Write down the relation between gamma function and factorial

The gamma function and the factorial function are related through the following equation:

Gamma(n+1) = n!

Here, the gamma function is denoted as Γ(n+1) and represents the extension of the factorial function to real and complex numbers. The factorial function, denoted as n!, is defined only for non-negative integers.

In other words, the gamma function is a continuous and smooth function that interpolates the factorial function for non-negative integer values. It provides a way to calculate the factorial of any positive real number or complex number.

This relation between the gamma function and the factorial function is useful in various mathematical and statistical applications, such as calculating probabilities, solving differential equations, and evaluating integrals involving factorials.