Calculating factorials of real numbers with the Gamma function

Abstract

Functions play a fundamental role in mathematics. Beyond traditional functions, there are several special functions, among which is the Gamma function. This is an important special function that extends the concept of factorial to positive real numbers (and to complex numbers). Through a definition based on an improper integral, it allows for the continuous calculation of factorials, enabling its application in contexts where the traditional factorial is not defined, such as for non-integer numbers. This work aims to present a brief introductory study on the Gamma function, including one of its constructions, a study of the function’s domain, some of its main properties, and brief comments on applications. Although rarely addressed in school curricula, the Gamma function proves to be essential in advanced areas of mathematics such as real analysis, statistics, theoretical physics, and number theory, demonstrating its versatility, conceptual depth, and historical importance in the development of mathematics.