Abstract
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 Z. Some properties of gamma function are discussed by using the properties of delta function.
Highlights
The problem of giving s! a useful meaning when s is any complex number was solved by Euler (1707-1783), who defined what is called the gamma function,∫ Γ ( s) :=∞e−tt s−1dt 0(s = σ + iτ,σ = R(s) > 0). (1.1)He extended the domain of gamma function from natural numbers, to complex numbers
Some properties of gamma function are discussed by using the properties of delta function
There is more than one representation for all special functions, for example, the series representation, the asymptotic representation, the integral representation, etc
Summary
The problem of giving s! a useful meaning when s is any complex number was solved by Euler (1707-1783), who defined what is called the gamma function,. He extended the domain of gamma function from natural numbers, , to complex numbers. This is an extensively studied classical function and can be considered as a basic special function. It can be written as a sum of two incomplete gamma functions. The function Γ (s) itself, and its incomplete versions γ ( s, x) and Γ (s, x) , are known to play important role in the study of the analytic solutions of a variety of problems in diverse areas of science and engineering (see, for example, [1] [2] and [3]; see the recent papers [4] [5] [6] and [7]).
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