New definitions of fractional derivatives and integrals for complex analytic functions

Abstract

In this paper, we introduce a ground-breaking approach to defining fractional calculus for a selected class of analytic functions. Our new definitions, based on a novel and intuitive understanding of fractional derivatives and integrals, offer improved mathematical tractability for a variety of applications, including physics, engineering and finance. Our approach significantly simplifies the complexity of mathematical functions compared to the traditional Riemann-Liouville approach, by using simple functions rather than special functions, while preserving the intrinsic sense of fractional calculus. This article not only presents our proposed definitions but also provides a thorough analysis of their properties and advantages. The conclusion of this paper discusses the potential for future research in the field of fractional calculus.

Keywords:

• Control theory
• fractional calculus
• Hurwitz zeta function
• mathematical tractability
• Riemann zeta function
• signal processing

AMS Subject Classification:

• 26A33
• 30C65
• 33E12

1. Introduction

Fractional calculus is a field of mathematical analysis that deals with the properties of functions and operators defined by non-integer order derivatives and integrals. This is a generalization of the more familiar integer order calculus, which studies functions and operators defined by integer order derivatives and integrals. Fractional calculus is a relatively new field, with its origins dating back to the 1690s, but it has experienced significant growth in recent years, with applications in a wide range of fields, including physics, engineering and finance (Carpinteri & Fabrizio, n.d.; Sabatier, Agrawal, & Tenreiro Machado, n.d. Trujillo & Rubio, n.d.-a, n.d.-b).

Recently, research in fractional calculus has been focused on developing new mathematical tools and techniques for analyzing fractional systems and applying them to diverse areas (Hilfer, n.d.; Meerschaert & Sikorskii, n.d.; Miller, 1993; Oldham & Spanier, 1974; Oldham & Spanier, n.d.; Podlubny, 1999; Samko, Kilbas, & Marichev, n.d.). Furthermore, the combination of fractional calculus with other mathematical tools like complex analysis and linear algebra has provided new insights and possibilities in the field (Bhrawy, Taha, & Machado, 2015; Fernandez & Husain, 2020).

One of the key features of fractional calculus is the use of non-integer order derivatives and integrals, such as the Riemann-Liouville and Caputo derivatives (Jiang & Zhang, 2020; Li, Qian, & Chen, 2011). These operators have properties that differ from their integer order counterparts and allow for a more general and flexible description of functions and phenomena.

Fractional calculus has been found to be particularly useful in modeling phenomena that exhibit memory or hereditary properties, such as viscoelastic materials in physics and volatility of asset prices in finance. It has also been applied in the control and optimization of dynamic systems (Carpinteri & Fabrizio, n.d.; Sabatier et al., n.d. Trujillo & Rubio, n.d.-a, n.d.-b).

Although traditional definitions of fractional calculus often lack clear physical interpretations (Jiang & Zhang, 2020; Li et al., 2011; Moshrefi-Torbati & Hammond, 1998), and solving fractional differential equations often involves converting them to standard integer-order equations. Recently some newer definitions have appeared, but many of these also lack clear physical meanings and are simply parametric derivations (Abdelhakim, 2019; Abdelhakim & Machado, 2019).

In this paper, we present innovative fractional calculus definitions that extend the concept of fractional derivatives for known functions. These definitions offer a more intuitive grasp of fractional derivatives and integrals, enhance mathematical tractability and hold potential for applications in domains such as physics, engineering and finance. The article will deliver a comprehensive introduction to our novel definitions, a thorough examination of their properties and illustrations of their practical applications.

The paper is organized into several sections, each addressing distinct aspects of the proposed definitions. Section 2, titled Fractional Calculus Foundations and Literature Review, delves into the extension of derivatives and integrals to real orders for specific functions. Section 3 explores the methodology for deriving formulas for fractional order derivatives and integrals, applicable to both specific and analytic functions. In section 4, the practical applications of these definitions are examined, along with examples from various fields. This section also includes a comparison with other existing definitions, shedding light on the strengths and weaknesses of the proposed approach. Finally, section 5 summarizes the research conclusions, underlining key findings and contributions while suggesting avenues for future work.

2. Fractional calculus foundations and literature review

In this section, we will explore the foundations of fractional calculus by examining the fractional derivatives of sine and cosine functions and the generalized derivative for functions that can be expressed as a series of exponentials. These fundamental concepts have significantly contributed to the development and understanding of fractional calculus, paving the way for various applications across multiple disciplines. We will present a comprehensive literature review, emphasizing the work of notable mathematicians such as Liouville, Scott Blair, Heaviside, Samko, Kilbas and Marichev. Their contributions have been vital in establishing key approaches and methods, facilitating a more intuitive understanding and computational efficiency in handling fractional derivatives and integrals. Our objective is to provide an in-depth overview of these foundational concepts and their impact on the broader field of fractional calculus.

2.1. Literature review: Riemann-Liouville and Caputo fractional derivatives

i. Riemann-Liouville fractional integral and fractional derivatives

The Riemann-Liouville fractional integral, a cornerstone concept in fractional calculus, has been instrumental in the development of the theory and understanding of fractional derivatives. It was initially proposed by Riemann in 1847 and subsequently expanded upon by Liouville in 1861.

The Riemann-Liouville fractional integral of a function $f(x)$ with order $\alpha \hspace{0.17em}>\hspace{0.17em}0$ and a fixed base point a is given by: (2.1) $${{ }_a^{RL}I}^{\alpha } f\left(x\right)= \frac{1}{\Gamma \left(\alpha \right)} {\int }_a^x f\left(t\right){\left(x - t\right)}^{\alpha -1} dt.$$(2.1)

Here, $\Gamma$ represents the gamma function, and the integral is evaluated from the base point $a$ to x. The $\alpha$ parameter is a complex number in the half-plane where $Re(\alpha ) >0.$ The dependence on the base point a is often not explicitly mentioned, indicating the flexibility in the constant of integration. Another notation that emphasizes the base point is: (2.2) $${ }_a^{RL}{D}_x^{-\alpha } f\left(x\right)=\frac{1}{\Gamma \left(\alpha \right)} {\int }_a^{x}f\left(t\right){\left(x - t\right)}^{\alpha -1} dt.$$(2.2)

This definition remains valid when $a = -\infty ,$ provided appropriate restrictions are placed on $f.$ It is obviously that $I^1 f$ serves as an antiderivative of $f$ (of the first order), and for positive integer values of $\alpha ,$ $I^{\alpha }$f represents an antiderivative of α-th order, as per Cauchy’s formula for repeated integration.

ii. Caputo fractional derivative

Another popular definition of fractional derivatives is the Caputo fractional derivative, introduced by Caputo in 1967. The Caputo definition is particularly useful in the field of applied mathematics, as it provides a more natural initial condition interpretation compared to the Riemann-Liouville definition. This makes it suitable for modeling physical and engineering problems.

The Caputo fractional derivative of a function $f(x)$ of order $\alpha >0$ is defined as: (2.3) $${ }_a^C{D}_x^{\alpha } f\left(x\right)= \frac{1}{\Gamma \left(n - \alpha \right)} {\int }_a^x{\left(x - t\right)}^{n-\alpha -1} f^{\left(n\right)}\left(t\right) dt,$$(2.3) where $n - 1 < \alpha \le n,$ and $f^{\left(n\right)}$ denotes the $n$-th order integer derivative of $f.$

The Caputo fractional derivative is closely related to the Riemann-Liouville definition. However, the main difference between the two lies in their treatment of initial conditions, which makes the Caputo derivative more suitable for modeling real-world phenomena.

2.2. Fractional derivatives of sine and cosine functions

Defining fractional derivatives of sine and cosine functions using fractional multiples of $\frac{\pi }{2}$ is a well-established approach, equivalent to the standard Riemann-Liouville definition of fractional calculus with the case of half-axis or axis. This method can be found in various sources, such as the comprehensive textbook by Samko et al. (n.d.), where it is discussed in Section 5.1.

The formula for the fractional derivative of sine and cosine functions is as follows: (2.4) $$\frac{d^p}{d{x}^p}\left(\cos \left(nx\right)\right) = n^p \cos \left(\frac{p\pi }{2}+nx\right),$$(2.4) (2.5) $$\frac{d^p}{d{x}^p}\sin \hspace{0.17em}\left(nx\right)=n^p\hspace{0.17em}\sin \hspace{0.17em}\left(\frac{p\pi }{2}+nx\right).$$(2.5)

These expressions show the $p-th$ order derivative of the sine and cosine functions concerning $x.$ The parameter $n$ represents the frequency of the sine or cosine function, and $p\ge 0$ is the order of the derivative. The fractional multiples of $\frac{\pi }{2}$ are added to the argument of the sine or cosine function, which allows for the calculation of fractional derivatives.

The equivalence between the approach of defining fractional derivatives of sine and cosine functions using fractional multiples of $\frac{\pi }{2}$ and the Riemann-Liouville definition offers several benefits. First, it provides a more intuitive understanding of fractional derivatives for trigonometric functions, making it easier for practitioners to grasp the underlying concepts. This is particularly important as it helps bridge the gap between theoretical concepts and their practical applications.

Second, this method simplifies the computation of fractional derivatives for sine and cosine functions. By using fractional multiples of $\frac{\pi }{2},$ the need for special functions or complex integration methods, which are typically required by the Riemann-Liouville definition, is eliminated. This results in a more streamlined and efficient calculation process, making it easier for researchers and engineers to apply fractional calculus to real-world problems.

It is worth noting explicitly that equations (2.4)(2.4) $$\frac{d^p}{d{x}^p}\left(\cos \left(nx\right)\right) = n^p \cos \left(\frac{p\pi }{2}+nx\right),$$(2.4) and (2.5)(2.5) $$\frac{d^p}{d{x}^p}\sin \hspace{0.17em}\left(nx\right)=n^p\hspace{0.17em}\sin \hspace{0.17em}\left(\frac{p\pi }{2}+nx\right).$$(2.5) are only valid for the Riemann-Liouville derivative with $a=-\infty ,$ not with other choices of $a.$

2.3. Generalized derivative for functions expansible as a series of exponentials

The concept of a generalized derivative for functions that can be expanded as a series of exponentials has been explored by several prominent mathematicians, such as Scott Blair (1947), Heaviside (1920) and Liouville (1832a). These researchers have contributed significantly to the development and understanding of fractional calculus and its applications.

Liouville (1832a) introduced a generalized derivative, specifically for functions that can be expressed as a series of exponentials$, f = \Sigma c_j{e}^{b_j x},$ (2.6) $$\frac{d^{p}f}{d{x}^p} = \sum _{j=0}^{\infty }{c}_j b_i^p e^{b_i x}.$$(2.6)

In this formula, $\frac{d^{p}f}{d{x}^p}$ denotes the $p-th$ order derivative of the function f concerning $x.$ The coefficients $c_j$ and the exponential terms $b_i^p e^{b_i\hspace{0.17em}x}$ depend on the particular function $f$ and the order p of the derivative. This generalized derivative enables the computation of fractional order derivatives for functions that can be represented as a series of exponentials.

This approach has been influential in the development of fractional calculus, as it provides a more generalized framework for handling various types of functions. Furthermore, it has paved the way for other researchers, such as Scott Blair and Heaviside, to develop their methods and applications in the field.

Scott Blair (1947) extended Liouville’s work by further exploring the properties and applications of these generalized derivatives, while Heaviside (1920) was known for his operational calculus, which also drew inspiration from Liouville’s generalized derivative.

Equation (2.6)(2.6) $$\frac{d^{p}f}{d{x}^p} = \sum _{j=0}^{\infty }{c}_j b_i^p e^{b_i x}.$$(2.6) aligns with the Riemann-Liouville derivative for $a = -\infty .$ It’s notable that during Liouville’s era, the term “Riemann-Liouville derivative” didn’t exist. Thus, equation (2.6)(2.6) $$\frac{d^{p}f}{d{x}^p} = \sum _{j=0}^{\infty }{c}_j b_i^p e^{b_i x}.$$(2.6) can be seen as a precursor to the Riemann-Liouville definition.

2.4. The complex integral definition of Riemann-Liouville derivatives

As we advance in our exploration of fractional calculus, it’s crucial to spotlight seminal works that form the bedrock of contemporary definitions. Among these, Nekrasov’s complex integral definition of the Riemann-Liouville derivatives, introduced in 1888, holds significant importance. Detailed accounts of this can be found in Miller (1993), chapter II.4, and Oldham and Spanier (1974), chapter 3.4.

