper, our main objective is to establish, in the fractional context, important results on the existence of antiderivatives of a function of a complex variable in a certain domain, such as Cauchy-Goursat Theorem. In this way, we extend the research done in [2].
In this complex analysis fractional formulation, we will use the definitions of conformable derivatives and integrals, introduced by Khalil et al. [3]. From this new definition of derivative, important elements of the mathematical analysis of functions of a real variable have been established, among which we can highlight: Rolle's Theorem, Mean Value Theorem, chain rule, fractional power series expansion, fractional integration by parts formulas, and fractional single Laplace transform and double Laplace transform definitions, [3][4][5]25]. This new definition is a limit-based definition that satisfies the properties of the usual derivatives.
The conformable partial derivative of the order ∈ 0,1] of the real-valued functions of several variables and conformable gradient vector are defined, as well as the conformable version of Clairaut's Theorem for partial derivatives of conformable fractional orders is proven in [6]. In [7], the conformable Jacobian matrix is defined; the chain rule for multivariable conformable derivative is given; the relation between conformable Jacobian matrix and conformable partial derivatives is revealed. In [8], two new results on homogeneous functions involving their conformable partial derivatives are introduced, specifically, the homogeneity of the conformable partial derivatives of a homogeneous function and the conformable version of Euler's Theorem. Furthermore, in a short time, several research studies have been conducted on the theory and applications of fractional differential equations based on this newly defined fractional derivative, [9][10][11][12][13][14][15][16][17][18][19][20][21]25]. In addition, both physical and geometrical meanings of conformable derivatives have been interpreted in [24,26], respectively. While many of the classical and well-known fractional derivatives can offer many advantages in modelling various physical systems, there are some disadvantages associated with those fractional derivatives such as the difficulty of obtaining analytical solutions and the need to implement special numerical techniques to overcome this challenge. However, conformable derivative provides a great help in finding analytical solutions in a very simple and effective way comparing to the classical fractional derivatives such as Grünwald-Letnikov, Caputo, and many others (we refer to [25] for more information about some analytical methods for solving wave equation in the context of conformable derivative).
In relation to complex fractional analysis, a theory of analytic functions in the conformable sense is developed in [22,23]. Furthermore, the definition of conformable integral along a contour and its first properties are introduced in [2].
The paper is organized as follows: In Section 2, the main concepts of conformable fractional calculus are presented. In Section 3, an important property of moduli of fractional contour integrals is presented. In Section 4, we establish the necessary and sufficient conditions for a continuous function to have − antiderivative. In Section 5, we develop the conformable version of some of the well-known Cauchy Theorems, specifically, the Cauchy-Goursat theorem, the Cauchy theorem for star-shaped domains, and the Morera theorem. In Section 6, we present several examples that show how the physical concepts, such as the circulation and the net flow of the velocity vector field of a two-dimensional fluid flow, can be modified appropriately to directly acclimatize the perception of fractional contour integrals.

Basic definitions and tools
We state in this section some definitions, remarks, and theorems which are important for our research study: Remark 2.2. [27]. On one hand, ∀ ∈ Ν, assume that = and Φ = where ∈ , then the De Moivre's formula can be defined as: = + . On the other hand, assume now that = Ψ + Ψ where ≠ 0; ∈ , then there exists a number of such that the roots of = can be expressed as follows: Definition 2.1. [22]. Let : → be a given function and 0 ∈ − 0 (which is an open set). Then, In this case, we write ( ) 0 for such fractional derivative. Also, we use 0 as an alternative way of writing the fractional derivative. If is -differentiable in some neighborhood of = 0, and lim →0 exists, then we define: As a consequence of the above definition, the following useful theorem is obtained: Theorem 2.1. [23]. Let : → be a given function and 0 ∈ − 0 (which is an open set). If is -differentiable at 0 > 0, then is continuous at 0 . One can easily prove the following: Theorem 2.2. [22,23]. The fractional derivative in Definition 2.1 satisfies the following: (i) = 0, for all constant functions such that = .
Motivated by the research work done in [27], to have a good visualization for some the selected functions in Theorem 2.     Definition 2.2. [22]. Let : ⊂ → such that is −differentiable at 0 and in some disk around 0 . Then, is called −analytic at 0 . If is −analytic at each in a region , then we say is −analytic on .
The following definition is the -fractional integral of a complex-valued function of a complex variable: Definition 2.3. Let : , → be a contour, with > 0 and * ⊂ − −∞. 0 ]. Let : * → be continuous. We define the contour − integral of along as soon as we understand the complex number [2], Note that, since γ is a contour, T α γ t is also piecewise continuous on the closed interval a, b ; hence, the existence of integral (2) is ensured.

