Definition 
The gamma function is defined by the following integral that shows up frequently in many pure and applied mathematical settings: 
Note that the gamma function with a negative argument is defined by utilizing the recursion formula explained in the next section. 
Important Properties 
Recursion Formula: Given the following formula, a gamma function at one point can be evaluated recursively in terms of its value at another point: Generalized Factorial: When x is a positive integer, one can easily prove the following by using the recursive formula: In fact, many formulas involving n! can be extended to noninteger cases by replacing n! with , and that is why the gamma function is also commonly referred to as the generalized factorial function. Reflection Formula: Multiplication Formula: 
Some Fractional Values  
