The gamma function, a fundamental mathematical function with applications across various fields, is initially defined for positive real numbers. However, its significance extends far beyond this initial domain. By analytically continuing the gamma function, we can extend its definition to a much larger portion of the complex plane. This analytic continuation not only provides a deeper understanding of the gamma function's properties but also enables its application to diverse problems in mathematics, physics and statistics. In this research paper, we delve into the intricacies of the gamma function and its analytic continuation. We begin by introducing the gamma function as an integral representation and explore its basic properties, such as the functional equation and its relationship to the factorial function. Subsequently, we discuss the concept of analytic continuation and demonstrate how it can be applied to extend the gamma function beyond its initial domain.