Upper bounds for moduli of contour −integrals
In this section, we consider a contour : , → , with > 0 and * ⊂ − −∞. 0 ]. Theorem 3.1. Let : , → be a contour. For a continuous function : * → , we have: where is the length of the contour .

Proof. By theorem 4.4 in [2]
, we have: Remark 3.1. In practice, it is not necessary to determine because it is very often to have an easy estimate as follows: Hence, we have max 1− : ∈ * ≤ . By using (3), we obtain the following: which in practice is just as useful as (3). Solution. The midpoint of is clearly the closest on to the origin. The distance of that midpoint from the origin is clearly 2, and the length of is 2 2.
Hence, if is any point of , ≥ 2 . This means that, for such a point Consequently, by taking = 16 and = 2 2, we have:

−Antiderivatives
Although the value of a contour −integral of a function from a fixed point 1 to a fixed point 2 depends generally on the path taken, there are certain functions whose −integrals from 1 to 2 have values that are independent of path. The following theorem is useful in determining when fractional integration is independent of path, and moreover, when an −integral around a closed path has a value of zero.
To prove the theorem, we shall discover a fractional extension of the fundamental theorem of calculus that simplifies the evaluation of many contour −integrals. That extension involves the concept of an −antiderivative of a continuous function in a domain ⊂ − −∞. 0 ], which we define as follows: The −integrals of around closed contours lying entirely in all have zero value. Proof. To prove the above theorem, it is sufficient to show that the statement (i) implies statement (ii), which also implies statement (iii), and finally that statement (iii) implies statement (i). Let assume that statement (i) is true. If a contour from 1 to 2 , lying in , is just a smooth curve, with parametric representation : , → , with > 0. By the chain rule, [4], we have: By the fundamental theorem of calculus which can be extended to be applicable to complex-valued functions of a real variable [2], so it follows that Since = 1 and = 2 , the value of this contour −integral can be written as: As a result, the above value is independent of the contour as long as extends from 1 to 2 and lies in . That is, where is smooth. Expression (4) is also valid when is any contour, not necessarily a smooth one, that lies in . If consists of a finite number of smooth curves = 1,2, … , , each is extending from point to a point +1 , then The fact that statement (ii) follows from statement (i) is now established. To see that statement (ii) implies statement (iii), we let 1 and 2 represent any two points on a closed contour lying in and form two paths, each with initial point 1 and final point 2 , such that = 1 + − 2 = 1 − 2 . By assuming that statement (ii) is true, one can write That is, the −integral of around the closed contour = 1 − 2 has a zero value. It remains to show that statement (iii) implies statement (i). We do this by assuming that statement (iii) is true, establishing the validity to statement (ii), and then arriving at statement (i). To see that statement (ii) is true, we let 1 and 2 represent any two contours, lying in , from a point 1 to a point 2 and let's observe that, in view of statement (iii), Equation (6) holds. Thus, Equation (5) holds. Therefore, integration is independent of path in , and we can define the function as: on . The proof of the theorem is complete once we show that = ( ) everywhere in . We do this by letting + ∆ • 1− be any point, distinct from , lying in some neighbourhood of that is small enough to be contained in . Then, we have: where the path of integration from to + ∆ • 1− may be selected as a line segment. Since Hence, we obtain the following: Consequently, since is continuous, we get ∆ →0

