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"Filling in" a function

Posted: Wed Jan 05, 2011 7:16 pm UTC
by dhokarena56
It is well known that the gamma function "fills in" the gaps left by the function f(x)=x!.

Consider, then, the function f(x)=∴(x), where ∴(n)= 1+2+3+4+5+6...+(n-2)+(n-1)+n.

What function would fill in the gaps, allowing us to evaluate (say) y=∴(π)?

Re: "Filling in" a function

Posted: Wed Jan 05, 2011 7:21 pm UTC
by ++$_
Well, [imath]\frac{1}{2}x(x+1)[/imath] would be a pretty good candidate.

Re: "Filling in" a function

Posted: Wed Jan 05, 2011 8:48 pm UTC
by firechicago
I remember reading a (likely apocryphal) story about Gauss that used that little fact. Little Carl Friedrich's school teacher wanted to shut the class up for a while, so he told them to add up all the numbers from 1 to 1000. He was very angry when Gauss had the temerity to announce that he was done after only a couple of minutes, and was even angrier when Gauss' answer was right.

Little Gauss had, of course, noticed that 1+2+...+1000 = (1+1000)+(2+999)+...+(500+501) = 500*1001 = 500,500

Re: "Filling in" a function

Posted: Wed Jan 05, 2011 10:23 pm UTC
by Eastwinn
There are plenty of functions that satisfy f(x+1) = f(x) + x + 1. You'll have to give another property that will define a unique function. For example, Gamma is the unique function that satisfies f(x+1)=xf(x), f(1)=1, and log convexity. Those properties together only belong to the Gamma function. You'll need a few more properties and then derive the formula just from those.

Re: "Filling in" a function

Posted: Thu Jan 06, 2011 3:25 am UTC
by Kurushimi
dhokarena56 wrote:It is well known that the gamma function "fills in" the gaps left by the function f(x)=x!.

Consider, then, the function f(x)=∴(x), where ∴(n)= 1+2+3+4+5+6...+(n-2)+(n-1)+n.

What function would fill in the gaps, allowing us to evaluate (say) y=∴(π)?

Wait, am I the only one that's confused here? What is y=∴(π)? I've never seen that before.

Re: "Filling in" a function

Posted: Thu Jan 06, 2011 3:26 am UTC
by ++$_
You haven't seen it before because he's just defined it in that post. By [imath]∴(\ \cdot\ )[/imath] he means the triangle number function, normally defined only on the natural numbers. He wants to know how to interpolate that function.

Re: "Filling in" a function

Posted: Sat Jan 08, 2011 2:05 am UTC
by elitekross
ive heard that story with 100, 50, bunches of numbers so :roll:

Re: "Filling in" a function

Posted: Sun Jan 09, 2011 3:56 pm UTC
by WarDaft
It's not really a terribly surprising story, I realized basically the same thing in grade 3. It was supposed to be a project to take up 30-45 minutes, and I had figured out before the teacher finished 'explaining' it. I didn't think of it as an equation in my head though... I hadn't even been introduced to the concept of solving for variables at that point. I'm pretty sure I'm not as smart as Guass was, so the story is totally believable to me.

Re: "Filling in" a function

Posted: Sun Jan 09, 2011 10:32 pm UTC
by Superisis
Same here. Though I added up all the numbers that became 100 (1+99, 2+98, etc) there where 49 such pairs, plus the 100 plus the 50 = 5050.