THE GAMMA FUNCTION:
A Superficial Analysis
Editor’s Note: This submission by Isaac E. ('14) is an article introducing and describing the gamma function.
Author's Note:
This is a description of how the great, grandiose, gamma function works, not why. An explanation of why it works requires a level of mathematical understanding that I in my current state have not yet achieve. If I may offer my input: I recommend that you read this article because it is a cool idea to think about regardless of whether or not you actually end up understanding why it can do what it does.
Introduction (read if you need to refresh yourself on what a factorial is):
Have you ever seen a number with an exclamation point next to it (e.g. 5!)? Chances are you have and you will know that that means you are taking that number’s factorial. If you haven’t, a numbers factorial is that number multiplied by every integer preceding it all the way to 1. For instance, 2! = 2 * 1 = 2, 3! = 3 * 2 * 1 = 6, 5! = 5 * 4 * 3 * 2 * 1 = 120, and so on. There are three things you should observe from taking a number’s (denoted by “n”) factorial: the first being that n! becomes very large very quickly, the second is that n! can only be taken for whole numbers (called integers), and the third is that every number on a graph of n! is a discrete point (i.e. they are not connected by a smooth curve).
The Gamma Function (actually):
The Gamma Function creates a curve which connects all of the discrete points of (n-1)! to create a smooth curve. This strange piece of mathemagic will be elucidated shortly, but let us first conceptualize the beauty of a function which can connect points we usually think of as simply being pretty much independent of one another.
Ok, now that our mental gawking is done, let’s look at how the gamma function works. Let us assume that Γ(n) = (n-1)! (that hangman-looking thing is actually the uppercase Greek letter “gamma), for all positive, non-zero integers. This function connects all of these discrete points from 1 until forever by taking this integral:
This is a description of how the great, grandiose, gamma function works, not why. An explanation of why it works requires a level of mathematical understanding that I in my current state have not yet achieve. If I may offer my input: I recommend that you read this article because it is a cool idea to think about regardless of whether or not you actually end up understanding why it can do what it does.
Introduction (read if you need to refresh yourself on what a factorial is):
Have you ever seen a number with an exclamation point next to it (e.g. 5!)? Chances are you have and you will know that that means you are taking that number’s factorial. If you haven’t, a numbers factorial is that number multiplied by every integer preceding it all the way to 1. For instance, 2! = 2 * 1 = 2, 3! = 3 * 2 * 1 = 6, 5! = 5 * 4 * 3 * 2 * 1 = 120, and so on. There are three things you should observe from taking a number’s (denoted by “n”) factorial: the first being that n! becomes very large very quickly, the second is that n! can only be taken for whole numbers (called integers), and the third is that every number on a graph of n! is a discrete point (i.e. they are not connected by a smooth curve).
The Gamma Function (actually):
The Gamma Function creates a curve which connects all of the discrete points of (n-1)! to create a smooth curve. This strange piece of mathemagic will be elucidated shortly, but let us first conceptualize the beauty of a function which can connect points we usually think of as simply being pretty much independent of one another.
Ok, now that our mental gawking is done, let’s look at how the gamma function works. Let us assume that Γ(n) = (n-1)! (that hangman-looking thing is actually the uppercase Greek letter “gamma), for all positive, non-zero integers. This function connects all of these discrete points from 1 until forever by taking this integral:
If you haven’t taken calculus, or even if you have taken calculus, a billion red lights must be going off in your head right now. What is this elongated curly-cue with a zero at the bottom and an infinity sign at the top? And, why are there two variables (x and t)? must be some of the questions you are asking right now. All will be explained.
The first thing you should understand is that the curly-cue thing is called an “integrand.” It basically tells you to find the area under the function located to its right (learning the calculus will allow you to perform this awesome feat). The second thing is that for some functions which are asymptotic at zero, the area under the curve approaches a finite value. In this case, the finite value is Γ(t), but more on that later.
The second question that must arise in your head is the fact that it looks like there are two variables in the function, and ostensibly there are. However, the “x” variable in this case merely serves a functionary purpose. Its sole reason for existence is that it creates the curve, under which we will find the area. It is the “t” values that truly matter. For instance, Γ(5) looks like:
The first thing you should understand is that the curly-cue thing is called an “integrand.” It basically tells you to find the area under the function located to its right (learning the calculus will allow you to perform this awesome feat). The second thing is that for some functions which are asymptotic at zero, the area under the curve approaches a finite value. In this case, the finite value is Γ(t), but more on that later.
The second question that must arise in your head is the fact that it looks like there are two variables in the function, and ostensibly there are. However, the “x” variable in this case merely serves a functionary purpose. Its sole reason for existence is that it creates the curve, under which we will find the area. It is the “t” values that truly matter. For instance, Γ(5) looks like:
The shaded area has a value of 24 units2. The curve shown varies in height based on the t value, but basically retains the same shape (all assuming that t is greater than or equal to 1). Look at the curve and shaded area for Γ(6):
Images created using Grapher application for Mac
The shaded area under this curve has a value of 120 units2. The cool thing about the gamma function though is that you can use it to find areas that are non-integer values. If you take Γ(5.5), you will get a value that is in between 24 units2 and 120 units2. In this case the value of Γ(5.5) = 52.3428. This applies for every real number (not only the integers) between 1 and infinity. The graph of the gamma function looks like this:
Image from Wikipedia
Once the t input values of Γ(t) become negative the graph has a lot of asymptotic behavior that can be further analyzed by using understanding how to manipulate the integral. The main point of the gamma function though, is to illustrate to you that there are ways of combining the factorial values in a smooth curve. Although this is somewhat of a superficial understanding, I invite you as a reader to explore the idea of the gamma function more deeply.
