If is an integer, then, using integration by parts repeatedly, we see that (see Exercise 2 in this post!). We can now use this integral form of the factorial of a non-negative integer to define the factorial of any real number which is not a negative integer.
Definition 1. The gamma function is defined by
Remark. By definition, for all integers and so for all integers So we may define for all real numbers but, in order to do that, we need to show that is well-defined, i.e. is convergent for all real numbers (Problem 2).
Example 1. Show that In other words,
Solution. By definition, and so the substitution gives (see this post!).
Problem 1. Show that for all
Solution. Intgeration by parts with and gives
Example 2. Show that for all integers In other words,
Solution. By Problem 1 and Example 2, we have
Problem 2. Show that is well-defined, i.e. is convergent for all real numbers
Solution. By Problem 1, we have and so we only need to show that is convergent for We have
The integral is a proper integral and so it is convergent. Now choose an integer such that Then
So, by the comparison test, is convergent. Thus, by is convergent too.
Definition 2. We now use Problem 1 to extend the domain of from to the set If then and so is defined; we then define If then and we just defined we then again define etc. Now if is a real number which is not a negative integer, we may define
Example 3. By the above definition and Example 1,
Exercise. Show that if then is divergent.