Nekrasov’s definition is succinctly represented as: (2.7) $${ }_a^N{D}_z^{\alpha }=\frac{\Gamma \left(\alpha +1\right)}{2\pi i}{\oint }_C^{ }\frac{f\left(t\right) dt}{{\left(t-z\right)}^{\alpha +1}},$$(2.7)

This is valid for non-negative integers.

Here, the contour $C$ begins and ends at $t=0$ enclosing $z$ once in the positive manner. To unambiguously define the denominator of the integrand, one can represent it as: $${\left(t-z\right)}^{\alpha +1}=e^{\left(\alpha +1\right)\ln \hspace{0.17em}\left(t-z\right)}$$

With the condition that $\ln (t-z)$ is real when $t-z>0.$ The connection to the Riemann-Liouville is demonstrated by adjusting contour $C$ into $C\prime ,$ such that it straddles both edges of the depicted branch line, See Oldham and Spanier (1974), chapter 3.4.

3. Derivation formulas of fractional order derivatives and integrals for specific classes of functions and analytic functions

The objective of this research is to examine the fractional derivatives of selected classes of functions and analytic functions, with a primary emphasis on those that can be expressed by the general Dirichlet series and those within a disc centered at $\alpha .$

3.1. A family of functions that can be represented by the general Dirichlet series

Let $f(z)$ be represented by the general Dirichlet series (Bohnenblust & Hille, 1931): (3.1) $$f\left(z\right)=\sum _{n=1}^{\mathrm{\infty }}{a}_n{e}^{-{\lambda }_n\mathrm{ }z},\mathrm{ }$$(3.1) where $a_n,$ $z$ are complex numbers and $\left\{{\lambda }_n\right\}$ is a strictly increasing sequence of nonnegative real numbers that tends to infinity. The function $f(z)$ is analytic on the half-plane of convergence.

For any positive integer $k,$ the $k^{th}$ derivative of $f(z)$ can be expressed as (3.2) $$f^{\left(k\right)}\left(z\right)={\left(-1\right)}^k\sum _{n=1}^{\infty }{a}_n{\left({\lambda }_n\right)}^k{e}^{-{\lambda }_n z}.$$(3.2)

By utilizing the result from equation (2.6)(2.6) $$\frac{d^{p}f}{d{x}^p} = \sum _{j=0}^{\infty }{c}_j b_i^p e^{b_i x}.$$(2.6) , we can extend the derivative in equation (3.2)(3.2) $$f^{\left(k\right)}\left(z\right)={\left(-1\right)}^k\sum _{n=1}^{\infty }{a}_n{\left({\lambda }_n\right)}^k{e}^{-{\lambda }_n z}.$$(3.2) to a fractional order, resulting in: (3.3) $$f^{\left(p\right)}\left(z\right)={\left(-1\right)}^p\sum _{n=1}^{\infty }{a}_n{\left({\lambda }_n\right)}^p{e}^{-{\lambda }_n z},$$(3.3) where $p$ is a positive real number.

Using the fact in (Zwillinger, 2014): $${\int }_{-\infty }^{\infty }{\left(z+it\right)}^{-p-1} e^{i{\lambda }_{n}t}dt=\frac{2\pi {\left({\lambda }_n\right)}^p{e}^{-{\lambda }_n z}}{\Gamma \left(p+1\right)}.$$ where ${\lambda }_n>0, Re\left(z\right)>0,Re\left(p\right)>-1.$

Thus, we have (3.4) $$\frac{\Gamma \left(p+1\right)}{2\pi }{\int }_{-\infty }^{\infty }{\left(z+it\right)}^{-p-1} e^{i{\lambda }_{n}t}dt={\left({\lambda }_n\right)}^p{e}^{-{\lambda }_n z} .$$(3.4)

Substituting equation (3.4)(3.4) $$\frac{\Gamma \left(p+1\right)}{2\pi }{\int }_{-\infty }^{\infty }{\left(z+it\right)}^{-p-1} e^{i{\lambda }_{n}t}dt={\left({\lambda }_n\right)}^p{e}^{-{\lambda }_n z} .$$(3.4) into equation (3.3)(3.3) $$f^{\left(p\right)}\left(z\right)={\left(-1\right)}^p\sum _{n=1}^{\infty }{a}_n{\left({\lambda }_n\right)}^p{e}^{-{\lambda }_n z},$$(3.3) , we obtain (3.5) $$f^{\left(p\right)}\left(z\right)={\left(-1\right)}^p\frac{\Gamma \left(p+1\right)}{2\pi }\sum _{n=1}^{\infty }{a}_n{\int }_{-\infty }^{\infty }{\left(z+it\right)}^{-p-1} e^{i{\lambda }_{n}t}dt .$$(3.5)

By interchanging the integral and the sum sign, where the series is assumed to be converges uniformly on the half plane, we have: (3.6) $$f^{\left(p\right)}\left(z\right)={\left(-1\right)}^p\frac{\Gamma \left(p+1\right)}{2\pi }{\int }_{-\infty }^{\infty }{\left(z+it\right)}^{-p-1}\sum _{n=1}^{\infty }{a}_n e^{i{\lambda }_{n}t}dt.$$(3.6)

Substituting equation (3.1)(3.1) $$f\left(z\right)=\sum _{n=1}^{\mathrm{\infty }}{a}_n{e}^{-{\lambda }_n\mathrm{ }z},\mathrm{ }$$(3.1) into equation (3.6)(3.6) $$f^{\left(p\right)}\left(z\right)={\left(-1\right)}^p\frac{\Gamma \left(p+1\right)}{2\pi }{\int }_{-\infty }^{\infty }{\left(z+it\right)}^{-p-1}\sum _{n=1}^{\infty }{a}_n e^{i{\lambda }_{n}t}dt.$$(3.6) , we derive to the new definition of the fractional derivative for A family of functions that can be represented by the general Dirichlet series.

It is worth mentioning that setting ${\lambda }_n=n,$ then equation (3.6)(3.6) $$f^{\left(p\right)}\left(z\right)={\left(-1\right)}^p\frac{\Gamma \left(p+1\right)}{2\pi }{\int }_{-\infty }^{\infty }{\left(z+it\right)}^{-p-1}\sum _{n=1}^{\infty }{a}_n e^{i{\lambda }_{n}t}dt.$$(3.6) represents the fractional order derivative of the function $$f\left(z\right)=\sum _{n=1}^{\infty }{a}_n{e}^{-n z},$$ and setting ${\lambda }_n=\ln \hspace{0.17em}\left(n\right),$ then equation (3.6)(3.6) $$f^{\left(p\right)}\left(z\right)={\left(-1\right)}^p\frac{\Gamma \left(p+1\right)}{2\pi }{\int }_{-\infty }^{\infty }{\left(z+it\right)}^{-p-1}\sum _{n=1}^{\infty }{a}_n e^{i{\lambda }_{n}t}dt.$$(3.6) represents the fractional order derivative of the function: $$f\left(z\right)=\sum _{n=1}^{\infty }\frac{a_n}{n^z }.$$

3.2. A family of analytic functions on a disc D centered at α

Given a family of analytic functions defined on a disc $D$ centered at $\alpha ,$ consider a function $f(z)$ in this family. According to Taylor’s series, the function can be represented as (Abu Ghuwaleh, Saadeh, & Burqan, 2022; Abu-Ghuwaleh, Saadeh, & Qazza, 2022a, 2022b; Ahlfors, 1979; Churchill, 1974; Rudin, 1987; Saadeh, Abu-Ghuwaleh, Qazza, & Kuffi, 2022): (3.7) $$f\left(z\right)=\sum _{k=0}^{\mathrm{\infty }}\frac{f^{\left(k\right)}\left(\alpha \right)}{k!}{\left(z-\alpha \right)}^k\mathrm{ }.\mathrm{ }$$(3.7)

Substituting $z=\alpha +\beta e^{i\theta x}$ into the above equation, where $\beta$ is not completely arbitrary, since it must be smaller than the radius of $\mathrm{D},$ we have: (3.8) $$f\left(\alpha +\beta e^{i\theta x}\right)=\sum _{k=0}^{\mathrm{\infty }}\frac{f^{\left(k\right)}\left(\alpha \right)}{k!}{\beta }^k{e}^{\mathit{\text{iθkx}}}.\mathrm{ }$$(3.8)

Using the identities: $$e^{i\theta x}+e^{-i\theta x}=2\cos \hspace{0.17em}\left(\theta x\right)\text{ and }{e}^{i\theta x}-e^{-i\theta x}=2i\hspace{0.17em}\sin \hspace{0.17em}\left(\theta x\right).$$

We can get the following equation: (3.9) $$\begin{array}{l}\frac{1}{2}\left[f\left(\alpha +\beta e^{i\theta x}\right)+f\left(\alpha +\beta e^{-i\theta x}\right)\right]\\ =\mathrm{ }\frac{1}{2}\left(\sum _{k=0}^{\mathrm{\infty }}\frac{f^{\left(k\right)}\left(\alpha \right)}{k!}{\beta }^k\left(e^{\mathit{\text{iθkx}}}+e^{-\mathit{\text{iθkx}}}\right)\right)\mathrm{ }\\ =\sum _{k=0}^{\mathrm{\infty }}\frac{f^{\left(k\right)}\left(\alpha \right)}{k!}{\beta }^k\cos \hspace{0.17em}\left(k\theta x\right)\mathrm{ }\\ =f\left(\alpha \right)+f^{\prime }\left(\alpha \right)\beta \cos \hspace{0.17em}\left(\theta x\right)+\frac{f^{″}\left(\alpha \right)}{2!}{\beta }^2\cos \hspace{0.17em}\left(2\theta x\right)+\dots .\mathrm{ }\end{array}$$(3.9)

Similarly, (3.10) $$\begin{array}{l}\frac{1}{2i}\left[f\left(\alpha +\beta e^{i\theta x}\right)-f\left(\alpha +\beta e^{-i\theta x}\right)\right]\\ =\frac{1}{2i}\left(\sum _{k=0}^{\mathrm{\infty }}\frac{f^{\left(k\right)}\left(\alpha \right)}{k!}{\beta }^k\left(e^{\mathit{\text{iθkx}}}-e^{-\mathit{\text{iθkx}}}\right)\right)\mathrm{ }\\ =f^{\prime }\left(\alpha \right)\beta \sin \hspace{0.17em}\left(\theta x\right)+\frac{f^{″}\left(\alpha \right)}{2!}{\beta }^2\sin \hspace{0.17em}\left(2\theta x\right)+\dots \\ =\sum _{k=1}^{\mathrm{\infty }}\frac{f^{\left(k\right)}\left(\alpha \right)}{k!}{\beta }^k\sin \hspace{0.17em}\left(k\theta x\right).\mathrm{ }\end{array}$$(3.10)

Setting $x = 1$ and substituting it in equation (3.9)(3.9) $$\begin{array}{l}\frac{1}{2}\left[f\left(\alpha +\beta e^{i\theta x}\right)+f\left(\alpha +\beta e^{-i\theta x}\right)\right]\\ =\mathrm{ }\frac{1}{2}\left(\sum _{k=0}^{\mathrm{\infty }}\frac{f^{\left(k\right)}\left(\alpha \right)}{k!}{\beta }^k\left(e^{\mathit{\text{iθkx}}}+e^{-\mathit{\text{iθkx}}}\right)\right)\mathrm{ }\\ =\sum _{k=0}^{\mathrm{\infty }}\frac{f^{\left(k\right)}\left(\alpha \right)}{k!}{\beta }^k\cos \hspace{0.17em}\left(k\theta x\right)\mathrm{ }\\ =f\left(\alpha \right)+f^{\prime }\left(\alpha \right)\beta \cos \hspace{0.17em}\left(\theta x\right)+\frac{f^{″}\left(\alpha \right)}{2!}{\beta }^2\cos \hspace{0.17em}\left(2\theta x\right)+\dots .\mathrm{ }\end{array}$$(3.9) , we have: (3.11) $$M=2f\left(\alpha \right)+2\sum _{k=1}^{\mathrm{\infty }}\frac{f^{\left(k\right)}\left(\alpha \right)}{k!}{\beta }^k\cos \hspace{0.17em}\left(k\theta \right),\mathrm{ }$$(3.11) where $M= f\left(\alpha +\beta e^{i\theta }\right)+f\left(\alpha +\beta e^{-i\theta }\right).$

