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bossk171

Member

Registered: 2007-07-16

Posts: 305

factorals for non intigers

How are they calcualted?

I understand n! = (n)*(n-1)*...*(2)*(1)

what is 2.5!

or e!

or -4!

and how do I do calculate it long hand?

-Nick

Last edited by bossk171 (2007-07-16 07:45:18)

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Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: factorals for non intigers

http://mathworld.wolfram.com/GammaFunction.html

It can be pretty complex stuff.  Let me know if there is anything in there you need clarified.

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

bossk171

Member

Registered: 2007-07-16

Posts: 305

Re: factorals for non intigers

Thanks, I've looked at this before, but I'm afraid it's a little over my head. I took AP calc (and did quite well) but this is too much.

Is there anyway to simplify things, or do I have to understand it completely to understand it at all?

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There are 10 types of people in the world, those who understand binary, those who don't, and those who can use induction.

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: factorals for non intigers

Ok, here is basically how things like this work.  We find a function, in this case a complex (difficult, not imaginary...) integral in which if we plug in the integers, we get factorials from it.  f(2) = 2, f(3) = 6, and so on.  But since it is an integral, we can also plug in any real number.  As such, we say that the gamma function is an expansion of the factorial function.  If you want a similar example, the Riemann zeta function is an expansion of the harmonic functions.

So all you need to know is that we plug in 2.5, get the value from the integral, and this is what we call 2.5!.

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

bossk171

Member

Registered: 2007-07-16

Posts: 305

Re: factorals for non intigers

Thanks.

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There are 10 types of people in the world, those who understand binary, those who don't, and those who can use induction.

mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: factorals for non intigers

That stuff is mostly over my head as well, but there's some easier stuff that can help you advance a bit. For example, a nice rule about factorials is that n! = n*(n-1)!.
That means that if you know r!, then you can find (r+z)! by induction. [r is real, z is integer]

That mostly doesn't help since the Gamma function is horrible for nearly all non-integer numbers that you put in, but f(0.5) gives the rather nice answer of √(π)/2.

You can then use the fact that 1.5! = 1.5*0.5! and 2.5! = 2.5*1.5!, etc. to find half-integer factorials.

That method also shows that there aren't negative integer factorials, because 0! = 0*(-1)!, which means that (-1)! = 0!/0 = 1/0, which is undefined.

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Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: factorals for non intigers

That method also shows that there aren't negative integer factorials, because 0! = 0*(-1)!, which means that (-1)! = 0!/0 = 1/0, which is undefined.

Actually mathsyperson, I wouldn't say that.  The n! = n*(n-1)! comes from the factorial function itself.  Since the factorial function is only defined on natural numbers, I would say that the rule doesn't apply to negative numbers.  But as you can see, the Gamma function is defined over the negatives.  Gamma(-0.5) ~ -3.54491.

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: factorals for non intigers

I'm only denying it for negative integers. All other negatives are fine.

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Why did the vector cross the road?
It wanted to be normal.

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: factorals for non intigers

Ah.  Excellent point then...

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

bossk171

Member

Registered: 2007-07-16

Posts: 305

Re: factorals for non intigers

Thanks, mathysperson, that was very enlightening.

I printted out a copy of the wolfram Mathworld page, and I'm trying again, but I have some questions:

What is "residue?" (line [2] on his page)

Is the Gamma function something we just accept (like Taylor series) because it's too difficult to understand, or is there a proof for why it works?

EDIT: I wikied "residue" and it says "In complex analysis, the residue is a complex number which describes the behavior of line integrals of a meromorphic function around a singularity."  That in No way helps me.

Last edited by bossk171 (2007-07-17 04:29:30)

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There are 10 types of people in the world, those who understand binary, those who don't, and those who can use induction.

krassi_holmz

Real Member

Registered: 2005-12-02

Posts: 1,905

Re: factorals for non intigers

"Is the Gamma function something we just accept (like Taylor series) because it's too difficult to understand..."

Noo!
In mathematics we don't "just accept" things! Somethimes the things can become complicated at first look, but they are done with concrete axiomatic mathematical logic. The Taylor series may look weird, but they are something very foundamental - the complex-valued function extensions are defined mostly by them!

The Gamma function isn't an exception.  It's just confusing in the beginning because there are new terms. And I thing it will be better if you start exploring the Gamma function with some more user-friendly text. I'm not saying that the Wolfram's page is not good- no, but it's mainly for reference and it's highly formalized.

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MathsIsFun

Administrator

Registered: 2005-01-21

Posts: 7,711

Re: factorals for non intigers

And I thing it will be better if you start exploring the Gamma function with some more user-friendly text.

Which someone here could volunteer to write ... ?

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bossk171

Member

Registered: 2007-07-16

Posts: 305

Re: factorals for non intigers

And I thing it will be better if you start exploring the Gamma function with some more user-friendly text.

Which someone here could volunteer to write ... ?

Yes, that would be nice.

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There are 10 types of people in the world, those who understand binary, those who don't, and those who can use induction.