"Filling in" a function

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dhokarena56
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"Filling in" a function

Postby dhokarena56 » Wed Jan 05, 2011 7:16 pm UTC

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)?
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Mo' Money
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Re: "Filling in" a function

Postby ++$_ » Wed Jan 05, 2011 7:21 pm UTC

Well, [imath]\frac{1}{2}x(x+1)[/imath] would be a pretty good candidate.

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Re: "Filling in" a function

Postby firechicago » Wed Jan 05, 2011 8:48 pm UTC

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

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Eastwinn
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Re: "Filling in" a function

Postby Eastwinn » Wed Jan 05, 2011 10:23 pm UTC

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.
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Kurushimi
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Re: "Filling in" a function

Postby Kurushimi » Thu Jan 06, 2011 3:25 am UTC

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.

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Mo' Money
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Re: "Filling in" a function

Postby ++$_ » Thu Jan 06, 2011 3:26 am UTC

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
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Re: "Filling in" a function

Postby elitekross » Sat Jan 08, 2011 2:05 am UTC

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

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WarDaft
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Re: "Filling in" a function

Postby WarDaft » Sun Jan 09, 2011 3:56 pm UTC

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
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Re: "Filling in" a function

Postby Superisis » Sun Jan 09, 2011 10:32 pm UTC

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.


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