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### x Factorial and the Gamma Function

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Date: 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
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Date: 05/30/98 at 15:48:49
From: Doctor Pete
Subject: Re: y=x!

Hi!

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,

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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Associated Topics:
High School Functions
High School Number Theory
Middle School Factorials

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