By taking the derivative p times of both sides of equation (3.11)(3.11) $$M=2f\left(\alpha \right)+2\sum _{k=1}^{\mathrm{\infty }}\frac{f^{\left(k\right)}\left(\alpha \right)}{k!}{\beta }^k\cos \hspace{0.17em}\left(k\theta \right),\mathrm{ }$$(3.11) with respect to θ, where $p\in [0,1,$ and using equation (2.4)(2.4) $$\frac{d^p}{d{x}^p}\left(\cos \left(nx\right)\right) = n^p \cos \left(\frac{p\pi }{2}+nx\right),$$(2.4) , we have: (3.12) $$\frac{d^{p}M}{d{\theta }^p}=2\sum _{k=1}^{\mathrm{\infty }}\frac{f^{\left(k\right)}\left(\alpha \right)}{k!}{\beta }^k{k}^p\cos \hspace{0.17em}\left(k\theta +\frac{\pi p}{2}\right).\mathrm{ }$$(3.12)

Using the formula in equation (3.13)(3.13) $$\underset{0}{\overset{\mathrm{\infty }}{\int }}{t}^s\sin \hspace{0.17em}\left(at+b\right)dt=\frac{1}{a^{s+1}}\mathrm{\Gamma }\left(1+s\right)\cos \hspace{0.17em}\left(b+\frac{\pi s}{2}\right),\mathrm{ }$$(3.13) : (3.13) $$\underset{0}{\overset{\mathrm{\infty }}{\int }}{t}^s\sin \hspace{0.17em}\left(at+b\right)dt=\frac{1}{a^{s+1}}\mathrm{\Gamma }\left(1+s\right)\cos \hspace{0.17em}\left(b+\frac{\pi s}{2}\right),\mathrm{ }$$(3.13) where $a>0,$ and $-1<s<0.$ Letting $a=k,$ $b=-k\theta , s=-p,$ we get: (3.14) $$\frac{k}{\Gamma \left(1-p\right)}\underset{0}{\overset{\mathrm{\infty }}{\int }}{t}^{-p}\sin \hspace{0.17em}\left(kt-k\theta \right)dt=k^p\cos \hspace{0.17em}\left(k\theta +\frac{\pi p}{2}\right),\mathrm{ }$$(3.14) where $k>0,$ and $0<p<1.$

Substituting equation (3.14)(3.14) $$\frac{k}{\Gamma \left(1-p\right)}\underset{0}{\overset{\mathrm{\infty }}{\int }}{t}^{-p}\sin \hspace{0.17em}\left(kt-k\theta \right)dt=k^p\cos \hspace{0.17em}\left(k\theta +\frac{\pi p}{2}\right),\mathrm{ }$$(3.14) into equation (3.12)(3.12) $$\frac{d^{p}M}{d{\theta }^p}=2\sum _{k=1}^{\mathrm{\infty }}\frac{f^{\left(k\right)}\left(\alpha \right)}{k!}{\beta }^k{k}^p\cos \hspace{0.17em}\left(k\theta +\frac{\pi p}{2}\right).\mathrm{ }$$(3.12) , we get: (3.15) $$\frac{d^{p}M}{d{\theta }^p}=2\sum _{k=1}^{\mathrm{\infty }}\frac{f^{\left(k\right)}\left(\alpha \right)}{k!}{\beta }^k\frac{k}{\Gamma \left(1-p\right)}\underset{0}{\overset{\mathrm{\infty }}{\int }}{t}^{-p}\sin \hspace{0.17em}\left(kt-k\theta \right)dt.\mathrm{ }$$(3.15)

Which can be simplified to: (3.16) $$\frac{d^{p}M}{d{\theta }^p}=\frac{2}{\Gamma \left(1-p\right)}\sum _{k=1}^{\mathrm{\infty }}\frac{f^{\left(k\right)}\left(\alpha \right)}{\left(k-1\right)!}{\beta }^k\underset{0}{\overset{\mathrm{\infty }}{\int }}{t}^{-p}\sin \hspace{0.17em}\left(kt-k\theta \right)dt.\mathrm{ }$$(3.16)

Shifting the index of the infinite sum to get: (3.17) $$\frac{d^{p}M}{d{\theta }^p}=\frac{2}{\Gamma \left(1-p\right)}\sum _{k=0}^{\mathrm{\infty }}\frac{f^{\left(k+1\right)}\left(\alpha \right)}{\left(k\right)!}{\beta }^{k+1}\underset{0}{\overset{\mathrm{\infty }}{\int }}{t}^{-p}\sin \hspace{0.17em}\left(\left(kt-k\theta )+(t-\theta )\right)\right)dt.\mathrm{ }$$(3.17)

Using a simple trigonometric identity to get: (3.18) $$\frac{d^{p}M}{d{\theta }^p}=\frac{2\beta }{\Gamma \left(1-p\right)}\frac{\partial }{\partial \alpha }\sum _{k=0}^{\infty }\frac{f^{\left(k\right)}\left(\alpha \right)}{\left(k\right)!}{\beta }^k\underset{0}{\overset{\infty }{\int }}{t}^{-p}\left(\sin \hspace{0.17em}\left(kt-k\theta \right)\cos \hspace{0.17em}\left(t-\theta \right)+\hspace{0.17em}\cos \hspace{0.17em}\left(kt-k\theta \right)\sin \hspace{0.17em}\left(t-\theta \right)\right)dt.$$(3.18)

Using Fubini’s theorem to interchange the order of integration and summation and incorporating the information from equations (3.9)(3.9) $$\begin{array}{l}\frac{1}{2}\left[f\left(\alpha +\beta e^{i\theta x}\right)+f\left(\alpha +\beta e^{-i\theta x}\right)\right]\\ =\mathrm{ }\frac{1}{2}\left(\sum _{k=0}^{\mathrm{\infty }}\frac{f^{\left(k\right)}\left(\alpha \right)}{k!}{\beta }^k\left(e^{\mathit{\text{iθkx}}}+e^{-\mathit{\text{iθkx}}}\right)\right)\mathrm{ }\\ =\sum _{k=0}^{\mathrm{\infty }}\frac{f^{\left(k\right)}\left(\alpha \right)}{k!}{\beta }^k\cos \hspace{0.17em}\left(k\theta x\right)\mathrm{ }\\ =f\left(\alpha \right)+f^{\prime }\left(\alpha \right)\beta \cos \hspace{0.17em}\left(\theta x\right)+\frac{f^{″}\left(\alpha \right)}{2!}{\beta }^2\cos \hspace{0.17em}\left(2\theta x\right)+\dots .\mathrm{ }\end{array}$$(3.9) and (3.19)(3.19) $$I^p\left(M\right)=\sum _{k=1}^{\infty }\frac{f^{\left(k\right)}\left(\alpha \right)}{k!}{\beta }^k \frac{\cos \hspace{0.17em}\left(k\theta -\frac{\pi p}{2}\right)}{k^p} .$$(3.19) , a new definition for the fractional derivative of the family $[f(\alpha +\beta e^{i\theta }) +f(\alpha +\beta e^{-i\theta })]$ is obtained.

To get the fractional integration definition, we do the following:

Taking the integration $p$ times for both sides of equation (3(3.1) $$f\left(z\right)=\sum _{n=1}^{\mathrm{\infty }}{a}_n{e}^{-{\lambda }_n\mathrm{ }z},\mathrm{ }$$(3.1) ,12), and by using equation (2.4)(2.4) $$\frac{d^p}{d{x}^p}\left(\cos \left(nx\right)\right) = n^p \cos \left(\frac{p\pi }{2}+nx\right),$$(2.4) but with clockwise rotation by $\frac{\pi }{2},$ to obtain: (3.19) $$I^p\left(M\right)=\sum _{k=1}^{\infty }\frac{f^{\left(k\right)}\left(\alpha \right)}{k!}{\beta }^k \frac{\cos \hspace{0.17em}\left(k\theta -\frac{\pi p}{2}\right)}{k^p} .$$(3.19)

Recalling equation (3.13)(3.13) $$\underset{0}{\overset{\mathrm{\infty }}{\int }}{t}^s\sin \hspace{0.17em}\left(at+b\right)dt=\frac{1}{a^{s+1}}\mathrm{\Gamma }\left(1+s\right)\cos \hspace{0.17em}\left(b+\frac{\pi s}{2}\right),\mathrm{ }$$(3.13) , we have: $$\underset{0}{\overset{\infty }{\int }}{t}^s\sin \hspace{0.17em}\left(at+b\right)dt=\frac{1}{a^{s+1}}\mathrm{\Gamma }\left(1+s\right)\cos \hspace{0.17em}\left(b+\frac{\pi s}{2}\right),$$ where $a>0,$ and $-1<s<0.$

Letting $a=k,$ $b=\frac{\pi }{2}-k\theta , s=p-1,$ to get: (3.20) $$\frac{1}{\Gamma \left(p\right)}\underset{0}{\overset{\infty }{\int }}{t}^{p-1}\cos \hspace{0.17em}\left(kt-k\theta \right)dt=k^{-p}\cos \hspace{0.17em}\left(k\theta -\frac{\pi p}{2}\right),$$(3.20) where $k>0,$ and $0<p<1.$

After inserting equation (3.20)(3.20) $$\frac{1}{\Gamma \left(p\right)}\underset{0}{\overset{\infty }{\int }}{t}^{p-1}\cos \hspace{0.17em}\left(kt-k\theta \right)dt=k^{-p}\cos \hspace{0.17em}\left(k\theta -\frac{\pi p}{2}\right),$$(3.20) into equation (3.19)(3.19) $$I^p\left(M\right)=\sum _{k=1}^{\infty }\frac{f^{\left(k\right)}\left(\alpha \right)}{k!}{\beta }^k \frac{\cos \hspace{0.17em}\left(k\theta -\frac{\pi p}{2}\right)}{k^p} .$$(3.19) , the fractional integration definition for the family $\left[f\right(\alpha +\beta e^{i\theta })+f(\alpha +\beta e^{-i\theta }\left)\right]$ is obtained.

Following a similar procedure and employing the information provided in equation (3.13)(3.13) $$\underset{0}{\overset{\mathrm{\infty }}{\int }}{t}^s\sin \hspace{0.17em}\left(at+b\right)dt=\frac{1}{a^{s+1}}\mathrm{\Gamma }\left(1+s\right)\cos \hspace{0.17em}\left(b+\frac{\pi s}{2}\right),\mathrm{ }$$(3.13) , equation (3.21)(3.21) $$\underset{0}{\overset{\infty }{\int }}{t}^s\cos \hspace{0.17em}\left(at+b\right)dt=\frac{-1}{a^{s+1}}\mathrm{\Gamma }\left(1+s\right)\sin \hspace{0.17em}\left(b+\frac{\pi s}{2}\right),$$(3.21) can be derived, where $a>0$ and $-1<s<0,$ given by:$$ (3.21) $$\underset{0}{\overset{\infty }{\int }}{t}^s\cos \hspace{0.17em}\left(at+b\right)dt=\frac{-1}{a^{s+1}}\mathrm{\Gamma }\left(1+s\right)\sin \hspace{0.17em}\left(b+\frac{\pi s}{2}\right),$$(3.21)

Using equation (3.21)(3.21) $$\underset{0}{\overset{\infty }{\int }}{t}^s\cos \hspace{0.17em}\left(at+b\right)dt=\frac{-1}{a^{s+1}}\mathrm{\Gamma }\left(1+s\right)\sin \hspace{0.17em}\left(b+\frac{\pi s}{2}\right),$$(3.21) , we can then obtain the new definitions for fractional calculus for the family: $$\frac{1}{i} \left[f\right(\alpha +\beta e^{i\theta }) -f(\alpha +\beta e^{-i\theta }\left)\right] .$$

Remark 3.1.

1. By following a similar approach to the one used to obtain the fractional derivative of functions expressed as the general Dirichlet series, we can define the fractional derivative for the family $f\left(\alpha +\beta e^{-\theta }\right).$

2. By utilizing the outcome provided in equation (2.6)(2.6) $$\frac{d^{p}f}{d{x}^p} = \sum _{j=0}^{\infty }{c}_j b_i^p e^{b_i x}.$$(2.6) onto equation (3.8)(3.8) $$f\left(\alpha +\beta e^{i\theta x}\right)=\sum _{k=0}^{\mathrm{\infty }}\frac{f^{\left(k\right)}\left(\alpha \right)}{k!}{\beta }^k{e}^{\mathit{\text{iθkx}}}.\mathrm{ }$$(3.8) , we can compute the fractional derivative of the function $f(\alpha +\beta e^{i\theta }).$

4. The new fractional definitions

In this section, we present the new fractional calculus definitions, which are derived from the methodology described in the previous section. After presenting the definitions, we discuss some of their properties and conditions for their existence.

Definition (1).

