Have You Ever Wondered?

About 6 months ago, I wrote a piece on Euler’s Identity, a mathematical formula of wondrous proportions. Two things came out of that. Firstly, all my friends are now adamant about not standing too close to me in public places (don’t want to be associated with nerds they say…), and more importantly, I have a great respect for divine creation that we call maths.

Here, I try to build on the latter. I’ll try to keep in short, I promise.

Let’s start by something called the Gamma function :

where z is a complex number with positive real parts (hence the integral from 0 to infinity and not minus infinity to infinity). t is some dummy variable which disappears after the integral is performed. But that’s not really important. In fact, the formula doesn’t even look particularly special. Why its called a Gamma function, and why it is so special that it has its own name is beyond me. Nor do I know why someone would even bother writing it down. Basically it looks pretty harmless.

However, some really cool urbane slick mathematician in his twenties showed, in his chic looking apartment with a verandah that overlooks the creeks and hills of his homeland (warning, overuse of creative licence is detected), that using integration by parts, that the Gamma function is identically

So what does this mean? It means that, say, Gamma(5) = 4 x Gamma(4), and Gamma(4) = 3 x Gamma(3) …

Uh huh. So?

Sooooooooooooo….. it basically means that Gamma(z+1) = z! , where z! is pronounced z-factorial. Factorials come up quite often, especially in probability and stuff like that. For example, if you had four shoes of different colour with four boxes of different colour, you would have 4! = 4×3x2×1 = 24 ways in which to arrange the shoes.

So, what’s so special about the Gamma function linking with factorials then? Well, normally, factorials are only defined for integer numbers, like 1,2,3…, and not for any other type of numbers - logically how can we have 2.3! or 7.771!  ? But hey, the Gamma function is defined for all z where the real part of z is positive. As long as we have positive numbers, we can define a factorial for it. That in itself, is pretty special. But in particular, think about this …

What is the factorial of half, i.e., 0.5! ? What do you think it is? Hell, I don’t even know where to begin guessing, but if you guessed this, you must be brilliant :

Yes, 0.5! = the square root of pi!!! (real exclamation marks, not factorials). What the hell man! Where did pi come from? Pi, a number that we cannot ever possibly determine exactly, due to its own nature, a number that is usually associated with the ratio of a circle’s perimeter over its diameter, has turned up in our weird half-factorial.

It’s a miracle.