x Factorial and the Gamma FunctionDate: 05/29/98 at 18:41:32 From: Ms. Curtis Basham's Math Class Subject: y=x! Our class was running a warm-up where the teacher forgot the function cards, and we were making up our own functions when Nick Gilbertson stumped the class. We were giving him values for x and he gave us values back. Nick got perplexed when we gave him zero and negative integers. We eventually discovered that his function was: y = x! Our teacher was so excited, and we guessed at what might be going on when x = 0, x = -1, and so on. Using calculators, we received error messages for negative integers but not for zero. Using the computer we graphed y = x! and saw its unique graph. Our questions: What is going on at x = -1, -2, -3, ... ? What is the rule for factorial when x = 0? Zero times anything is zero, so why is 0! not zero? How do you calculate 0.5! rather than read it off a graph? What about when x = -3.5? Help, Ms. Curtis Basham's 8th grade class Date: 05/30/98 at 15:48:49 From: Doctor Pete Subject: Re: y=x! Hi! I'm really glad you asked this question. Clearly, you're all familiar with the factorial function, y = F[x] = x!, which is defined for positive integers. What I'll present to you is something very, very few people ever learn or understand, but I think it is truly remarkable. The usual definition of the factorial function is: x! = x(x-1)! = x(x-1)(x-2)...(2)(1) But, as you noticed, this doesn't make sense when x = 0. You may be inclined to say 0! = 0, based on the above definition. However, you simply can't use this definition to make any meaning of 0!. Rather, you need another way to define x!, one that isn't limited to positive integers. To get around this problem, mathematicians introduce a new function, which is called the gamma function. It's written with the capital Greek letter gamma, but since I can't type Greek letters in e-mail, I'll use the letter G. It is defined for every number except the negative integers, and it also satisfies these properties: (A) G[1] = 1 (B) G[x+1] = x * G[x], for any number x (C) G[x] = (x-1)! , when x is a positive integer (D) G[1/2] = Sqrt[Pi], that is, G[1/2] is the square root of Pi. Now, I haven't really said what G[x] is in terms of x. In order for me to describe G[x] as a function of x, I would have to discuss calculus, in particular, things called improper integrals. You'll probably learn about these in senior high school mathematics courses. But the basic idea behind the gamma function is this: We want a function which generalizes the factorial function, so that property (B) is satisfied for as many numbers x as possible. So in particular, we have: (1/2)! = G[3/2] = (1/2) * G[1/2] = Sqrt[Pi]/2 We also want G to be "smooth," that is, if you were to graph G[x] as a function of x, you would find that G is a smooth line that passes through the points (x, (x-1)!) for every positive integer x. One can prove that G is the only such function that does this; that is, it is the unique generalization of the factorial function with these properties. Finally, there is an easy way to "see" why 0! = 1. Consider: (n+k)! / n! = (n+k)(n+k-1)(n+k-2)...(n+2)(n+1) Then let n = 0: k! / 0! = (k)(k-1)(k-2)...(2)(1) = k! It follows that for this to be true, 0! = 1. ---Note to the Teacher:--- Hi, I'm really glad your students are learning about the factorial function. It is an excellent introduction to arithmetical functions, concepts like domain, empty sums (which have natural meaning as 0), and empty products (which, contrary to many people's intuition, is actually 1, the multiplicative identity). Unfortunately, while a discussion of the gamma function is beyond the scope of an 8th grade class, it is nevertheless fascinating and stimulating to show the means by which functions can be often be generalized to cover a larger domain, and further this generalization can be made unique - this is analytic continuation. In particular: G[z] = Integral[t^(z-1) Exp[-t] dt, {t, 0, Infinity}] is defined for all complex number z with Re[z] > 0, and from using the relation G[z] = G[z+1]/z, this definition can be extended into the left half-plane, except when z = 0, -1, -2, ... . Alternatively, there are definitions of G using infinite products, limits, and identities involving trigonometric functions. As calculators become more sophisticated and available to younger minds, students will be introduced to strange and new concepts at earlier and earlier ages. We'll have a lot of explaining to do! -Doctor Pete, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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