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A Note on Gamma Function and Convolution

Francisco Mota

Abstract

In this note we explore the relationship between the operation of convolution of functions and the Eulerian integrals. This approach allow us to obtain some expressions for the convolution of a certain class of functions in terms of the Gamma Function as well as to derive some well known properties of the Gamma Function by using the concept and properties of the convolution.

A Note on Gamma Function and Convolution

Abstract

In this note we explore the relationship between the operation of convolution of functions and the Eulerian integrals. This approach allow us to obtain some expressions for the convolution of a certain class of functions in terms of the Gamma Function as well as to derive some well known properties of the Gamma Function by using the concept and properties of the convolution.
Paper Structure (6 sections, 13 theorems, 61 equations)

This paper contains 6 sections, 13 theorems, 61 equations.

Table of Contents

  1. Introduction
  2. Convolution of Functions
  3. Convolution Derivative Property, Units Step Function and its Derivative
  4. The Set of Functions $f_{\alpha}$
  5. The Eulerian Integrals and Convolution
  6. Conclusions

Key Result

Proposition 3.1

Let it be a set of real and non-negative (parameter dependent) functions "$f_{\alpha}$", $\alpha>0\in\mathbb{R}$, which satisfies the following properties: Then this set of functions is necessarily defined as:

Theorems & Definitions (32)

  • Definition 2.1
  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Proposition 3.1
  • proof
  • Corollary 3.2
  • proof
  • Proposition 4.1
  • proof
  • ...and 22 more