Mathematical equations homer energy results

Mathematical equations homer energy results series#

It is important to mention that this contribution remained unachievable without reviewing. Further detailed analysis of the gamma function can be found in and related bibliography therein.

Here, denotes the set of natural numbers or positive integers. It is a basic special function so that Pochhammer symbols have the following relationship with the gamma function: The focus point of the current investigation is to study a modified (generalized) form of gamma function, defined by Similarly, the hyperbolic functions can be defined as Similarly, the definition of fractional circular functions is which are obviously related to the Mittag-Leffler function defined in equation ( 3). For our purposes, we would be using the following form of Mittag-Leffler function : The decades of this century are mainly devoted to the analysis and applications of this function, see for example, and related references therein. Three-parameter Mittag-Leffler function is also known as Prabhakar function. Here, denotes the set of complex numbers and denotes the real part of any complex number. For example, two- and three-parameter Mittag-Leffler functions are defined by respectively. It is known as Mittag-Leffler function defined byДЇor one-parameter Mittag-Leffler function reduces to the exponential function There are many generalizations of Mittag-Leffler function.

Mittag-Leffler proposed a function that appears a natural substitute of exponential function in the case of fractional order. Before going on to our main results, we describe necessary background and preliminaries in the subsequent paragraphs. Taking motivation from these researches, the aim of the present paper is to study the modified gamma function by using Mittag-Leffler function. Following them, Paris has discussed asymptotic behavior of the modified exponential integral and studied its more general form by using two-parameter Mittag-Leffler functions. Referring to these phenomena, creep rule is generalized in the sense that Becker model involves Mittag-Leffler function of order in place of exponential function. This model represents the Maxwell body at one end point, namely,, and the Becker body at other end point, namely. have explored a new model which depends on. More recently, Mainardi and Masina have presented modified exponential integral by using Mittag-Leffler function. The review of literature demonstrates that many of the new researches depend on the involvement of Mittag-Leffler function in place of exponential function. This gives insights for the emergence of a new fractional operator as one of the natural consequences of this research. Hence, its action on a suitably chosen function over a specific domain is easily obtainable in view of the standard properties of the delta function. Hence, a basic motivation and a significant contribution of this research are to study a special function containing the Mittag-Leffler function (also known as вЂњqueen of fractional calculusвЂќ ), which is represented in terms of complex delta function. How the fractional operators are defined? It might be summarized as follows: вЂњeach fractional operator is the action of a particular special function on a general function, which usually represents the solution of a well-defined problem by using a suitable integral transform.вЂќ See for example. One attractive aspect of this approach is to model complex systems by taking into account the effects of filtration and memory. Use of fractional operators has become a remarkable tool to study new technology and researches. Similar properties of existing distributions can be deduced. Furthermore, the new function is used to generate a probability density function, and its statistical properties are explored. Hence, the results are also validated with the earlier obtained results for gamma function as special cases.

Mathematical equations homer energy results series#

Properties of this modified function are discussed by investigating a new series representation involving delta function. Present motivation is to define a new special function by modification in the original gamma function with Mittag-Leffler function. Recent applications of Mittag-Leffler function have reshaped the scientific literature due to its fractional effects that cannot be obtained by using exponential function. Mittag-Leffler function is a natural generalization of the exponential function.