Before I start, I should remind the reader that this blog is aimed at a non-mathematical audience so I hope any actual mathematicians who read this aren’t too upset with my hand-wavey explanations.

At some point during a high-school education, you have likely come across the mathematical notion of a factorial. Given a positive integer, we can define the factorial of it to be the product of the integers less than or equal to itself. A factorial is represented by an exclamation mark – for example $5!=5\times4\times3\times2\times1=120$. This makes sense for positive integers, but can we make sense of 2.5!, or something similar? Obviously since I’m posting about it, the answer will turn out to be yes, sort of… But how?

First let’s plot a graph of the factorials that we can make sense of.

The word “plot” is used very loosely…

Any line passing through all of these points (and all of the other factorials) will be the graph of a function which agrees with the factorial function. So really there are infinitely many ways we could define what 2.5! means; we should be asking if there is some canonical way of doing this. Again, there is. There is a hierarchy of “niceness” (regularity/smoothness) of functions, and there is a unique “nicest” way to join the dots in the graph above. On one side of the “niceness” scale are functions which can’t even be drawn without taking your pen off of the paper and then functions with sharp points like zig-zags; on the other side of the “niceness” scale, we have the usual nice functions like polynomials.

Having just said that polynomials are among the nicest functions, maybe we can build something that’s polynomial-like to agree with the factorial function. If we only wanted to write down a function that agrees with two points in the graph above, then I could just draw a straight line through it – easy! If we want to write down a function agreeing with three points in the above graph, then we could write down a parabola ($f(x)=ax^2+bx+c$) which exactly agrees with those three points. In fact, if we want a function to agree with one million different factorials, then we could do this perfectly with a polynomial – albeit one with one million different terms, and an $x^{999999}$ in there.

In some sense, we consider a polynomial with infinitely many terms so that we can make sure we hit every single (infinitely many) factorial in our graph above. This “infinite polynomial” is a called a power series, and like polynomials, power series are the “nicest” functions we’ve got. This particular function extends to most of the complex numbers too and is called the “gamma function” – technically the gamma function shifted by one.

These “nicest” functions are called analytic, and the idea of using series to analytically extend things is very important in complex analysis.