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The gamma function is defined by the improper integral

(1.1)

And is related to the factorial function by the formula

(1.2)

This relationship can be proved using a simple integration by parts.
Choose



After integrating, we obtain

(1.3)

Which is equivalent to

(1.4)

Or

(1.5)

Then note that




And so on. Thus, we have the relationship shown in (1.2).

So why is this of any importance whatsoever?

Think about it: now we can define the factorial function for non-integer arguments. To paraphrase: have you ever wondered what one-half factorial was?

Well, let's see. Plug 1/2 into (1.1) to obtain



Letting x = u^2 and integrating, we have



This is known as the Gaussian integral. It has many applications in probability theory. Consulting a table of integrals, we thus find



Then, by (1.5),



Any questions or comments are very much welcomed. 3nodding