کاربرد توابع بتا در حل انتیگرال‌های غیر عادی معین

Abstract

Improper integrals are those which, according to the general rules of integration, cannot be solved directly using standard or partial integration methods. In the twentieth century, mathematicians used the Beta function to solve certain highly complex and improper integrals. Therefore, by applying the Beta function B(x, y)—which itself is defined by an improper definite integral—many of these integrals can be evaluated. Beta functions are widely used in pure mathematics and physics, particularly in problems involving integrals, analytical mathematics, and even number theory. They are also frequently applied in solving improper integrals that arise in statistical mechanics and electrodynamics in physics. The aim of this study is to demonstrate how the Beta function can simplify and solve complex improper integrals. The research method employed in this study is library-based and descriptive–analytical. Hence, this paper attempts to solve improper integrals—those which cannot be easily evaluated—by means of the Beta function, which itself possesses an integral definition.