What is the factorial of some ?
First what's a factorial? The factorial of a number n is the number of ways of arranging n objects in a line. Mathematically,
So what is the value of ?
To find the answer, let us first see the graph of
Surprise! Surprise!! Turns out, exists for negative integers as well. How? For that, we return to the graph again.
We can see that the function satisfies -
It is a "smooth" graph. So is convex.
Turns out, there's actually another function that satisfies all the three above conditions - the gamma function.
This would converge for some values of x, for some values, it won't.
So this leads to
We can now find the value of
Importance of the gamma function-
It pops up in many applications like quantum physics, fluid dynamics, statistics, number theory.