An Exploration of Some Special Functions and their Applications

Abstract

Special functions are a class of mathematical functions that fall outside of ele-mentary functions. They have distinct properties and wide-ranging applications in areas such as engineering, applied mathematics, physics, statistics, economics, and finance. In this research paper, I study two popular special functions: the Gamma function Γ(m) and the Beta function β(m, n). I develop some of their important properties and results, prove these properties, and demonstrate some of their practical applications. Understanding the properties and applications of these functions is vital for solving mathematical problems that arise in practical modeling. This study sheds new light on the significance and versatility of Γ(m) and β(m, n), their applications, and mathematical properties.