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#9401 2013-03-22 01:24:51

zetafunc.
Guest

Re: Linear Interpolation FP1 Formula

But it looks so tedious to find the residues...

#9402 2013-03-22 01:33:02

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 81,394

Re: Linear Interpolation FP1 Formula

I am getting:


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

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#9403 2013-03-22 01:37:18

zetafunc.
Guest

Re: Linear Interpolation FP1 Formula

That is correct... did you use the residues?

#9404 2013-03-22 01:38:06

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 81,394

Re: Linear Interpolation FP1 Formula

Yes, there is a simple formula for computing them when the poles are simple.


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

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#9405 2013-03-22 01:38:34

zetafunc.
Guest

Re: Linear Interpolation FP1 Formula

What formula?

Also, what is a simple pole?

#9406 2013-03-22 01:40:12

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 81,394

Re: Linear Interpolation FP1 Formula

I am the worst at definitions so please check this. I think a simple pole is a pole with multiplicity of one.


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

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#9407 2013-03-22 01:43:52

zetafunc.
Guest

Re: Linear Interpolation FP1 Formula

I'm not sure what you mean by the pole having a multiplicity of one...

#9408 2013-03-22 01:45:21

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 81,394

Re: Linear Interpolation FP1 Formula

Multiplicity of one means there is only one pole like that. Of two, would mean two poles the same.


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

Offline

#9409 2013-03-22 01:49:51

zetafunc.
Guest

Re: Linear Interpolation FP1 Formula

Yes but I don't understand how two poles can be the same... how do you mean?

#9410 2013-03-22 01:51:32

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 81,394

Re: Linear Interpolation FP1 Formula

If the roots of the denominator are all different then the poles are all simple ones because they are of order 1.

Since all 4 roots are distinct they are all simple.

If the roots were 2,2,2,2 then 2 would be a pole of order 4 and not simple.


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

Offline

#9411 2013-03-22 01:53:25

zetafunc.
Guest

Re: Linear Interpolation FP1 Formula

Ohh I see, so it is sort of like having repeated roots of a polynomial?

#9412 2013-03-22 01:55:06

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 81,394

Re: Linear Interpolation FP1 Formula

Yes, something like that. It may be more complicated and I do not remember. But for our purposes that integral has 4 simple poles. We are interested in only two. The formula becomes very, very easy when the pole is simple.

Last edited by bobbym (2013-03-22 01:55:32)


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

Offline

#9413 2013-03-22 02:03:08

zetafunc.
Guest

Re: Linear Interpolation FP1 Formula

Are you using this formula?

I cannot get the formula to display properly, it is the one with the limit of z -> z[sub]0[/sub], and the derivative and the factorial.

#9414 2013-03-22 02:04:05

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 81,394

Re: Linear Interpolation FP1 Formula

Hi;

I can not see your latex. I could show you what I am doing, it does look like that, but I am not sure.

Last edited by bobbym (2013-03-22 02:05:24)


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

Offline

#9415 2013-03-22 02:13:05

zetafunc.
Guest

Re: Linear Interpolation FP1 Formula

Is there some sort of bug with the latex today, I have tried to type 3 formulae and none of them worked apart from a few...?

I'll take out the limit, maybe this works:

Let z[sub]0[/sub] be a pole of f, of order n. Then,

#9416 2013-03-22 02:21:51

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 81,394

Re: Linear Interpolation FP1 Formula

Hi;

n is the order of the pole. a is the pole.

Add that for each pole and then times by 2πi


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

Offline

#9417 2013-03-22 02:27:01

zetafunc.
Guest

Re: Linear Interpolation FP1 Formula

Yes, that is the formula I am using. But how do you use it?

For instance, consider

How would you then use your formula to find the residue?

#9418 2013-03-22 02:28:05

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 81,394

Re: Linear Interpolation FP1 Formula

Let's continue with the other one first. I will post the answer step by step.


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

Offline

#9419 2013-03-22 02:31:12

zetafunc.
Guest

Re: Linear Interpolation FP1 Formula

Okay, thank you.

#9420 2013-03-22 02:43:37

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 81,394

Re: Linear Interpolation FP1 Formula

We need two:


n = 1


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

Offline

#9421 2013-03-22 02:54:58

zetafunc.
Guest

Re: Linear Interpolation FP1 Formula

How did you evaluate that last line?

#9422 2013-03-22 03:01:09

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 81,394

Re: Linear Interpolation FP1 Formula

The limits? L'Hopitals rule and Alpha for the plugging in.


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

Offline

#9423 2013-03-22 03:19:04

zetafunc.
Guest

Re: Linear Interpolation FP1 Formula

I am not getting your answer... I am ending up with

#9424 2013-03-22 03:30:25

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 81,394

Re: Linear Interpolation FP1 Formula

What did you get for the first limit?


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

Offline

#9425 2013-03-22 03:50:15

zetafunc.
Guest

Re: Linear Interpolation FP1 Formula

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