Fractional Cauchy's integral theorem
The usefulness of Theorem 4.1 to prove the existence of −antiderivatives is doubtful, since to justify that a function has −antiderivatives in a certain domain , it is necessary to verify that its −integral along any closed curve in is zero, which does not seem an easy task in practice. This problem in classical complex analysis is solved through theorems that guarantee that under certain conditions, the integral of a function along any curve closed is null. Those theorems are called Cauchy theorems. In those theorems, we consider an open set and a closed curve in .
Additional hypotheses are assumed in or to conclude that = 0 for each analytic function in . In this section, we will first establish a conformable version of one of the most important classical integral theorems mentioned above, the so-called Cauchy-Goursat theorem.
Proof. We will call = , , , , ∆= ∆ , , and = 1− . The objective is to prove that = 0. For this, let's consider the midpoints of the sides ´= We can write in the following form: This equality is true since there are paths (the ones inside the triangle) which are travelled in opposite directions, so the respective integrals are cancelled (see Figure 5). If we call these four integrals 1 , 2 , 3 , 4 , and 1 one of the integrals with the largest modulus from them, and 1 = 1 , 1 , 1 , 1 to the path of 1 , we have: By repeating the same argument for the triangle ∆ 1 = ∆ 1 , 1 , 1 , we obtain a sequence of triangles ∆ = ∆ , , and polygonal = , , , with the property that ∆ ⊂ ∆ −1 and (which exists since we are considering the intersection of a sequence decreasing non-empty closed with a succession of diameters converging to zero in complete metric space). Clearly, 0 ∈ ∆ , , . Let's put which is a polynomial function. Therefore, it has a −antiderivative; then 1− = 0. Hence, we can write the following: If we take Given > 0, due to the −differentiability of at 0 , there exists > 0 such that the open disk 0 , is contained in , and for all ∈ 0 , , we have: If the ∆ < then ∆ ⊂ 0 , . Furthermore, if denotes the minimum value of 1− in * , we have: Since > 0 is chosen arbitrary, we get = 0.
1. An open set ⊂ − −∞. 0 ] is a star-shaped domain with respect to a point 0 ∈ if the line segment that unites 0 with any other point of stays inside , that is, 0 . * ⊂ for all ∈ . For example, a disk is a star-shaped domain with respect to any of its points. Of course, any convex set is a star-shaped domain over any of its points, but there are star-shaped domains that are not convex (such as the polygon shown in the Figure 6). Proof. Let ⊂ − −∞. 0 ] be a star-shaped domain with respect to 0 and let be −analytic on . We are looking for an −antiderivative of , and the most intuitive way to define it is as follows: We are going to show that the defined is certainly −antiderivative of in . Since is a star-shaped domain at 0 , the function is well defined. Let ∈ and > 0, such that , ⊂ . Let us take a point ∈ , . Since all the points of the segment , are contained in , then the segment that joins 0 with any of these points will be contained in as this is a star-shaped domain. Therefore, the triangle ∆ 0 , , is totally contained in (see Figure 6). From the previous equality, and by following the same reasoning that we use in the proof of the characterization of the existence of −antiderivatives (see Theorem 4.1), it is easily proven that is −differentiable at and = ( ), which concludes the proof. Finally, we will establish a result that is a reciprocal of the fractional Cauchy-Goursat theorem for triangles. for all ∈ 0 , . is an −antiderivative of on 0 , , that is, = ( ) for all ∈ 0 , (proof of this is identical to the proof of the existence of primitives in star-shaped domains, see theorem 5.2). Then f is the −derivative of −analytic on function on disk 0 , . Therefore, it is an −analytic function on the mentioned disk; is particulary an −differentiable at 0 . Since 0 is chosen arbitrary, it follows that is −analytic on .

Application: Circulation and net flow
It is a well-known result that if = , + ( , ) is the complex representation of the velocity vector field , = , + ( , ) of a two-dimensional fluid flow, [28], the circulation of around a closed contour is given by Likewise, the net flux of across a closed contour can be defined as ( ) .
In the following examples we will formulate these physical concepts in terms of fractional contour integrals: Thus, the circulation around is 0, and the net flux across is 2 .

Conclusions
The main objective of this work is to establish some generalizations in the field of fractional calculus, and to provide some important results on complex integration. The objective has been successfully achieved, so the definition of fractional contour integral has been used to construct some results, such as the necessary and sufficient conditions for a continuous function to have antiderivative in the conformable sense and the extension of some of the well-known Cauchy integral theorems. It seems that the results obtained in this work correspond to the results obtained in the classical case. Finally, we would like to indicate that this research work opens the door for further research studies on developing the complex fractional integration with applications to natural sciences and engineering.