A Distributional Representation of Gamma Function with Generalized Complex Domian

In this paper, we present a new representation of gamma function as a series of complex delta functions. We establish the convergence of this representation in the sense of distributions. It turns out that the gamma function can be defined over a space of complex test functions of slow growth denoted by  . Some properties of gamma function are discussed by using the properties of delta function.


Introduction
The problem of giving ! s a useful meaning when s is any complex number was solved by Euler , who defined what is now called the gamma function,   [1] [2] and [3]; see also the recent papers [4] [5] [6] and [7]). By using the How to cite this paper: Tassaddiq was obtained in [8]. This led to new integral identities involving the gamma function.
At first glance, it seemed to give useful results but the "distributional representation" presented was not mathematically well defined there. As such, we could not be entirely sure of the validity of any new results presented in [8]. Similarly, the results presented were generalized in [9] ( ) ( Again some new formulas were derived, which were also checked by using classical means but the representation was not mathematically well defined. Such type of representations are also obtained for other special functions, see for example [10] and [11].
In this paper, we present a new representation of gamma function and establish the convergence of this representation in the sense of distributions. By doing so it turns out that the domain of gamma function can be extended from complex numbers  to complex functions  . Some properties of gamma function are discussed by using the properties of delta function.
The plan of this paper is as follows. After giving a necessary and brief introduction to distributions and test functions in Section 2, we discuss the distributional representation of gamma function in Section 3. The convergence of the series along with its properties are also discussed here. Some properties of gamma function w.r.t Fourier transformation are discussed in Section 4.

Distributions and Test Functions
The space of all continuous linear functionals acting on the space of test functions with compact support and its dual are denoted by  and ′  respectively. Distributions can be generated by using the following method. Let ( ) f t be a locally integrable function then corresponding to it, one can define a distribution f through the convergent integral It is easy to show that it is a continuous linear functional by using the definition of φ and the details are omitted here, [12]. Distributions that can be contain representations of locally integrable functions but also include many other entities that are not regular distributions. Therefore many operations like limits, integration and differentiation, which were originally defined for functions, can be extended to these new entities. All distributions that are not regular are called singular distributions, for example, the delta function is a singular distribution.
The Fourier transform of an arbitrary distribution in ′  is not, in general, a distribution but is instead another kind of continuous linear functional which is defined over a new space of test functions. Such a functional is called an ultradistribution, for example delta function of complex argument is an ultradistribution ( [12], Section (7.7)). Some properties of delta function include Therefore, we first discuss the space of test functions on which such distributions act. The space of test functions denoted by  consists of all those entire functions whose Fourier transforms are the elements of  , ( [12], Section (7.6)). Since φ is an entire function, it cannot be zero on any interval a t b and Here  is the space of testing functions of rapid descent and ′  is the space of distributions of slow growth, which is also called the space of tempered distributions. An important example of ultradistributions is ( [12], p. 204, Equation (7)) where α is a complex number. Hence, delta function of complex argument is defined on the space of test functions  .  Note that φ ∈  implies that the series is convergent. Conversely if the series is convergent then φ may or may not belong to  , therefore the condition is necessary and not sufficient.  Since complex delta is linear over  , this implies that gamma function is also linear over  . It gives us

Distributional Representation of the Gamma Function and Some Properties
is known to be linear, definition of continuity will be wane to as "the  6) The product of gamma function with a regular distribution ( ) s ψ is defined over  .
7) The product is also continuous linear functional over  .
Proof. Results (1-7) can easily be proved by using the properties of complex delta functions. Theorem 5. Prove that distributional derivatives of gamma function also exist.

Properties of Distributional Representation of Gamma Function w.r.t Fourier Transformation
Before stating the properties, we write the necessary notations used for this is an infinitely smooth distribution.
Proof. By following the lines of above theorem which shows that derivative of Fourier transform of ( ) s Γ exist upto k th order in distributional sense. So ( ) t Γ  is a distribution that is infinitely smooth.

Concluding Remarks
One normally thinks of a function as defined by a series or an integral of some variable or in terms of those functions that we regard as "elementary". However, the function needs to be regarded as an entity in itself, which is represented by a series or an integral. Only in this sense can we continue the function beyond its original domain of definition. This is essential for applying the function beyond dealing with the problem it was originally defined to deal with. This consideration becomes especially important when discussing the theory of (special) functions. There is more than one representation for all special functions, for example, the series representation, the asymptotic representation, the integral representation, etc. Further there can be more than one integral or series representation, which help to define the function in different regions. For example the integral representation (1.1) of the gamma function is defined in the positive half of the complex plane. It leads to another representation, which is defined in the whole complex plane except at negative integers. One can also use a different representation to express the function in terms of simpler functions. It will help to know the behavior of the original function at certain points of its domain. One representation can give simpler proofs of some known properties as compared to other. Here, in our present investigation, we have obtained a new representation of gamma function, which defines it over a space of complex entire test functions of slow growth denoted by  . Some properties of gamma function are also discussed over this space. It is hoped that various new properties and applications of gamma function can also be defined using this generalized domain. It will extend the use of gamma function in distribution theory and other fields of Engineering and Science.