Alternate Definitions of the Gamma Function

For the discussion of math. Duh.

Moderators: gmalivuk, Moderators General, Prelates

User avatar
Eastwinn
Posts: 303
Joined: Thu Jun 19, 2008 12:36 am UTC
Location: Maryland

Alternate Definitions of the Gamma Function

Postby Eastwinn » Tue Nov 23, 2010 6:22 pm UTC

I was reading Gamma by Julian Havil and I got hung up on one of the alternate definitions it gave for the Gamma function. It then used that definition to derive an extension of Gamma that converges for all C sans negative integers.

It says that Euler (hey, BTW, what is the proper way to pronounce his name?) proposed this convoluted definition in 1729 in a letter:
[math]\Gamma _{r}(x)=\frac{r!r^{x}}{x(x+1)(x+2)...(x+r)}[/math]
[math]=\frac{r^{x}}{x(x+\frac{x}{1})(x+\frac{x}{2})...(x+\frac{x}{r})}[/math]
and related it to Gamma in this way:
[math]\Gamma(x) =\lim_{r \to \infty }\Gamma _{r}(x)[/math]

It justified this by showing that [imath]\lim_{r \to \infty }\Gamma _{r}(x)[/imath] satisfied the identity [imath]\lim_{r \to \infty }\Gamma _{r}(x+1) = x\lim_{r \to \infty }\Gamma _{r}(x)[/imath] just like the Gamma function, and then it showed that [imath]\Gamma (1) = \lim_{r \to \infty }\Gamma _{r}(1)[/imath].

This is sufficient proof that the two definitions are equal for natural numbers, but it doesn't prove to me that they are the same for all x > 0. Is anyone familiar with this definition? Could anyone point me in the direction of a proof or offer insight so I could prove it myself? Thanks :D
http://aselliedraws.tumblr.com/ - surreal sketches and characters.

User avatar
jestingrabbit
Factoids are just Datas that haven't grown up yet
Posts: 5967
Joined: Tue Nov 28, 2006 9:50 pm UTC
Location: Sydney

Re: Alternate Definitions of the Gamma Function

Postby jestingrabbit » Tue Nov 23, 2010 8:30 pm UTC

Does your text anywhere mention that Gamma is the unique analytic function which satisfies the recurrence formula and [imath]\Gamma(1)=1[/imath]? Its true, even if it doesn't prove it, though I suspect it does.
ameretrifle wrote:Magic space feudalism is therefore a viable idea.

User avatar
Blatm
Posts: 638
Joined: Mon Jun 04, 2007 1:43 am UTC

Re: Alternate Definitions of the Gamma Function

Postby Blatm » Tue Nov 23, 2010 9:09 pm UTC

Eastwinn wrote:Euler (hey, BTW, what is the proper way to pronounce his name?)


Oil-er

User avatar
Eastwinn
Posts: 303
Joined: Thu Jun 19, 2008 12:36 am UTC
Location: Maryland

Re: Alternate Definitions of the Gamma Function

Postby Eastwinn » Tue Nov 23, 2010 11:24 pm UTC

^ I've been pronouncing it wrong then :oops:

jestingrabbit wrote:Does your text anywhere mention that Gamma is the unique analytic function which satisfies the recurrence formula and [imath]\Gamma(1)=1[/imath]? Its true, even if it doesn't prove it, though I suspect it does.


Thank for the response. Yes, it does. It names a theorem and mentions that it's log convex. I'll look into that. If I could show that the limit is log convex then all matters would be settled.

I found a paper that proves it using the fact that Gamma is log convex and increasing for input greater than zero but I didn't quite understand it because I'm not too hip on this log convex stuff (yet).
http://aselliedraws.tumblr.com/ - surreal sketches and characters.

skullturf
Posts: 556
Joined: Thu Dec 07, 2006 8:37 pm UTC
Location: Chicago
Contact:

Re: Alternate Definitions of the Gamma Function

Postby skullturf » Wed Nov 24, 2010 12:57 am UTC

jestingrabbit wrote:Does your text anywhere mention that Gamma is the unique analytic function which satisfies the recurrence formula and [imath]\Gamma(1)=1[/imath]? Its true, even if it doesn't prove it, though I suspect it does.


Strictly speaking, is this correct as written, if you don't also say something about log-convexity? Couldn't you add an analytic function that's zero at the positive integers?

Tyl.
Posts: 32
Joined: Thu Sep 11, 2008 9:23 pm UTC

Re: Alternate Definitions of the Gamma Function

Postby Tyl. » Wed Nov 24, 2010 1:05 am UTC

Blatm wrote:
Eastwinn wrote:Euler (hey, BTW, what is the proper way to pronounce his name?)


Oil-er

Actually, I would say it's 'Oi-ler'.

