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#101 2013-03-30 14:49:55

anonimnystefy
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From: The Foundation
Registered: 2011-05-23
Posts: 14,812

Re: Contour integration

I think the problem is that you are treating a regular integral as a contour integral.


“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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#102 2013-03-30 14:53:03

bobbym
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From: Bumpkinland
Registered: 2009-04-12
Posts: 81,445

Re: Contour integration

In order to evaluate real integrals, the residue theorem is used in the following manner: the integrand is extended to the complex plane and its residues are computed (which is usually easy), and a part of the real axis is extended to a closed curve by attaching a half-circle in the upper or lower half-plane, forming a semicircle. The integral over this curve can then be computed using the residue theorem. Often, the half-circle part of the integral will tend towards zero as the radius of the half-circle grows, leaving only the real-axis part of the integral, the one we were originally interested in.

It is not a problem, looks like you convert it into a contour integral or something like that. Anyway the method can be used for real integrals.

Do you think we have done 5 of these and got the right answer by accident? Sure we are lacking in rigor in the approach but it does work for real integrals of that type.


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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#103 2013-03-30 15:01:18

anonimnystefy
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From: The Foundation
Registered: 2011-05-23
Posts: 14,812

Re: Contour integration

Do you see the part after "Often". That's where your problem is. It will not always tend to zero. Also, there are some things called branch points, which I am trying to figure out, which cannot be handled regularly, So a different path must be chosen.


“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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#104 2013-03-30 15:04:05

bobbym
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From: Bumpkinland
Registered: 2009-04-12
Posts: 81,445

Re: Contour integration

You are missing the point. This will obviously not do every integral. No method does. But often is good enough. The whole integral is reduced to a line on the complex plane. That page I sent you uses the same method  we are using to do an integral. We are lacking the knowledge of when this can applied.

I think you will find that it depends on the principal part of the Laurent series of the integrand.

http://mathworld.wolfram.com/ContourIntegration.html

Look at equation 12. I remember saying for rational functions only. You will also see it is a definite integral. The method can be extended to some other forms, see (14),(15) and (16).


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.

Online

#105 2013-03-30 15:42:11

anonimnystefy
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From: The Foundation
Registered: 2011-05-23
Posts: 14,812

Re: Contour integration

It seems to me the only condition is that the function is holomorphic.


“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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#106 2013-03-30 17:26:07

bobbym
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From: Bumpkinland
Registered: 2009-04-12
Posts: 81,445

Re: Contour integration

(14)(15) and (16) show that it is only for those forms.


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.

Online

#107 2013-03-30 18:38:37

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 14,812

Re: Contour integration

It nowhere says that it is for those forms only.


“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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#108 2013-03-30 20:29:53

bobbym
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From: Bumpkinland
Registered: 2009-04-12
Posts: 81,445

Re: Contour integration

it only mentions 3 forms, (14)(15) and (16) have the exact method I am using. Your integral is not of that form, so other methods have to be used.


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.

Online

#109 2013-03-31 03:38:42

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 14,812

Re: Contour integration

Well, contour integration works on that one.


“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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#110 2013-03-31 10:08:29

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 81,445

Re: Contour integration

Yes, it does but not using residues. There are a couple of ways to do a contour integration.


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.

Online

#111 2013-03-31 10:19:07

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 14,812

Re: Contour integration

Where'd you get that idea?


“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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#112 2013-03-31 10:20:26

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 81,445

Re: Contour integration

From a little pdf I downloaded.


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.

Online

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