Students are typically introduced to the factorial symbol when they encounter combinatorics in their high school math classes. Many mistake it for a punctuation mark, shouting the number out to the amusement of their math teachers the first time they see it written on the blackboard. To the cognoscenti, however, the symbol is just shorthand for saying the product of all the positive integers up to the number whose factorial is being calculated:
This concise symbol is used to define combinations and permutations, which in turn allow us to determine how many ways one can choose an appetizer, entrée, and dessert at their favorite SOHO restaurant. Along similar lines of reasoning the factorial appears in probability theory and in the Calculus to perform Taylor expansions of smooth functions. It seems simple enough, right? Just do a bunch of multiplication, and voila! there’s your answer. If mathematics has taught us anything over its millennia of existence, it is that structure hides within simplicity. We’re only beginning to scratch the surface of the deep…
An interesting question to ask is whether this function can be generalized to non-positive integer values of n i.e. n∉Z . What would the factorial of π be? or of -2? What is the factorial of a fraction? There is a simple solution that one can stumble upon within a few moments of first starting to think about the question:
Though a good guess, this definition has a serious flaw; it only extends the definition of the factorial to positive real values of the argument. Negative numbers, let alone Complex numbers, are still beyond its reach. To achieve this much wanted generality we must resort to the Calculus, and begin with a non-obvious definition. Beware what follows, it may get a little heavy, but the fruits of that labor will be quite beautiful. We begin with the rather strange definition of an integral function:
The first thing we discover, after integrating by parts, is that it satisfies an interesting recursion relation:
This relation is just what we want. If n is an integer then we find that I reduces to the standard factorial. In fact, once the function is known for all values between 0 and 1, all other positive values of the function can be determined. Furthermore, the recursion relation can be inverted to extend the definition to negative values of the argument as well:
We quickly find from there that the function is undefined for all the negative integers. This function is not new, and was first studied by Euler in 1729, though he did name it differently. To mathematicians it is known as the Euler Gamma Function, and is defined by:
The graph of the Gamma function is pretty interesting; a far cry from the simplicity of the simpler functions one learns about in their basic college math courses.
But why stop with reality? The integral definition of the Gamma function can be analytically extended beyond the real axis to the entirety of the complex plane. For complex values of the argument the Gamma function returns complex values, numbers with both a real part and an imaginary. One can visualize both the real and imaginary parts with three dimensional graphs, as well as the magnitude of Gamma. The resulting pictures display much of the structure behind this simple, yet elegant, function. Here is a picture of the magnitude of Gamma graphed on the complex plane:
The fun is only just beginning. Armed with the power of a factorial function for non-integers, it is possible to define fractional derivatives, and connect both differentiation and integration into a single diffeointegral operator which forms the basis of a fascinating branch of mathematics known as the Umbral Calculus. For those of you who are interested about learning more about the history of the Gamma function you can check here. Wolfram’s Mathworld has a fantastic exposition of the mathematical properties of the function, as well as an interesting page which allows you to graph Gamma on different parts of the complex plane.
As one final treat, I want to calculate for you the value of one-half factorial. First from the definition we begin with:
At first this integral seems a bit intractable, however a simple coordinate transformation reveals an old friend. Using…
and changing the limits of integration (which happen to be the same as before), we find…
This is just a weighted Gaussian. It can be evaluated by using a simple trick involving the introduction of a new variable in the exponential and using the derivative with respect to that variable to transform this integral into a standard Gaussian…
Who would have thought that the factorial of 1/2 is related to π? That’s what I find so incredibly fascinating about mathematics: how seemingly different mathematical topics, the factorial which is used in combinatorics, and π which comes from studying the circle, are actually related when one begins to probe the mathematical depths. These connections are not imposed by the mathematician, they are not constructed, they lie in hiding, inherent in the definitions of the objects that are imagined. They can be discovered by the adventurous, by those who are curious and ready to whittle away the darkness of the unknown armed with only the light of reason. Mathematics is an unexplored continent I hope everyone has the opportunity to visit. I hope you’ve enjoyed the few connections I’ve shown you here. Hopefully more little exposes on various mathematical topics will find their way into this section, and I can take you on more excursions into the unknown.