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#9376 2013-03-20 23:56:18

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 89,107

Re: Linear Interpolation FP1 Formula

Hi;

I am not getting that, let me check my notes.

The Laurent series are generated from the Taylor series with a substitution thrown in. Maybe we should do one that has a real pole first.

Last edited by bobbym (2013-03-21 00:09:01)


In mathematics, you don't understand things. You just get used to them.

I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.

Online

#9377 2013-03-21 00:33:34

zetafunc.
Guest

Re: Linear Interpolation FP1 Formula

I looked up a handout, apparently you just take the reciprocal of z^n and sum it...

#9378 2013-03-21 01:19:05

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 89,107

Re: Linear Interpolation FP1 Formula

Do you have the link?


In mathematics, you don't understand things. You just get used to them.

I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.

Online

#9379 2013-03-21 01:23:26

zetafunc.
Guest

Re: Linear Interpolation FP1 Formula

Google "calculating laurent series" and click on the first link, titled "[PDF] 1 What is a Laurent series? 2 Calculating the Laurent series..."

#9380 2013-03-21 01:33:16

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 89,107

Re: Linear Interpolation FP1 Formula

I see your problem there but how is that answer the Laurent series. It has no principal part!

Last edited by bobbym (2013-03-21 01:41:00)


In mathematics, you don't understand things. You just get used to them.

I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.

Online

#9381 2013-03-21 01:34:01

zetafunc.
Guest

Re: Linear Interpolation FP1 Formula

Principle part?

#9382 2013-03-21 01:39:21

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 89,107

Re: Linear Interpolation FP1 Formula

The part of a Laurent series that has the terms involving negative exponents.


In mathematics, you don't understand things. You just get used to them.

I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.

Online

#9383 2013-03-21 01:40:57

zetafunc.
Guest

Re: Linear Interpolation FP1 Formula

According W|A there was a 1/(z - a) term...

#9384 2013-03-21 01:45:57

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 89,107

Re: Linear Interpolation FP1 Formula

On the last page they give,

but it has no negative exponent.

Oh boy! It is staring me right in the face, I am sorry.

Last edited by bobbym (2013-03-21 01:48:10)


In mathematics, you don't understand things. You just get used to them.

I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.

Online

#9385 2013-03-21 03:30:05

zetafunc.
Guest

Re: Linear Interpolation FP1 Formula

Yes, that one has negative powers of z.

Do you know anything about the contours themselves? I've heard of a few methods, like the "keyhole" method, or drawing squares or rectangles instead of curves...

#9386 2013-03-21 03:36:59

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 89,107

Re: Linear Interpolation FP1 Formula

I have some examples of how to do contour integrals but my understanding of the theory is not good.


In mathematics, you don't understand things. You just get used to them.

I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.

Online

#9387 2013-03-21 03:39:31

zetafunc.
Guest

Re: Linear Interpolation FP1 Formula

Which examples do you have?

#9388 2013-03-21 03:43:27

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 89,107

Re: Linear Interpolation FP1 Formula

A couple of hopefully not too difficult ones. They were easier back when I did them, now they might be impossible.


In mathematics, you don't understand things. You just get used to them.

I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.

Online

#9389 2013-03-21 03:48:34

zetafunc.
Guest

Re: Linear Interpolation FP1 Formula

Hopefully it will not look like Greek to me... (in which case, I can just get adriana to help)

#9390 2013-03-21 03:52:34

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 89,107

Re: Linear Interpolation FP1 Formula

If you have one to post go right ahead. Or do you want me to post one?


In mathematics, you don't understand things. You just get used to them.

I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.

Online

#9391 2013-03-21 03:56:30

zetafunc.
Guest

Re: Linear Interpolation FP1 Formula

I thought you were going to post one -- I would like to see one of your examples, if possible, thanks.

#9392 2013-03-21 04:03:08

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 89,107

Re: Linear Interpolation FP1 Formula

How about this one?


In mathematics, you don't understand things. You just get used to them.

I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.

Online

#9393 2013-03-22 00:03:46

zetafunc.
Guest

Re: Linear Interpolation FP1 Formula

Would it be appropriate to use partial fractions with imaginary co-efficients here, to generate the Laurent series?

My chosen contour would be the upper half of the plane (a semi-circle from negative infinity to positive infinity), since it encloses the real integral...

I think I can generate the partial fractions but it gets messy. Maybe the method of residues is not the way to go.

#9394 2013-03-22 00:18:38

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 89,107

Re: Linear Interpolation FP1 Formula

Hi;

Let's see what we can do first with the residues.

The poles are

Since we are going from 0 to ∞ we only take the second and fourth poles. Are we okay up to here?


In mathematics, you don't understand things. You just get used to them.

I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.

Online

#9395 2013-03-22 00:25:35

zetafunc.
Guest

Re: Linear Interpolation FP1 Formula

Using Cauchy's integral formula I got the wrong answer... I said

Let the contour enclose the part of the plane from 0 to infinity, so only sqrt(i) is the pole included in the contour, right?

Then Cauchy's integral formula says:

Letting n = 1, a = sqrt(i) gives me:

evaluated at z = sqrt(i) gives me an answer of pi/2... but the correct answer is

I am stuck.

---

Sorry, just saw your post. Okay that makes sense, we use those two poles since they are in our region of interest (the real integral doesn't exist below the axis, where the other poles are).

#9396 2013-03-22 00:41:49

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 89,107

Re: Linear Interpolation FP1 Formula

Hi;

so only sqrt(i) is the pole included in the contour, right?

Post #9394, there are two poles. The positive ones count.


In mathematics, you don't understand things. You just get used to them.

I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.

Online

#9397 2013-03-22 01:00:52

zetafunc.
Guest

Re: Linear Interpolation FP1 Formula

Okay, so that means we are taking two contours?

I just tried with Cauchy's integral formula and I am getting a division by zero...

#9398 2013-03-22 01:08:41

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 89,107

Re: Linear Interpolation FP1 Formula

Maybe there are conditions for the Cauchy integral that are being violated.


In mathematics, you don't understand things. You just get used to them.

I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.

Online

#9399 2013-03-22 01:19:19

zetafunc.
Guest

Re: Linear Interpolation FP1 Formula

I tried again and no division by zero, but I'm getting

which is obviously wrong...

#9400 2013-03-22 01:23:08

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 89,107

Re: Linear Interpolation FP1 Formula

Maybe we can work this with this formula here for the residues it looks easy.


In mathematics, you don't understand things. You just get used to them.

I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.

Online

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