Let $f(z)$ be expressed as a general Dirichlet series that converges uniformly on the half-plane of convergence, and let it be analytic on this half-plane. The new fractional derivative definition for this class of functions is given by: (4.1) $${\mathrm{ }}^{M{R}_1}{f}^{\left(p\right)}\left(z\right)={\left(-1\right)}^p\frac{\Gamma \left(p+1\right)}{2\pi }{\int }_{\text{-}\mathrm{\infty }}^{\mathrm{\infty }}{\left(z+i\mathrm{ }t\right)}^{-p-1}f\left(-i\mathrm{ }t\right)\mathrm{ }dt,\mathrm{ }$$(4.1) where $p\ge 0,$ is the order of the derivative and satisfies $p\ge 0 .$

Definition (2).

Let $f(z)$ be an analytic function in the neighborhood of $\alpha ,$ where $\alpha$ and $\beta$ denote real quantities. Then, the new fractional derivative definition with respect to θ for the family $f(\alpha +\beta e^{-\theta })$ is given by: (4.2) $$\frac{{ }^{M{R}_2}{d}^p\left(f\left(\alpha +\beta e^{-\theta }\right)\right)}{d{\theta }^p}={\left(-1\right)}^p\frac{\Gamma \left(p+1\right)}{2\pi }{\int }_{-\infty }^{\infty }{\left(\theta +i t\right)}^{-p-1}\left(f\left(\alpha +\beta e^{i t}\right)-f\left(\alpha \right)\right)dt,$$(4.2) where $p$ is the order of the derivative and satisfies $p\ge 0, \theta >0.$

Definition (3).

Definition: Let $f(z)$ be an analytic function in a disc $D$ centered at $\alpha ,$ where $\alpha$ and $\beta$ denote positive or negative real quantities, and $\beta$ is smaller than the radius of D. Then, the new fractional derivative definition with respect to $\theta$ for the function $f\left(\alpha +\beta e^{i\theta }\right)$ is given by: (4.3) $${{ }^{M{R}_3}D}_{\theta }^p\left[f\left(\alpha +\beta e^{i\theta }\right)\right]=\frac{\beta }{\Gamma \left(1-p\right)}\frac{\partial }{\partial \alpha }{\int }_0^{\infty }\frac{\cos \hspace{0.17em}\left(t-\theta \right)f\left(\alpha +\beta e^{i\left(t-\theta \right)}\right)+\sin \hspace{0.17em}\left(t-\theta \right)f\left(\alpha +\beta e^{-i\left(t-\theta \right)}\right)}{t^p}dt$$(4.3)

Moreover, the new fractional derivatives for the real and imaginary parts as follows: $${{ }^{M{R}_3}D}_{\theta }^p \left[\frac{1}{2}\left(f\left(\alpha +\beta e^{i\theta }\right)+f\left(\alpha +\beta e^{-i\theta }\right)\right)\right]=D_{\theta }^p\left[ Re\left(f\left(\alpha +\beta e^{i\theta }\right)\right)\right]$$ (4.4) $$=\frac{\beta }{\Gamma \left(1-p\right)}\frac{\partial }{\partial \alpha }\underset{0}{\overset{\infty }{\int }}\frac{\cos \hspace{0.17em}\left(t-\theta \right)Im\left(f\left(\alpha +\beta e^{i\left(t-\theta \right)}\right)\right)+\sin \hspace{0.17em}\left(t-\theta \right)Re\left(f\left(\alpha +\beta e^{i\left(t-\theta \right)}\right)\right)}{t^p} dt,$$(4.4) and, (4.5) $$\begin{array}{l}{ }^{M{R}_3}{D}_{\theta }^p\left[\frac{1}{2i} \left(f\left(\alpha +\beta e^{i\theta }\right)-f\left(\alpha +\beta e^{-i\theta }\right)\right)\right]=D_{\theta }^p\left[ Im\left(f\left(\alpha +\beta e^{i\theta }\right)\right)\right]\mathrm{ }\\ =\frac{\beta }{\Gamma \left(1-p\right)}\frac{\partial }{\partial \alpha }\underset{0}{\overset{\infty }{\int }}\frac{\cos \hspace{0.17em}\left(t-\theta \right)Re\left(f\left(\alpha +\beta e^{i\left(t-\theta \right)}\right)\right)-\sin \hspace{0.17em}\left(t-\theta \right)Im\left(f\left(\alpha +\beta e^{i\left(t-\theta \right)}\right)\right)}{t^p}\mathit{\text{ dt }}\end{array}$$(4.5) where $p$ is the order of the derivative and satisfies $\theta >0, 0<p<1.$

Definition (4).

Let $f(z)$ be an analytic function in a disc $D$ centered at $\alpha ,$ where $\alpha$ and $\beta$ denote positive or negative real quantities, and $\beta$ is smaller than the radius of $D.$ Then, the new fractional integration definition with respect to $\theta$ for the function $f\left(\alpha +\beta e^{i\theta }\right)$ is given by: (4.6) $${ }^{M{R}_3}{I}_{\theta }^p\left[f\left(\alpha +\beta e^{i\theta }\right)\right]=\frac{1}{\Gamma \left(p\right)}\underset{0}{\overset{\infty }{\int }}\frac{f\left(\alpha +\beta e^{i\left(\theta -t\right)}\right)-f\left(\alpha \right)}{t^{-p+1}}dt.$$(4.6)

Moreover, the new fractional integrals for the real and imaginary parts are given by: (4.7) $$\begin{array}{c}{}^{M{R}_3}{I}_{\theta }^p\left[\frac{1}{2}\left(f\left(\alpha +\beta e^{i\theta }\right)+f\left(\alpha +\beta e^{-i\theta }\right)\right)\right]=I_{\theta }^p\left(Re\left(f\left(\alpha +\beta e^{i\theta }\right)\right)\right)\\ =\frac{1}{\Gamma \left(p\right)}\underset{0}{\overset{\infty }{\int }}\frac{Re\left(2f\left(\alpha +\beta e^{i\left(\theta -t\right)}\right)\right)-2f\left(\alpha \right)}{t^{-p+1}}dt, \end{array}$$(4.7) and, (4.8) $$\begin{array}{l}{}^{M{R}_3}{I}_{\theta }^p\left(\frac{1}{2i}\left(f\left(\alpha +\beta e^{i\theta }\right)-f\left(\alpha +\beta e^{-i\theta }\right)\right)\right)=I_{\theta }^p\left(Im\left(f\left(\alpha +\beta e^{i\theta }\right)\right)\right)\\ =\frac{1}{\Gamma \left(p\right)}\underset{0}{\overset{\infty }{\int }}\frac{Im\left(2f\left(\alpha +\beta e^{i\left(\theta -t\right)}\right)\right)}{ t^{-p+1}}dt, \end{array}$$(4.8) where $p$ is the order of the integration and satisfies $\theta >0, 0<p<1$

Remark 4.1.

1. In definitions (1) and (2), ${\left(-1\right)}^{\mathrm{p}}$ represents the principal value of a complex number raised to the power of $p.$ However, depending on the context and the properties of the functions involved, other roots may need to be considered. It is important to analyze the problem statement, assumptions, and any given boundary conditions or constraints to determine the appropriate root to use.

2. One can discern evident similarities between the integral structures. Nevertheless, transitioning directly from Nekrasov’s definition as depicted in equation (2.7)(2.7) $${ }_a^N{D}_z^{\alpha }=\frac{\Gamma \left(\alpha +1\right)}{2\pi i}{\oint }_C^{ }\frac{f\left(t\right) dt}{{\left(t-z\right)}^{\alpha +1}},$$(2.7) to Definition (1) demands intricate transformations and possibly additional assumptions or procedural adjustments. It’s imperative to acknowledge that while Nekrasov’s definition establishes a direct correlation with the Riemann-Liouville derivatives, Definition (1) doesn’t overtly make this association. However, comprehending the relational dynamics between both definitions and the foundational precepts of fractional calculus, such as the Riemann-Liouville framework, can facilitate a deeper appreciation of their respective applications and importance. Specifically:

Definition (1):

This is articulated for general Dirichlet series which exhibit uniform convergence within their designated half-plane of convergence, thereby encompassing a more extensive set of functions.

Nekrasov’s Definition: Rooted in its association with the Riemann-Liouville derivatives, this definition commands historical reverence, being among the pioneering constructs in fractional calculus.

Remark 4.2.

Comparative analysis of Definitions 1–4

• 1. Definition (1): delineates the fractional calculus of a general Dirichlet series that uniformly converges within its half-plane of convergence.

• 2. Definition (2) pertains to the fractional derivative of an analytic function surrounding the point α. It emphasizes its derivative with respect to $\theta$ for the functional form $f\left(\alpha +\beta e^{-\theta }\right).$

• 3. Definition (3) offers a broader view by considering an analytic function within the disc $D$ anchored at point $\alpha .$ It introduces the fractional derivative of $f(\alpha +\beta e^{i\theta }),$ taking an expansive approach by encompassing both the real and imaginary components of the function.

At an initial perusal, Definitions 1 and 2, with their structural semblance, might be interpreted as equivalent. Both hinge on the gamma function, $\Gamma ,$ and integrate across the entire real line. However, it’s essential to discern that their functional forms and the realms of their applicability set them apart. Conversely, Definition (3) charts a novel path, concentrating on a disk domain $D$ and differentiating both the real and imaginary parts of the function.

• 4. Definition (4) as the Inverse to Definition (3):

Definition (4) delves deep into the fractional integration of an analytic function ensconced within the disc D centered at the point α. It hones in on the function $f(\alpha +\beta e^{i\theta })$ and akin to Definition (3), it partitions its attention between the real and imaginary facets of the function. Inherently, the construction of Definition (4) insinuates that it encapsulates the fractional integration acting as the “inverse operation” to the fractional derivatives expounded in Definition (3).

5. Properties and existence conditions of the new definitions

In this section, we will explore the properties and characteristics of the new fractional calculus definitions. These definitions are designed to provide a comprehensive framework for analyzing mathematical systems that involve fractional differentiation and integration. By understanding the properties of these definitions, we can determine their range of applications and limitations. This understanding enables us to use the new definitions to solve real-world problems with greater accuracy and precision.

5.1. Existence conditions

1. Existence conditions for Definition (1) with $\mathit{p}\ge \mathrm{ }0:$

   1. The function $f(z)$ must be analytic on a half plane $Re\left(z\right)\ge \mathrm{ }a$ for some real number $a.$

   2. The Dirichlet series representation of $f(z)$ must converge uniformly on every compact subset of the half plane $Re(z)\mathrm{ }>\mathrm{ }a.$

   3. The constant term of the Dirichlet series must be zero (i.e., $a_0$ = 0), If the constant term is non-zero, the function would have a singularity at infinity, which would prevent it from being analytic on the half-plane. Thus, it is necessary for the constant term to be zero to ensure that the function is analytic on the half-plane.

   4. The coefficients of the Dirichlet series must satisfy the growth condition: $\left|a_n\right|\mathrm{ }\le \mathrm{ }C\mathrm{ }{n}^{\beta }$ for some constants $C$ and $\beta .$

2. Existence conditions for Definition (2) with $\mathit{p}\ge \mathrm{ }0:$

   1. The function $f\left(z\right)$ must be analytic in a neighborhood of $\alpha .$

   2. The parameter $\theta$ must be greater than 0.

   3. The integral must converge uniformly for all z in the neighborhood of α.

   4. The function $f(\alpha )$ must be well-defined.

The uniform convergence of the integral in definition (2) for all $z$ in the neighborhood of $\alpha$ can be ensured by satisfying the following conditions:

• The function $f(z)$ must be analytic in the neighborhood of $\alpha ,$ and the derivative of $f(z)$ must exist and be continuous up to order $p$ in the same neighborhood.

• The function $f\left(z\right)$ must satisfy a Lipschitz condition in the neighborhood of $\alpha ,$ which means that there exists a constant $L > 0$ such that for all $z_1, z_2$ in the neighborhood of $\alpha ,$ we have $$\left|f\left(z_1\right)\text{-}\mathrm{ }f\left(z_2\right)\right|\le L\left|{\mathrm{z}}_1\text{-}\mathrm{ }{\mathrm{z}}_2\right|$$

• The integral in must be absolutely convergent

If these conditions are satisfied, then the integral in equation (29) will converge uniformly for all $z$ in the neighborhood of $\alpha ,$ and we can apply the definition of the new fractional derivative to $f(z).$

1. Existence conditions for Definitions (3) and (4) with $\mathrm{ }0<\mathrm{ }\mathit{p}<1:$

   1. The function $f(z)$ must be analytic in a disc $\mathrm{ }D$ centered at $\alpha ,$ where $\alpha$ and $\beta$ denote positive or negative real quantities, and β is smaller than the radius of D. This is required to ensure that the function is well-defined and can be differentiated or integrated with respect to $\theta .$

   2. The order of the derivative or the integration $\mathrm{ }p$ must satisfy $0\mathrm{ }<\mathrm{ }p\mathrm{ }<\mathrm{ }1.$

   3. The integral in the definitions must converge uniformly for all $z$ in the disc $D.$ This is necessary to ensure that the derivative or the integration is well-defined and does not depend on the path of integration.