User avatar
Blatm
Posts: 638
Joined: Mon Jun 04, 2007 1:43 am UTC

Re: Alternate Definitions of the Gamma Function

Postby Blatm » Wed Nov 24, 2010 1:14 am UTC

Tyl. wrote:
Blatm wrote:
Eastwinn wrote:Euler (hey, BTW, what is the proper way to pronounce his name?)


Oil-er

Actually, I would say it's 'Oi-ler'.


You're right, that's probably a bit closer.

mike-l
Posts: 2758
Joined: Tue Sep 04, 2007 2:16 am UTC

Re: Alternate Definitions of the Gamma Function

Postby mike-l » Wed Nov 24, 2010 1:15 am UTC

skullturf wrote:
jestingrabbit wrote:Does your text anywhere mention that Gamma is the unique analytic function which satisfies the recurrence formula and [imath]\Gamma(1)=1[/imath]? Its true, even if it doesn't prove it, though I suspect it does.


Strictly speaking, is this correct as written, if you don't also say something about log-convexity? Couldn't you add an analytic function that's zero at the positive integers?


I don't know whether it's true or not, but the theorem that says the gamma function is the unique function that's log-convex that satisfies the recurrence doesn't require analyticity (analyticness?), and the gamma function certainly is analytic, so that may be an equivalent classification.

The recurrence is true for all x, so you can't just add an arbitrary analytic function that's zero on the positive integers, so the statement is actually equivalent to saying that if an analytic function satisfies the recurrence and is 0 on the positive integers (or equivalently at 1, by the recurrence) then it is 0. Is this true? (JR suggests it is)
addams wrote:This forum has some very well educated people typing away in loops with Sourmilk. He is a lucky Sourmilk.

User avatar
Eastwinn
Posts: 303
Joined: Thu Jun 19, 2008 12:36 am UTC
Location: Maryland

Re: Alternate Definitions of the Gamma Function

Postby Eastwinn » Wed Nov 24, 2010 1:25 am UTC

The Bohr-Mollerup Theorem says that there is only one function that satisfies f(x+1) = x f(x), f(1) = 1, and log-convexity, and that function is Gamma. I'm yet to understand the proof, however, but still if I can show that the alternative definition in the OP is log-convex then that will be sufficient proof.

Edit: It was simple to find that

[math]\frac{\mathrm{d}^{2} ln\Gamma_{r} (x)}{\mathrm{d}^{2} x} = \sum_{k=0}^{r}\frac{1}{(x+k)^{2}}[/math]

which is positive when x is positive, so that settles the issue -- the alternate definition in my original post is equivalent to the integral definition where convergence allows.

Here's something interesting though. Take the limit as r->infinity...

[math]\frac{\mathrm{d}^{2} ln\Gamma (x)}{\mathrm{d}^{2} x} = \sum_{k=0}^{\infty }\frac{1}{(x+k)^{2}}[/math]

And plug in 1 for x...

[math]\frac{\mathrm{d}^{2} ln\Gamma (1)}{\mathrm{d}^{2} x} = \sum_{k=0}^{\infty }\frac{1}{(1+k)^{2}} = \sum_{k=1}^{\infty }\frac{1}{k^{2}} = \frac{\pi ^{2}}{6}[/math]

Cool!
http://aselliedraws.tumblr.com/ - surreal sketches and characters.

User avatar
jestingrabbit
Factoids are just Datas that haven't grown up yet
Posts: 5967
Joined: Tue Nov 28, 2006 9:50 pm UTC
Location: Sydney

Re: Alternate Definitions of the Gamma Function

Postby jestingrabbit » Wed Nov 24, 2010 7:18 am UTC

skullturf wrote:
jestingrabbit wrote:Does your text anywhere mention that Gamma is the unique analytic function which satisfies the recurrence formula and [imath]\Gamma(1)=1[/imath]? Its true, even if it doesn't prove it, though I suspect it does.


Strictly speaking, is this correct as written, if you don't also say something about log-convexity? Couldn't you add an analytic function that's zero at the positive integers?


You're right, you do need the convexity condition, or something like it. Something as weak as being defined for all positive reals might be enough, but you do need more than just analytic, the recurrence and the value. My mistake.

Edit: after thinking about this for a while, I think that you need something as strong as the log convexity, and nothing weaker will get you close. if you have some periodic function that's 1 on the integers, like [imath]f(z) = \exp(\sin(\pi z))[/imath], then [imath]g(z)= \Gamma(z)f(z)[/imath] will still be analytic everywhere that Gamma is and in this case zero nowhere just like Gamma. You need the log convexity, for sure.
ameretrifle wrote:Magic space feudalism is therefore a viable idea.


Return to “Mathematics”

Who is online

Users browsing this forum: No registered users and 17 guests