## "Filling in" a function

For the discussion of math. Duh.

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dhokarena56
Posts: 179
Joined: Fri Mar 27, 2009 11:52 pm UTC

### "Filling in" a function

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)?

++$_ Mo' Money Posts: 2370 Joined: Thu Nov 01, 2007 4:06 am UTC ### Re: "Filling in" a function Well, [imath]\frac{1}{2}x(x+1)[/imath] would be a pretty good candidate. firechicago Posts: 621 Joined: Mon Jan 11, 2010 12:27 pm UTC Location: One time, I put a snowglobe in the microwave and pushed "Hot Dog" ### Re: "Filling in" a function I remember reading a (likely apocryphal) story about Gauss that used that little fact. Little Carl Friedrich's school teacher wanted to shut the class up for a while, so he told them to add up all the numbers from 1 to 1000. He was very angry when Gauss had the temerity to announce that he was done after only a couple of minutes, and was even angrier when Gauss' answer was right. Little Gauss had, of course, noticed that 1+2+...+1000 = (1+1000)+(2+999)+...+(500+501) = 500*1001 = 500,500 Eastwinn Posts: 303 Joined: Thu Jun 19, 2008 12:36 am UTC Location: Maryland ### Re: "Filling in" a function There are plenty of functions that satisfy f(x+1) = f(x) + x + 1. You'll have to give another property that will define a unique function. For example, Gamma is the unique function that satisfies f(x+1)=xf(x), f(1)=1, and log convexity. Those properties together only belong to the Gamma function. You'll need a few more properties and then derive the formula just from those. http://aselliedraws.tumblr.com/ - surreal sketches and characters. Kurushimi Posts: 841 Joined: Thu Oct 02, 2008 12:06 am UTC ### Re: "Filling in" a function dhokarena56 wrote:It is well known that the gamma function "fills in" the gaps left by the function f(x)=x!. Consider, then, the function f(x)=∴(x), where ∴(n)= 1+2+3+4+5+6...+(n-2)+(n-1)+n. What function would fill in the gaps, allowing us to evaluate (say) y=∴(π)? Wait, am I the only one that's confused here? What is y=∴(π)? I've never seen that before. ++$_
Mo' Money
Posts: 2370
Joined: Thu Nov 01, 2007 4:06 am UTC

### Re: "Filling in" a function

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.

elitekross
Posts: 10
Joined: Tue Nov 10, 2009 7:39 pm UTC

### Re: "Filling in" a function

ive heard that story with 100, 50, bunches of numbers so

WarDaft
Posts: 1583
Joined: Thu Jul 30, 2009 3:16 pm UTC

### Re: "Filling in" a function

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.
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Superisis
Posts: 48
Joined: Fri Sep 17, 2010 8:48 am UTC

### Re: "Filling in" a function

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.