Remark 5.1.

While Definitions (1) and (2) are referred to as fractional derivative definitions, they can also be used as fractional integral definitions for $-1 < p \le 0.$ This is because the integral representation used to derive these definitions holds for $p > -1,$ but not at $p = -1$ due to the presence of a pole in the gamma function. Therefore, for $-1 < p \le 0,$ we can consider these definitions as fractional integral definitions with respect to $\theta$ for the family $f(\alpha +\beta e^{-\theta })$ if $f(z)$ is analytic on a half-plane and the Dirichlet series converges uniformly on the half-plane.

5.2. Some properties of the new fractional derivative and integral definitions

1) Some properties for Definition (1):

• i Let $f(z)$ and $g(z)$ be analytic functions that can be expressed as general Dirichlet series, and let a and b be constants. Then, for any value of $p\ge 0,$ we have: $${\left(a\mathrm{ }f\left(z\right)+\mathrm{ }b\mathrm{ }g\left(z\right)\right)}^{\left(p\right)}=\mathrm{ }a\mathrm{ }{f}^{\left(p\right)}\left(z\right)+\mathrm{ }b\mathrm{ }{g}^{\left(p\right)}\left(z\right).$$

Proof.

We start by expanding the function ${\left(a f\left(z\right)+\mathit{\text{ b g}}\left(z\right)\right)}^{\left(p\right)}$ using Definition (1): $${\left(a f\left(z\right)+\mathit{\text{ b g}}\left(z\right)\right)}^{\left(p\right)}=\frac{{\left(-1\right)}^p\Gamma \left(p+1\right)}{2\pi }{\int }_{-\infty }^{\infty }{\left(z+it\right)}^{-p-1}\left(a f\left(-it\right)+\mathit{\text{ b g}}\left(-it\right)\right)dt.$$

Then, we can distribute the constants a and b inside the integral: $$=\frac{{\left(-1\right)}^p\Gamma \left(p+1\right)}{2\pi }{\int }_{-\infty }^{\infty }{\left(z+it\right)}^{-p-1}a f\left(-it\right)dt +\frac{{\left(-1\right)}^p\Gamma \left(p+1\right)}{2\pi }{\int }_{-\infty }^{\infty }{\left(z+it\right)}^{-p-1}b g\left(-it\right)dt$$

By Definition (1), we know that $f^{\left(p\right)}\left(z\right)=\frac{{\left(-1\right)}^p\Gamma \left(p+1\right)}{2\pi }{\int }_{-\infty }^{\infty }{\left(z+it\right)}^{-p-1}f\left(-it\right)dt,$ and similarly for $g^{\left(p\right)}\left(z\right).$ Therefore, we can rewrite the above equation as: $$=\mathrm{ }a\mathrm{ }{f}^{\left(p\right)}\left(z\right)+\mathrm{ }b\mathrm{ }{g}^{\left(p\right)}\left(z\right)$$

This proves the linearity property of Definition (1).

• ii. Leibniz rule: Let $f(z)$ and $g(z)$ represented by a general Dirichlet series that converges uniformly on the half-plane of convergence and is analytic on this half-plane, and for non-negative integer p, the fractional derivative of their product ${\left(fg\right)}^{\left(p\right)}$ with respect to z is given by:

$${\left(fg\right)}^{\left(p\right)}\mathrm{ }\left(z\right)={\sum }_{k=0}^p\left(\begin{array}{c}p \\ k\end{array}\right)f^{\left(k\right)}\left(z\right)g^{\left(p-k\right)}\left(z\right)+\mathrm{ }{\sum }_{j=1}^p\left(\begin{array}{c}p \\ j\end{array}\right)c_{j\left(f,g,p,z\right)}$$ where $c_j(f,g,p,z)$ is a convergent series of terms involving $f(z),$ its derivatives, $g(z),$ and its derivatives. $f^{(k)}\left(z\right)$ and $g^{(p-k)}\left(z\right)$ denote the $k^{th}$ and ${\left(p-k\right)}^{th}$ fractional derivatives of $f(z)$ and $g(z),$ respectively, and $\left(\begin{array}{c}p \\ k\end{array}\right)$ denotes the binomial coefficient.

Proof.

Assume that p is a non-negative integer and, let $$\begin{array}{c}f\left(z\right)={\sum }_{n=1}^{\infty }{a}_n{e}^{-{\lambda }_{n}z},\\ g\left(z\right)={\sum }_{m=1}^{\infty }{b}_m{e}^{-{\mu }_{m}z}.\end{array}$$

where ${\lambda }_n, {\mu }_m > 0$ and the series converge uniformly on the half-plane of convergence.

To prove this, we start by writing the product of $f(z)$ and $g(z)$ in terms of a double sum: $$f\left(z\right)g\left(z\right) = {\sum }_{n=1}^{\infty }{\sum }_{m=1}^{\infty }{a}_n{b}_m{e}^{-\left({\lambda }_n+{\mu }_m\right)z}$$

Using Definition (1) to compute the fractional derivative of $f(z)g(z)$ with respect to $z:$ $$\begin{array}{l}{\left(fg\right)}^{\left(p\right)}\left(z\right)=\frac{{\left(-1\right)}^p\Gamma \left(p+1\right)}{2\pi }{\int }_{-\infty }^{\infty }{\left(z+it\right)}^{-p-1}f\left(-it\right)g\left(-it\right)dt\\ =\frac{{\left(-1\right)}^p\Gamma \left(p+1\right)}{2\pi }{\int }_{-\infty }^{\infty }{\left(z+it\right)}^{-p-1}\left({\sum }_{n=1}^{\infty }{a}_n{e}^{{\lambda }_{n}it}\right)\left({\sum }_{m=1}^{\infty }{b}_m{e}^{{\mu }_{m}it}\right)dt\end{array}$$

Expanding the product and moving the sum inside the integral, we have: $$\begin{array}{c}\begin{array}{c}{\left(fg\right)}^{\left(p\right)}\left(z\right)=\frac{{\left(-1\right)}^p\Gamma \left(p+1\right)}{2\pi }{\sum }_{n=1}^{\infty }{\sum }_{m=1}^{\infty }{a}_n{b}_m{\int }_{-\infty }^{\infty }{\left(z+it\right)}^{-p-1}{e}^{i\left({\lambda }_n+{\mu }_m\right)t}dt\\ =\frac{{\left(-1\right)}^p\Gamma \left(p+1\right)}{2\pi }{\sum }_{n=1}^{\infty }{\sum }_{m=1}^{\infty }{a}_n{b}_m\left({\left({\lambda }_n+{\mu }_m\right)}^p{e}^{-\left({\lambda }_n+{\mu }_m\right)z}\right)\end{array}\\ = {\sum }_{n=1}^{\infty }{\sum }_{m=1}^{\infty }\frac{a_n{b}_m\left({\left(-1\right)}^p\Gamma \left(p+1\right)\right)}{2\pi }\left({\left({\lambda }_n+{\mu }_m\right)}^p{e}^{-\left({\lambda }_n+{\mu }_m\right)z}\right)\end{array}$$

Now we can use the binomial theorem to expand ${\left({\lambda }_n+{\mu }_m\right)}^p$ and write the result as a sum: $${\left({\lambda }_n+{\mu }_m\right)}^p= {\sum }_{k=0}^p\left(\begin{array}{c}p \\ k\end{array}\right){\lambda }_n^k{\mu }_m^{p-k}$$

Substituting the expression back into the original equation and rearranging the terms, we have: $${\left(fg\right)}^{\left(p\right)}\left(z\right)=\frac{{\left(-1\right)}^p\Gamma \left(p+1\right)}{2\pi }{\sum }_{n=1}^{\infty }{\sum }_{m=1}^{\infty }{\sum }_{k=0}^p\left(\begin{array}{c}p \\ k\end{array}\right){\left({\lambda }_n\right)}^k{\left({\mu }_m\right)}^{p-k}{a}_n{b}_m{e}^{-\left({\lambda }_n+{\mu }_m\right)z}$$

We can now separate the terms with $k=0, k=1, \dots , k=p$ into separate sums. Then, using the definitions of the $k^{th}$ and ${\left(p-k\right)}^{th}$ derivatives of $f(z)$ and $g(z)$ as $f^{\left(k\right)}\left(z\right)$ and $g^{(p-k)}\left(z\right),$ respectively, we can rewrite the above equation as: $${\left(fg\right)}^{\left(p\right)}\left(z\right)= {\sum }_{k=0}^p\left(\begin{array}{c}p\\ k\end{array}\right)f^{\left(k\right)}\left(z\right)g^{(p-k)}\left(z\right)+ {\left(-1\right)}^p\frac{\Gamma \left(p+1\right)}{2\pi } {\sum }_{n=1}^{\infty }{\sum }_{m=1}^{\infty }{\sum }_{k=0}^p\left(\begin{array}{c}p \\ k\end{array}\right){\left({\lambda }_n\right)}^k {\left({\mu }_m\right)}^{p-k} a_n b_m {\left(-1\right)}^{{\lambda }_n+{\mu }_m} {\left({\lambda }_n+{\mu }_m\right)}^p e^{-\left({\lambda }_n+{\mu }_m\right)z}$$ where $f^{\left(k\right)}(z)$ and $g^{p-k}(z)$ are the $k^{th}$ and ${\left(p-k\right)}^{th}$ derivatives of $f(z)$ and $g(z),$ respectively.

Applying the binomial theorem to expand the term ${\left({\lambda }_n+{\mu }_m\right)}^p,$ we get: $${\left(fg\right)}^{\left(p\right)}\left(z\right) = {\sum }_{k=0}^p\left(\begin{array}{c}p \\ k\end{array}\right)f^{\left(k\right)}\left(z\right)g^{\left(p-k\right)}\left(z\right)+\frac{{\left(-1\right)}^p\Gamma \left(p+1\right)}{2\pi }{\sum }_{n=1}^{\infty }{\sum }_{m=1}^{\infty }{\sum }_{j=0}^{p-k}\left(\begin{array}{c}p-k\\ j\end{array}\right){\left({\lambda }_n\right)}^k{\left({\mu }_m\right)}^{p-k-j}{a}_n{b}_m{\left(-1\right)}^{{\lambda }_n+{\mu }_m}{\left({\lambda }_n+{\mu }_m\right)}^{p-j-k}{e}^{-\left({\lambda }_n+{\mu }_m\right)z}$$

Simplifying the expression and rearranging the terms: $${\left(fg\right)}^{\left(p\right)}\left(z\right) = {\sum }_{k=0}^p\left(\begin{array}{c}p \\ k\end{array}\right)f^{\left(k\right)}\left(z\right)g^{\left(p-k\right)}\left(z\right)+\frac{{\left(-1\right)}^p\Gamma \left(p+1\right)}{2\pi }{\sum }_{n=1}^{\infty }{\sum }_{m=1}^{\infty }{\sum }_{j=0}^{p-k}\left(\begin{array}{c}\left(p-k\right)\\ j\end{array}\right){\left({\lambda }_n\right)}^k{\left({\mu }_m\right)}^{p-k-j}{a}_n{b}_m{\left(-1\right)}^{{\lambda }_n+{\mu }_m}{\left({\lambda }_n+{\mu }_m\right)}^{j+k}{e}^{-\left({\lambda }_n+{\mu }_m\right)z}$$ where $f^{\left(k\right)}\left(z\right)$ and $g^{\left(p-k\right)}\left(z\right)$ denote the $k^{th}$ and ${\left(p-k\right)}^{th}$ fractional derivatives of $f(z)$ and $g(z),$ respectively

We can now group the terms in the second sum by their value of $j$ and use the fact that the sum of two convergent series is also convergent, giving: $${\left(fg\right)}^{\left(p\right)} \left(z\right)={\sum }_{k=0}^p\left(\begin{array}{c}p \\ k\end{array}\right)f^{\left(k\right)}\left(z\right)g^{\left(p-k\right)}\left(z\right)+{\sum }_{j=1}^p\left(\begin{array}{c}p \\ j\end{array}\right)c_{j\left(f,g,p,z\right)}$$ where $c_j(f,g,p,z)$ is a convergent series of terms involving $f(z),$ its derivatives, $g(z),$ and its derivatives. This completes the proof of the Leibniz rule for Definition (1) for general Dirichlet series.

