The gamma function is a mathematical function that generalizes the factorial function to non-integer values, including complex numbers. It is denoted by the Greek letter gamma (Γ) and is defined as Γ(z) = ∫∞0 xz-1 e-x dx, where z is a complex number.
The gamma function is an important tool in complex analysis because it extends the concept of factorial to non-integer values, allowing for the evaluation of integrals and series involving complex numbers. It also has many applications in fields such as physics, engineering, and statistics.
The gamma function is a generalization of the factorial function, meaning that for positive integers n, Γ(n) = (n-1)! This relationship allows for the extension of the factorial function to non-integer values.
The gamma function has many properties, including the reflection formula, the duplication formula, and the recurrence relation. It is also an entire function, meaning that it is analytic everywhere in the complex plane.
The gamma function is used to evaluate integrals and series involving complex numbers, as well as to solve equations and differential equations. It also has applications in probability and number theory. Additionally, the gamma function is used in the definition of other important functions, such as the beta function and the zeta function.