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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,544

Can somebody explain contour integration to me a little bit?

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 87,256

Do you want to see it work or you want a theoretical discussion?

**In mathematics, you don't understand things. You just get used to them.Of course that result can be rigorously obtained, but who cares?Combinatorics is Algebra and Algebra is Combinatorics.**

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
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An example would be nice.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
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Please integrate that.

**In mathematics, you don't understand things. You just get used to them.Of course that result can be rigorously obtained, but who cares?Combinatorics is Algebra and Algebra is Combinatorics.**

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**ShivamS****Member**- Registered: 2011-02-07
- Posts: 3,557

Stefy, check this out:

http://walet.phy.umist.ac.uk/MaMe/MMA/Contour.pdf

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
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bobbym wrote:

Please integrate that.

How do I do that?

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Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and Punishment

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
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Get the poles first.

**In mathematics, you don't understand things. You just get used to them.Of course that result can be rigorously obtained, but who cares?Combinatorics is Algebra and Algebra is Combinatorics.**

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
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Okay, that is the easy part.

*Last edited by anonimnystefy (2013-03-27 03:45:16)*

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
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Take the positive 2, do you know why?

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,544

No...

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
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The limits of integration are positive so we only take the positive poles.

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
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Since when can complex numbers be positive and negative? Do you want their real parts to be positive or...?

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
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I get the poles like this

You take the ones that do not have a minus sign in front. See the drawing provided later to tell which poles to use.

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
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I think those are the 1. and 3. in my list.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
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Now get the residue of those two poles. This is done in a couple of ways. I prefer the formula. If you like your own way use it.

If you chose the right ones (the ones in red ) you will get:

Once you have the residues you are almost done. Tell me when you get mine.

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 6,426

hi bobbym,

Would you mind explaining how to get the 'residues' ? Thanks.

Bob

You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
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Hi;

One way uses the Laurent series. I do not know it offhand. The other way zetafunc and I were using just a couple of days ago. It is just a formula. Hold on while I get it.

As usual I did not write it down but Wiki has it:

where c is a pole and n is its order.

Ex:

has 2 poles i and -i. To get the residue of i we say n =1 and c = i.

The whole formula simplifies to

which equals - i / 2

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 6,426

Many thanks.

Give me a week or so to get my brain around all of this and I may have more questions.

Bob

You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 87,256

Hi;

I added some stuff to post #17.

Give me a week or so to get my brain around all of this

If you can get it in a week then you will have far surpassed me. I never did get it, despite having it explained to me at least 5 times!

Rule 1 of my signature applies! To some, it only applies maybe 3 or 4 times. For me I stopped counting after 10000.

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,544

Hi bobbym

Yes, I am getting those residues. What now?

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**zetafunc.****Guest**

Sum the residues, and multiply the sum by 2iπ.

**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 87,256

Hi zeta;

I was going to ask you to come in here and help out. We were just working on this!

Hi anonimnystefy;

Normally 2 π i but here we are only taking half the contour so it is π i

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,544

pi*sqrt(2)/4?

*Last edited by anonimnystefy (2013-03-27 23:40:39)*

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
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Correct! Wunderbar!

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,544

Yes, I have. Did you see post #23?

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