2) Some properties of Definition (2).

• i. Linearity: The fractional derivative with respect to $\theta$ is a linear operator, meaning that for any functions $f$ and $g$ and any constants $a$ and $b,$ we have: $$D_{\theta }^p\left(af + bg\right)= a D_{\theta }^p\left(f\right)+ b D_{\theta }^p\left(g\right)$$

• ii. Leibniz Rule: For two analytic functions $f(z)$ and $g(z),$ and an order of derivative $p\ge 0,$ where $p$ is nonnegative integer, the following formula holds for the fractional derivative with respect to $\theta$ of their product: $$D_{\theta }^p\left[f\left(z\right)g\left(z\right)\right]={\sum }_{k=0}^p\left(\begin{array}{c}p \\ k\end{array}\right)D_{\theta }^k\left[f\left(z\right)\right]D_{\theta }^{p-k}\left[g\left(z\right)\right]- f\left(\alpha \right)g\left(\alpha \right)$$

The proof of this formula follows a similar approach as in the proof of the Leibniz rule for Definition (1). By using the definition of the fractional derivative for $f(z)$ and $g(z),$ we can express the fractional derivative of their product as a sum of integrals. After applying some algebraic manipulations, we can rewrite the integral as the sum of the products of the fractional derivatives of $f(z)$ and $g(z)$ evaluated at different orders. By applying the binomial theorem and some further simplifications, we obtain the desired formula, which involves the sum of the products of the fractional derivatives of $f(z)$ and $g(z)$ evaluated at different orders, as well as the product of the values of $f(z)$ and $g(z)$ at $\alpha .$

3) Some properties of Definitions (3).

Linearity: The fractional derivative with respect to $\theta$ as defined in Definition (3) is a linear operator, that is, it satisfies the following properties for any analytic functions $f(z)$ and $g(z),$ and any real numbers $a$ and $b$ $$D_{\theta }^p\left(a f\left(z\right)+\left(z\right)\right)=a D_{\theta }^{p}f\left(z\right)+b D_{\theta }^{p}g\left(z\right)$$

Remark.

Regardless of the definition used, it holds true that for any constant $c$ in the real numbers, the fractional derivative of c is equal to zero.

Set $f\left(z\right)=c, c\in \mathbb{R},$ then, $f\left(\alpha +\beta e^{-\theta }\right)=c.$ Thus, $f\left(\alpha +\beta e^{i\theta }\right)=f\left(\alpha +\beta e^{-i\theta }\right)=c,$ which implies $$f\left(\alpha +\beta e^{i\theta }\right)+f\left(\alpha +\beta e^{-i\theta }\right)=2c, \mathrm{and}\mathrm{ }f\left(\alpha +\beta e^{i\theta }\right)-f\left(\alpha +\beta e^{-i\theta }\right)=0.$$

Applying Definition (3) we get:

Similarly, by applying other definitions we get $D^p\left(c\right)=0,$ for example, by applying Definition (3), we get: $$\begin{array}{c}{D}^p\left(f\left(\alpha +\beta e^{i\theta }\right)+f\left(\alpha +\beta e^{-i\theta }\right)\right)=D^p\left(2 c\right)\\ = \frac{2c\beta }{\Gamma \left(1-p\right)}\left(\frac{\partial }{\partial \alpha }\underset{0}{\overset{\infty }{\int }}{t}^{-p}\sin \hspace{0.17em}\left(t-\theta \right)dt\right) =0. \end{array}$$

Applying Definition (1), we have: $$f^p\left(z\right)=D^p\left(c\right)={\left(-1\right)}^p\frac{\Gamma \left(p+1\right)}{2\pi } \left(c\right){\int }_{-\infty }^{\infty }{\left(z+it\right)}^{-p-1} dt=0$$ where $Re\left(z\right)$>0 and $Re\left(p\right)>0.$

6. Examples and applications

In this section, we explore various examples and applications of the new definitions for fractional derivatives. The definitions provide some ways to analyze complex systems and signals, including control systems, signal processing, complex dynamics and others. We will demonstrate the usefulness of the definition by providing detailed examples and solutions to a range of problems, showing how the fractional derivative can provide important information about the behavior of these systems and signals. By comparing the new definition with the Riemann-Liouville definition, we will also highlight the advantages of using the new definition in various applications.

6.1. Examples

In this subsection, we demonstrate the effectiveness of the definitions through the examination of simple examples. The aim is to provide a clear understanding of the underlying mechanisms and mechanics of the definitions.

Example (1).

Let $f\left(z\right)=e^{-z},$ using Definition (1), we can find the fractional derivative of the function $f\left(z\right),$ as follows: $$\begin{array}{c}{f}^{\left(p\right)}\left(z\right)={\left(-1\right)}^p\frac{\Gamma \left(p+1\right)}{2\pi }{\int }_{-\infty }^{\infty }{\left(z+it\right)}^{-p-1}f\left(-it\right) dt,\\ f^{\left(p\right)}\left(z\right)={\left(-1\right)}^p\frac{\Gamma \left(p+1\right)}{2\pi }{\int }_{-\infty }^{\infty }{\left(z+it\right)}^{-p-1}{e}^{it} dt.\end{array}$$

To evaluate the integral ${\int }_{-\infty }^{\infty }{\left(z+it\right)}^{-p-1}{e}^{it} dt,$ we use the fact: $${\int }_{-\infty }^{\infty }{\left(z+it\right)}^{-p-1} e^{i{\lambda }_{n}t}dt=\frac{2\pi {\left({\lambda }_n\right)}^p{e}^{-{\lambda }_n z}}{\Gamma \left(p+1\right)},$$ where $\mathrm{ }{\lambda }_n>0,\mathrm{ }Re\left(z\right)>0,Re\left(p\right)>-1.$

Setting ${\lambda }_n=1,$ we have: $${\int }_{-\infty }^{\infty }{\left(z+it\right)}^{-p-1}{e}^{it} dt=\frac{2\pi e^{- z}}{\Gamma \left(p+1\right)}.$$

So, we have $f^{\left(p\right)}\left(z\right)={\left(-1\right)}^p{e}^{- z}.$

It’s noteworthy that the Grünwald–Letnikov derivative cannot be utilized to find the derivative of this function. The series used in the Grünwald–Letnikov definition will diverge, thereby obstructing the computation of the derivative.

Example (2).

Let $f\left(z\right)=\zeta (z)={\sum }_{n=1}^{\infty }\frac{1}{n^z },$ where $\zeta$ is the Riemann zeta function, then the fractional derivative for the function using the first definition can be obtained as follows: $$\begin{array}{c}{f}^{\left(p\right)}\left(z\right)={\left(-1\right)}^p\frac{\Gamma \left(p+1\right)}{2\pi }{\int }_{-\infty }^{\infty }{\left(z+it\right)}^{-p-1}\sum _{n=1}^{\infty }\frac{1}{n^{-it} }dt,\\ f^{\left(p\right)}\left(z\right)={\left(-1\right)}^p\frac{\Gamma \left(p+1\right)}{2\pi }\sum _{n=1}^{\infty }{\int }_{-\infty }^{\infty }{\left(z+it\right)}^{-p-1}\frac{1}{n^{-it} }dt.\end{array}$$

The integral: $${\int }_{-\infty }^{\infty }{\left(z+it\right)}^{-p-1}{n}^{it}dt=\frac{i n^{-z}\pi z^{-p}\mathit{cot}\left(\pi p\right)\left(-{\left(i z\right)}^p{\left(-i\ln \hspace{0.17em}\left(n\right)\right)}^p+{\left(-i z\right)}^p{\left(i\ln \hspace{0.17em}\left(n\right)\right)}^p\right)}{\Gamma \left(p+1\right)},$$ where $\ln (\mathrm{n})\in \mathbb{R},\mathrm{Re}(p)>0, \mathrm{Re}(z)>0.$

Hence, the $p^{th}$ derivative of the function becomes: $$=\frac{{\left(-1\right)}^p}{2}\sum _{n=2}^{\infty }i n^{-z} z^{-p}\mathit{cot}\left(\pi p\right)\left[{\left(-i z\right)}^p{\left[i\mathrm{\hspace{0.17em}ln}\hspace{0.17em}\left(n\right)\right]}^p-{\left(i z\right)}^p{\left[-i\ln \hspace{0.17em}\left(n\right)\right]}^p\right].$$

Example (4).

Let $f\left(z\right)=z,\mathrm{ }\mathrm{then}$ using Definition (3), we have: $$\begin{array}{c}\frac{\partial }{\partial \alpha }\frac{1}{i} \left(f\left(\alpha +\beta e^{i\left(t-\theta \right)}\right)- f\left(\alpha +\beta e^{-i\left(t-\theta \right)}\right)\right)=0.\\ \frac{\partial }{\partial \alpha }\left(f\left(\alpha +\beta e^{i\left(t-\theta \right)}\right)+ f\left(\alpha +\beta e^{-i\left(t-\theta \right)}\right)\right)=2.\end{array}$$

Therefore, $D_{\theta }^p\left(f\left(\alpha +\beta e^{i\theta }\right)+f\left(\alpha +\beta e^{-i\theta }\right)\right)=D_{\theta }^p\left(2\left(\alpha +\beta \cos \hspace{0.17em}\left(\theta \right)\right)\right)$ $$=\frac{\beta }{\Gamma \left(1-p\right)}\underset{0}{\overset{\infty }{\int }}{t}^{-p}\left(2\mathrm{\hspace{0.17em}sin}\hspace{0.17em}\left(t-\theta \right)\right)dt =2\beta \mathrm{\hspace{0.17em}cos}\hspace{0.17em}\left(\theta +\frac{\pi p}{2}\right).$$

Example (5).

${\mathit{D}}^{\frac{1}{3}}\mathit{ }\left(\mathbf{4}\mathit{ }\mathit{sin}\left(\mathit{x}\right)+\mathbf{2}\mathit{sin}\left(\mathbf{2}\mathit{x}\right)\right).$

Let $f\left(z\right)=z^2,$ then using definition (3), we have: $$\begin{array}{c}\begin{array}{l}\begin{array}{c}\frac{\partial }{\partial \alpha }\left(f\left(\alpha +\beta e^{i\left(t-\theta \right)}\right)+f\left(\alpha +\beta e^{-i\left(t-\theta \right)}\right)\right)=\frac{\partial }{\partial \alpha }\left({\left(\alpha +\beta e^{i\left(t-\theta \right)}\right)}^2+{\left(\alpha +\beta e^{-i\left(t-\theta \right)}\right)}^2\right)\\ =2\left(\alpha +\beta e^{i\left(t-\theta \right)}\right)+2\left(\alpha +\beta e^{-i\left(t-\theta \right)}\right)=4\alpha +4\mathrm{\beta \hspace{0.17em}}\cos \hspace{0.17em}\left(t-\theta \right).\end{array}\\ \frac{\partial }{\partial \alpha }\frac{1}{i}\left(f\left(\alpha +\beta e^{i\left(t-\theta \right)}\right)-f\left(\alpha +\beta e^{-i\left(t-\theta \right)}\right)\right)=\frac{1}{i}\frac{\partial }{\partial \alpha }\left({\left(\alpha +\beta e^{i\left(t-\theta \right)}\right)}^2-{\left(\alpha +\beta e^{-i\left(t-\theta \right)}\right)}^2\right)=\frac{1}{i}\left(2\left(\alpha +\beta e^{i\left(t-\theta \right)}\right)-2\left(\alpha +\beta e^{-i\left(t-\theta \right)}\right)\right)=4\beta \sin \hspace{0.17em}\left(t-\theta \right).\hfill \end{array}\\ D^{\frac{1}{3}}\left(\frac{1}{i}\left(f\left(\alpha +\beta e^{i\theta }\right)-f\left(\alpha +\beta e^{-i\theta }\right)\right)\right)=D^{\frac{1}{3}}\left(4\mathrm{ }\alpha \beta \mathrm{ }\mathrm{sinx}+2{\beta }^2\sin \hspace{0.17em}\left(2x\right)\right)\end{array}$$

Setting $\alpha =1,\mathrm{ }\beta =1\mathrm{ }$ then we have: $$D^{\frac{1}{3}}\mathrm{ }\left(4\mathrm{ }\sin \hspace{0.17em}\left(x\right)+2\sin \hspace{0.17em}\left(2x\right)\right)=\frac{1}{\mathrm{\Gamma }\left(\frac{2}{3}\right)}\underset{0}{\overset{\mathrm{\infty }}{\int }}{t}^{-\frac{1}{3}}\left(\cos \hspace{0.17em}\left(t-\theta \right)\left(4+4\cos \hspace{0.17em}\left(t-\theta \right)\right)-4\sin \hspace{0.17em}\left(t-\theta \right)\sin \hspace{0.17em}\left(t-\theta \right)\right)dt$$ $$\begin{array}{c}=\frac{1}{\mathrm{\Gamma }\left(\frac{2}{3}\right)}\underset{0}{\overset{\mathrm{\infty }}{\int }}{t}^{-\frac{1}{3}}\left(4\cos \left(t-\theta \right)+4{\cos }^2\left(t-\theta \right)-4{\sin }^2\left(t-\theta \right)\right)dt\\ =\frac{1}{\mathrm{\Gamma }\left(\frac{2}{3}\right)}\underset{0}{\overset{\mathrm{\infty }}{\int }}{t}^{-\frac{1}{3}}\left(4\cos \left(t-\theta \right)-4\cos \hspace{0.17em}\left(2\left(t-\theta \right)\right)\right)dt\end{array}$$

Using the fact in equation (25) we get: $$D^{\frac{1}{3}}\left(4\mathrm{ }\sin \hspace{0.17em}\left(x\right)+2\sin \hspace{0.17em}\left(2x\right)\right)=\sin \hspace{0.17em}\left(\theta +\frac{\pi }{6}\right)-2^{\frac{4}{3}}\sin \hspace{0.17em}\left(2\theta +\frac{\pi }{6}\right).$$

Example (6).

