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#1 2013-03-27 02:54:04

anonimnystefy
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Registered: 2011-05-23
Posts: 15,522

Contour integration

Can somebody explain contour integration to me a little bit? smile


“Here lies the reader who will never open this book. He is forever dead.

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

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#2 2013-03-27 02:57:40

bobbym
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From: Bumpkinland
Registered: 2009-04-12
Posts: 86,371

Re: Contour integration

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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#3 2013-03-27 03:09:02

anonimnystefy
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Posts: 15,522

Re: Contour integration

An example would be nice.


“Here lies the reader who will never open this book. He is forever dead.

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

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#4 2013-03-27 03:19:17

bobbym
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From: Bumpkinland
Registered: 2009-04-12
Posts: 86,371

Re: Contour integration

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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#5 2013-03-27 03:21:54

ShivamS
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Registered: 2011-02-07
Posts: 3,531

Re: Contour integration

Stefy, check this out:
http://walet.phy.umist.ac.uk/MaMe/MMA/Contour.pdf

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#6 2013-03-27 03:24:54

anonimnystefy
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Posts: 15,522

Re: Contour integration

bobbym wrote:

Please integrate that.

How do I do that?


“Here lies the reader who will never open this book. He is forever dead.

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

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#7 2013-03-27 03:26:21

bobbym
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From: Bumpkinland
Registered: 2009-04-12
Posts: 86,371

Re: Contour integration

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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#8 2013-03-27 03:44:33

anonimnystefy
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Registered: 2011-05-23
Posts: 15,522

Re: Contour integration

Okay, that is the easy part.

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


“Here lies the reader who will never open this book. He is forever dead.

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

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#9 2013-03-27 03:46:47

bobbym
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From: Bumpkinland
Registered: 2009-04-12
Posts: 86,371

Re: Contour integration

Take the positive 2, do you know why?


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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#10 2013-03-27 03:56:22

anonimnystefy
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Registered: 2011-05-23
Posts: 15,522

Re: Contour integration

No...


“Here lies the reader who will never open this book. He is forever dead.

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

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#11 2013-03-27 03:57:39

bobbym
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From: Bumpkinland
Registered: 2009-04-12
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Re: Contour integration

The limits of integration are positive so we only take the positive poles.


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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#12 2013-03-27 04:05:38

anonimnystefy
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Registered: 2011-05-23
Posts: 15,522

Re: Contour integration

Since when can complex numbers be positive and negative? Do you want their real parts to be positive or...?


“Here lies the reader who will never open this book. He is forever dead.

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

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#13 2013-03-27 04:11:47

bobbym
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From: Bumpkinland
Registered: 2009-04-12
Posts: 86,371

Re: Contour integration

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.


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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#14 2013-03-27 13:19:37

anonimnystefy
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Posts: 15,522

Re: Contour integration

I think those are the 1. and 3. in my list.


“Here lies the reader who will never open this book. He is forever dead.

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

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#15 2013-03-27 22:40:32

bobbym
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From: Bumpkinland
Registered: 2009-04-12
Posts: 86,371

Re: Contour integration

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.

View Image: 2013-03-28_033058.gif

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.

Online

#16 2013-03-27 22:50:43

bob bundy
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Registered: 2010-06-20
Posts: 6,380

Re: Contour integration

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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#17 2013-03-27 22:57:19

bobbym
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From: Bumpkinland
Registered: 2009-04-12
Posts: 86,371

Re: Contour integration

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


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.

Online

#18 2013-03-27 23:16:04

bob bundy
Moderator
Registered: 2010-06-20
Posts: 6,380

Re: Contour integration

Many thanks. 

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

Bob


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

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#19 2013-03-27 23:17:33

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 86,371

Re: Contour integration

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.


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.

Online

#20 2013-03-27 23:33:21

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 15,522

Re: Contour integration

Hi bobbym

Yes, I am getting those residues. What now?


“Here lies the reader who will never open this book. He is forever dead.

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

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#21 2013-03-27 23:35:16

zetafunc.
Guest

Re: Contour integration

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

#22 2013-03-27 23:39:13

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 86,371

Re: Contour integration

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


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.

Online

#23 2013-03-27 23:40:08

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 15,522

Re: Contour integration

pi*sqrt(2)/4?

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


“Here lies the reader who will never open this book. He is forever dead.

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

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#24 2013-03-27 23:41:26

bobbym
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From: Bumpkinland
Registered: 2009-04-12
Posts: 86,371

Re: Contour integration

Correct! Wunderbar!


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.

Online

#25 2013-03-27 23:43:40

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 15,522

Re: Contour integration

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


“Here lies the reader who will never open this book. He is forever dead.

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

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