Free Online Class – Special Functions: The Gamma Function

The Gamma function is one of the “special” functions in math… special due to its significance in analysis.

Lets start by defining the gamma function. Now, if you know the factorial, you know the gamma function. The gamma function is simply a generalization of our good old factorial.

Firstly, this is how we can define it mathematically…

(n being positive)

Now, how is it related to the factorial function…Lets have a look…

We integrate the above definition by parts & it gives us…

When we plug in the upper limit(infinity), the resulting form calls for the L Hopital’s rule & the value of the limit comes out to be 0.

Which gives us…

Now, take up the RHS…& apply the above definition furthur…
(assuming that n is a positive integer)

So, we have…