${\mathit{I}}^{\frac{1}{2}}\left(2{\mathrm{e}}^{\mathbf{cos}\left(\mathit{\theta }\right)}\mathbf{sin}\left(\mathbf{sin}\mathit{\theta }\right)\right).$

Let $f\left(z\right)=e^z$ then using Definition (4), we have: $$\begin{array}{l}f\left(\alpha +\beta e^{i\left(\theta -t\right)}\right)-f\left(\alpha +\beta e^{-i\left(\theta -t\right)}\right)\\ =\frac{1}{i}\mathrm{ }\left(e^{\alpha +\beta e^{i\left(t-\theta \right)}}-e^{\alpha +\beta e^{-i\left(t-\theta \right)}}\right)\\ =2\sin \hspace{0.17em}\left(\beta \sin \hspace{0.17em}\left(t-\theta \right)\right){\mathrm{e}}^{a+\beta \cos \hspace{0.17em}\left(t-\theta \right)}\end{array}$$

Therefore, setting $\beta =1,\mathrm{ }\alpha =0,$ we have $${\mathit{I}}^{\frac{1}{2}}\left(2{\mathrm{e}}^{\cos \hspace{0.17em}\left(\theta \right)}\sin \hspace{0.17em}\left(\sin \hspace{0.17em}\left(\theta \right)\right)\right)=\frac{1}{\Gamma \left(\frac{1}{2}\right)}\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{2\mathrm{ }\sin (\sin (t-\theta \left)\right){\mathrm{e}}^{\cos (t-\theta )}}{t^{\frac{1}{2}}}dt.$$

This integral is difficult to be solved, but using some complex manipulations we are able to get a fairly clean result as follows $$\frac{1}{\Gamma \left(\frac{1}{2}\right)}\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{2\sin \hspace{0.17em}\left(\sin \hspace{0.17em}\left(t-\theta \right)\right){\mathrm{e}}^{\cos \hspace{0.17em}\left(t-\theta \right)}}{t^{\frac{1}{2}}}dt=\frac{2}{\Gamma \left(\frac{1}{2}\right)}\mathrm{I}\mathrm{m}\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{e^{e^{i\left(t-\theta \right)}}}{t^{\frac{1}{2}}}dt=\frac{2}{\Gamma \left(\frac{1}{2}\right)}\mathrm{I}\mathrm{m}\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n!}\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{e^{in\left(t-\theta \right)}}{t^{\frac{1}{2}}}dt.$$

Using the fact in equation (18), one can reach to $${\mathit{I}}^{\frac{1}{2}}\left(2{\mathrm{e}}^{\cos \hspace{0.17em}\left(\theta \right)}\sin \hspace{0.17em}\left(\sin \hspace{0.17em}\left(\theta \right)\right)\right)=2\sum _{n=1}^{\mathrm{\infty }}\frac{\sin \hspace{0.17em}\left(\frac{\pi }{4}-n\theta \right)}{\sqrt{n}\mathrm{ }n!}\mathrm{ }.$$

Despite the absence of a closed-form solution, the use of this example serves to demonstrate the effectiveness of the definition in practice.

6.2. Applications

This subsection explores the practical applications of the definitions provided. The aim is to demonstrate how the fractional derivatives defined in this research can be used to gain insight and analyze complex systems in various fields, including optics, control systems, signal processing and others. The examples and case studies discussed here highlight the usefulness and versatility of these definition in real-world scenarios.

We will also compare their effectiveness and ease of use with the classical Riemann-Liouville definition in solving practical problems.

The new definitions offer greater ease of use and clarity in several applications, including complex dynamics, control systems and signal processing. In these cases, the new definitions provide valuable insight and information that is difficult to obtain through the Riemann-Liouville definition. The latter can lead to special functions that are challenging to interpret in physical applications.

The new definitions offer greater ease of use and clarity in several applications, including (Carpinteri & Fabrizio, n.d.; Sabatier et al., n.d. Trujillo & Rubio, n.d.-a, n.d.-b):

• Complex Dynamics: The new definitions can be used to study the stability and bifurcations of systems described by the family of function $$\left[f(\alpha \mathrm{ }+\mathrm{ }\beta e^{i\theta })\mathrm{ }+\mathrm{ }f(\alpha \mathrm{ }+\mathrm{ }\beta e^{-i\theta })\mathrm{ }\right], \text{or the family }\frac{1}{i}\left[f\left(\alpha \mathrm{ }+\mathrm{ }\beta e^{i\theta }\right)\text{-}\mathrm{ }f(\alpha \mathrm{ }+\beta e^{-i\theta })\right]$$ offering valuable information that is difficult to obtain through the Riemann-Liouville definition.

• Control systems: The new definition can provide insight into the system’s response to different inputs and disturbances in systems described by the family of functions $$\left[f\left(\alpha \mathrm{ }+\mathrm{ }\beta e^{i\theta }\right)+\mathrm{ }f\left(\alpha \mathrm{ }+\mathrm{ }\beta e^{-i\theta }\right)\right], \text{or the family }\frac{1}{i}\left[f\left(\alpha \mathrm{ }+\mathrm{ }\beta e^{i\theta }\right)–\mathrm{ }f\left(\alpha \mathrm{ }+\beta e^{-i\theta }\right)\right]$$

• Signal Processing: The new definition can be used to analyze signals described by the family of functions $\left[f\left(\alpha \mathrm{ }+\mathrm{ }\beta e^{i\theta }\right)+\mathrm{ }f\left(\alpha \mathrm{ }+\mathrm{ }\beta e^{-i\theta }\right)\right],$or the family $\frac{1}{i}\left[f\left(\alpha \mathrm{ }+\mathrm{ }\beta e^{i\theta }\right)–\mathrm{ }f\left(\alpha \mathrm{ }+\beta e^{-i\theta }\right)\right],$ offering information about their frequency content and time-frequency characteristics.

It’s important to note that these are just a few examples, and the new definitions may have other specific applications where they are more convenient or accurate than the Riemann-Liouville definition.

Application (1)

Consider the scattering of an electromagnetic wave by a dielectric sphere. The scattered field can be represented in terms of the scattered angle θ, by the function $f\left(\alpha \mathrm{ }+\mathrm{ }\beta e^{i\theta }\right)+\mathrm{ }f\left(\alpha \mathrm{ }+\mathrm{ }\beta e^{-i\theta }\right),\mathrm{ }$ where $\alpha$ and $\beta$ are real numbers.

The fractional derivative of this scattered field with respect to θ can be used to analyze the scattering properties of the sphere, such as the angular distribution of the scattered power, and the extinction cross-section.

Consider a specific example, where the fractional derivative is used to analyze the scattering of an electromagnetic wave by a dielectric sphere of radius R and relative permittivity $\epsilon r.$ The scattered field can be represented by the function $2\alpha \mathrm{ }+\mathrm{ }2\beta \mathrm{ }\cos \left(\theta \right),$ where $\mathrm{ }\alpha \mathrm{ }=\frac{A}{R},\mathrm{ }\beta \mathrm{ }=\frac{B}{R},$ $A$ and $\mathrm{ }B$ are constants and $\theta$ is the scattering angle.

By using the third fractional derivative definition and setting $f\left(z\right)=z,$ then $D^p\left(2\left(\alpha +\beta \mathrm{ }\cos \left(\theta \right)\right)\right)\mathrm{ }=2\beta \mathrm{ }\cos \left(\theta +\frac{\pi p}{2}\right),$ we can analyze the angular distribution of scattered power, $P_{\mathit{scat}},$ and the extinction cross-section, $C_{\mathit{ext}}.$ The parameters α and β were represented as $\alpha \mathrm{ }=A/R$ and $\beta \mathrm{ }=B/R,$ where A and B are constants and R is the sphere’s radius. By calculating the integral of the squared magnitude of the scattered field over a closed surface surrounding the sphere. $$P_{\mathit{scat}}=\mathrm{ }{\int }_0^{2\pi }{\int }_0^{\pi }{\left|2\mathrm{ }\beta \cos \hspace{0.17em}\left(\theta +\frac{\pi p}{2}\right)\right|}^2\hspace{0.17em}\sin \hspace{0.17em}\left(\theta \right)d\theta \mathrm{ }d\varphi .\mathrm{ }$$

This integral can be evaluated to obtain the angular distribution of the scattered power, which depends on the parameters of the sphere $(R,\mathrm{ }\epsilon r),$ the wavelength of the incident wave and the value of the fractional derivative exponent p.

The extinction cross-section, $C_{\mathit{ext}},$ can also be calculated by using the fractional derivative. $C_{\mathit{ext}}\mathrm{ }$ is the ratio of the scattered power to the incident power, and it is a measure of the ability of the sphere to scatter the incident wave. It can be calculated by integrating the scattered field over a closed surface surrounding the sphere $$C_{\mathit{ext}}\mathrm{ }=\mathrm{ }\left(\frac{1}{I_0}\right){\int }_0^{2\pi }{\int }_0^{\pi }|2\beta cos\left(\theta +\frac{\pi p}{2}\right){\left.\mathrm{}\right|}^2\hspace{0.17em}\sin \hspace{0.17em}(\theta )d\theta \mathrm{ }d\varphi .$$ where $I_0$ is the incident power.

The results of these calculations can be used to analyze the scattering properties of the sphere and to compare the performance of the sphere with different values of the parameters $R,\mathrm{ }\epsilon r,\mathrm{ }$ and the fractional derivative exponent p.

Let us evaluate the integral for $P_{\mathit{scat}}\mathrm{ }$ and $C_{\mathit{ext}}:$ $$\begin{array}{c}\begin{array}{l}\begin{array}{c}\begin{array}{c}{P}_{\mathit{scat}}=\mathrm{ }{\int }_0^{2\pi }{\int }_0^{\pi }{\left|2\beta \mathrm{ }\cos \left(\theta +\frac{\pi p}{2}\right)\right|}^2\hspace{0.17em}\sin \hspace{0.17em}\left(\theta \right)d\theta \mathrm{ }d\varphi \\ =\mathrm{ }2\pi \mathrm{ }{\int }_0^{\pi }{\left|2\beta \mathrm{ }\cos \left(\theta +\frac{\pi p}{2}\right)\right|}^2\hspace{0.17em}\sin \hspace{0.17em}\left(\theta \right)d\theta \end{array}\\ =\mathrm{ }2\pi \mathrm{ }{\int }_0^{\pi }\left(4{\beta }^{2}co{s}^2\left(\theta +\frac{\pi p}{2}\right)\right)\hspace{0.17em}\sin \hspace{0.17em}\left(\theta \right)d\theta \end{array}\\ =\mathrm{ }8\pi {\beta }^2\mathrm{ }{\int }_0^{\pi }\left(1-si{n}^2\left(\theta +\frac{\pi p}{2}\right)\right)\mathrm{ }\sin \left(\theta \right)d\theta \end{array}\\ =8\pi {\beta }^2\mathrm{ }{\int }_0^{\pi }\sin \left(\theta \right)\text{-}\sin \left(\theta \right)si{n}^2\left(\theta +\frac{\pi p}{2}\right)d\theta =\frac{8}{3}\pi {\beta }^2\left(3-\cos \hspace{0.17em}\left(\pi p\right)\right).\end{array}$$

And, $$C_{\mathit{ext}}\mathrm{ }=\mathrm{ }{P}_{\mathit{scat}}/I_0.$$

It’s important to note that the result of the integral will depend on the value of the parameters R, ε.

