A Generalized Function defined by the Euler first kind integral and its connection with the Dirac delta function

TL;DR

This paper demonstrates that the Euler integral of the first kind can define a generalized function in regions of divergence and explores its connection with the Dirac delta function.

Contribution

It introduces a new perspective on the Euler integral of the first kind as a generalized function and establishes its relationship with the Dirac delta function.

Findings

Euler integral of the first kind can define a generalized function in divergent regions

The connection between this generalized function and the Dirac delta function is established

Provides a new framework for understanding Euler integrals and delta functions

Abstract

We have shown that in some region where the Euler integral of the first kind diverges, the Euler formula defines a generalized function. The connected of this generalized function with the Dirac delta function is found.