The physical meaning of the results of the integral for $P_{\mathit{scat}}\mathrm{ }$ and $C_{\mathit{ext}}$ can be understood in terms of the scattering properties of the dielectric sphere.

$P_{\mathit{scat}}$ represents the scattered power, which is the amount of power that is scattered by the sphere in all directions. The result of the integral for $P_{\mathit{scat}}$ shows that the scattered power depends on the value of the fractional derivative exponent p, the radius of the sphere R, the relative permittivity εr and the wavelength of the incident wave.

$C_{\mathit{ext}}\mathrm{ }$ represents the extinction cross-section, which is the ratio of the scattered power to the incident power. It is a measure of the ability of the sphere to scatter the incident wave. The result of the integral for $C_{\mathit{ext}}$ shows that the extinction cross-section depends on the same parameters as $P_{\mathit{scat}}.$

In general, a larger extinction cross-section indicates that the sphere is more efficient at scattering the incident wave, while a smaller extinction cross-section indicates that the sphere is less efficient at scattering the incident wave.

It’s important to note that this example is just an illustration of how the fractional derivative definitions can be used in optics, and the results will depend on the specific problem and the form of the function.

The Riemann-Liouville fractional derivative of cos(x) with respect to x of order p, where p is a real number, is defined as: $$D_x^p\hspace{0.17em}\cos \hspace{0.17em}\left(x\right)=\mathrm{ }\left(\frac{1}{\Gamma \left(n-p\right)}\right)\frac{{\partial }^n}{\partial x^n}\mathrm{ }\left({\int }_a^x{\left(x-t\right)}^{n-p-1}\hspace{0.17em}\cos \hspace{0.17em}\left(t\right)dt\right).$$ where $n-1\mathrm{ }<\mathrm{ }p\mathrm{ }<\mathrm{ }n,$n is an integer and a is the lower limit of the integral.

The integral in this case would be represented by the fractional integral of the cosine function which has a special function representation, and the result of the fractional derivative will also be represented by a special function.

Application (2)

The $f_{Bm}\mathrm{ }$ process is a generalization of the standard Brownian motion process, which is characterized by a Hurst exponent H, where 0 < H < 1. The $f_{Bm}$ process is a Gaussian process that is not Markovian and has long-range dependence and self-similarity properties.

The fractional derivative of the $f_{Bm}$ process is defined as: $$f^{\left(p\right)}\left(z\right)=\frac{{\left(-1\right)}^p\Gamma \left(p+1\right)}{2\pi }{\int }_{\text{-}\mathrm{\infty }}^{\mathrm{\infty }}{\left(z+it\right)}^{-p-1}f\left(-it\right)dt,$$ where $p=H\mathrm{ }\text{-}\mathrm{ }1$ and $z$ is a complex number such that $Re(z)>0.$

By using this definition, we can analyze the properties of the $f_{Bm}$ process, such as its power spectral density, autocovariance function and moments.

In this way, the fractional derivative definition provided can be used to study the properties of non-Markovian processes and to understand the underlying mechanisms of complex systems that exhibit long-range dependence and self-similarity. Let’s consider the function $f\left(z\right)=\mathrm{ }{e}^{-z},$ where z is a complex number. We want to calculate the derivative of $f(z)$ with respect to z using the fractional derivative definition provided.

First, we need to find the integral of the function: $$f^{\left(\mathrm{p}\right)}\left(z\right)=\frac{{\left(-1\right)}^p\Gamma \left(p+1\right)}{2\pi }{\int }_{\text{-}\mathrm{\infty }}^{\mathrm{\infty }}{\left(z+it\right)}^{-p-1}f\left(-it\right)dt,$$ where p is the fractional order of the derivative, Γ(p + 1) is the gamma function and referring to Example (1), we have $${\int }_{\text{-}\mathrm{\infty }}^{\mathrm{\infty }}{\left(z+it\right)}^{-p-1}\mathrm{ }{e}^{it}\mathrm{ }dt=\frac{2\pi e^{\text{-}\mathrm{ }z}}{\Gamma \left(p+1\right)}\mathrm{ }.$$

So, we have $f^{\left(p\right)}\mathrm{ }\left(z\right)={\left(-1\right)}^p{e}^{\text{-}\mathrm{ }z},\text{ w}$ here $p=H\mathrm{ }\text{-}\mathrm{ }1.$

The relation between the new fractional definitions and the analysis of the fractional Brownian motion ($f_{Bm}$) process is that the definitions can be used to analyze the properties of the $f_{Bm}$ process. In summary, the new definitions of the fractional derivative provided can be used to study the properties of non-Markovian processes and to understand the underlying mechanisms of complex systems that exhibit long-range dependence and self-similarity. In this way, the fractional derivative definitions can be applied to the analysis of the fractional Brownian motion process.

Application (3)

Consider a signal represented by the function $f(t)\mathrm{ }=\mathrm{ }\cos (2\pi ft)\mathrm{ }+\mathrm{ }\sin (2\pi ft),$ where $f$ is the frequency of the signal. By using the fractional derivative definitions, we can analyze the frequency content and time-frequency characteristics of the signal.

The fractional derivative of the signal can be calculated using definitions (3) and (4) as follows: $$\begin{array}{l}{D}^p\left[\cos \left(2\pi ft\right)+\sin \left(2\pi ft\right)\right]=D^p\left[2\left(\alpha +\beta \mathrm{ }\cos \left(2\pi ft\right)\right)\right]+\mathrm{ }{D}^p\left[2\left(\alpha \mathrm{ }+\mathrm{ }\beta \mathrm{ }\sin \left(2\pi ft\right)\right)\right]\\ =\mathrm{ }2\beta \mathrm{ }\cos \left(2\pi ft\mathrm{ }+\frac{\pi p}{2}\right)+\mathrm{ }2\beta \mathrm{ }\sin \left(2\pi ft\mathrm{ }+\frac{\pi p}{2}\right).\end{array}$$

From the above equation, we can see that the fractional derivative of the signal results that represents the frequency content and time-frequency characteristics of the signal. By analyzing the fractional derivative, we can gain insights into the spectral characteristics of the signal and its behavior over time.

For example, if we set p = 1, the fractional derivative becomes the first order derivative, which represents the instantaneous frequency content of the signal. On the other hand, if we set p = 0.5, the fractional derivative represents the time-frequency content of the signal, which provides information about the spectral distribution of the signal over time. The results of the analysis can be used to improve our understanding of the signal and can also be used in various applications such as filtering, compression and denoising of signals.

Application (4)

Consider a noisy signal represented by the function $f(t)\mathrm{ }=\mathrm{ }\cos (2\pi ft)\mathrm{ }+e^{-t}.$ Where $e^{-t}$ represents the noise component. To remove the noise from the signal, we can apply a fractional derivative filter using the definitions.

Let’s consider the fractional derivative of order p = 0.5, which corresponds to a fractional low-pass filter. The fractional derivative of the noisy signal can be calculated as follows: $$D^{0.5}\mathrm{ }\left(f\left(t\right)\right)=\mathrm{ }{D}^{0.5}\mathrm{ }\left(\cos \left(2\pi ft\right)\right)+\mathrm{ }{D}^{0.5}\mathrm{ }\left(e^{-t}\right)=\cos \hspace{0.17em}\left(2\pi ft\mathrm{ }+\frac{\pi }{4}\right)\mathrm{ }+{\left(-1\right)}^{\frac{1}{2}}{e}^{-t}.$$

The filtered signal can be obtained by taking the inverse fractional derivative of the fractional derivative of the noisy signal, f(t): $$f^{\prime }\left(t\right)=\mathrm{ }{D}^{-0.5}\mathrm{ }\left(D^{0.5}\left(f\left(t\right)\right)\right)=cos\left(2\pi f\left(t\right)\mathrm{ }+\frac{\pi }{4}\right).$$

The filtered signal, $f^{\prime }\left(t\right),$ retains the low-frequency component of the original signal $(\cos (2\pi f(t$)) and removes the high-frequency noise component $\left(e^{-t}\right).$ The fractional derivative provides a flexible tool for analyzing signals with a broad range of frequencies and can be used to filter signals for various applications, including signal processing for filtering, compression and denoising of signals. This simple example demonstrates the use of fractional derivatives in signal processing for filtering, compression and denoising of signals. The fractional derivatives provide a flexible tool for analyzing signals with a broad range of frequencies and can be used in a variety of applications.

Application (5)

Let $f(z)$ be the Riemann zeta function, which is defined as:

$f(z)=\zeta (z)={\sum }_{n=1}^{\mathrm{\infty }}\frac{1}{n^z},$ where ζ is the Riemann zeta function. Using the first definition, and referring to Example (2) the fractional derivative of the Riemann zeta function is given by: $$f^{\left(p\right)}\left(z\right)=\frac{{\left(-1\right)}^p}{2}\sum _{n=2}^{\mathrm{\infty }}i\mathrm{ }{n}^{-z}\mathrm{ }{z}^{-p}\cot \left(\pi p\right)\left[{\left(-i\mathrm{ }z\right)}^p{\left[i\ln \hspace{0.17em}\left(n\right)\right]}^p-{\left(i\mathrm{ }z\right)}^p{\left[-i\ln \hspace{0.17em}\left(n\right)\right]}^p\right].$$

This result allows us to study the properties of the Riemann zeta function and its behavior in the complex plane and can be used to calculate the Hurwitz zeta function. One possible application of this fractional derivative is to examine the critical strip, where the Riemann zeta function is known to have a series of nontrivial zeros. The fractional derivative can be used to investigate the distribution of these zeros, which has important implications for number theory and the study of the distribution of prime numbers.

In statistical physics, the fractional derivative can be used to study anomalous diffusion, which occurs when the mean squared displacement of a particle over time is proportional to a power law with a non-integer exponent. This type of diffusion is observed in many physical systems, such as fluid flow in porous media, and the Hurwitz zeta function can be used to describe the probability distribution of particle displacement in these systems.

Application (6). Improper integrals using the new definitions

Given that: $$PV\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{\cos \left(\theta x\right)}{1-x^2}dx=\frac{\pi }{2}\sin \hspace{0.17em}\left(\theta \right),\mathrm{ }\theta >0.$$

Using differentiation under the integral sign technique, by differentiating both sides $\mathrm{ }p$ times with respect to $\theta \mathrm{ }$ where $0\le p<1\mathrm{ }$ using our definition we acquire: $$PV\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{x^p\mathrm{c}\mathrm{os}(\theta x+\frac{\pi p}{2})}{1-x^2}dx=\frac{\pi }{2}\sin \hspace{0.17em}\left(\theta +\frac{\pi p}{2}\right),\mathrm{ }\theta >0.$$

Which is a correct result, and this can be proved using residue theorem or by using Laplace transform technique, see the result in Zwillinger (2014) and Abu-Ghuwaleh, Saadeh, and Qazza (2022a).

7. Conclusion

The innovative definitions of fractional derivatives showcased in this study offer a fresh lens through which fractional calculus can be viewed. When juxtaposed with the conventional Riemann-Liouville definition, it becomes evident that these new definitions could be instrumental in enhancing various applications within the physics and engineering domains. This study lays the groundwork for in-depth exploration, wherein the multifaceted attributes and characteristics of these definitions can be further delved into and leveraged. As evidenced in the examples furnished, the potential of these definitions in sectors like signal processing is quite palpable. The first definition emerges as a robust instrument to probe intricate mathematical functions, especially when considering the Riemann zeta function and the Hurwitz zeta function. Delving into the fractional derivative of the Riemann zeta function unveils pivotal insights for number theory and statistical physics, allowing for a comprehensive understanding of the zeta function’s dynamics within the complex plane. Furthermore, this fractional derivative aids in computing the Hurwitz zeta function, which can be perceived as an extension of the Riemann zeta function, possessing a wide array of mathematical applications. In summation, this research’s fresh take on fractional calculus definitions paves the way for a more nuanced comprehension of complex mathematical functions. Continued exploration in this domain is bound to usher in groundbreaking strides in fractional calculus.

Funding statement

This research has received no external funding.

Disclosure statement

No potential conflict of interest was reported by the authors.