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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 104,366

Ah ha, you did not see post #24...

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.** **A number by itself is useful, but it is far more useful to know how accurate or certain that number is.**

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**anonimnystefy****Real Member**- From: Harlan's World
- Registered: 2011-05-23
- Posts: 16,015

You edited it after I posted.

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

The knowledge of some things as a function of age is a delta function.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 104,366

Ditto! Are you clear on the contour integration. Want to do another?

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.** **A number by itself is useful, but it is far more useful to know how accurate or certain that number is.**

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**anonimnystefy****Real Member**- From: Harlan's World
- Registered: 2011-05-23
- Posts: 16,015

Wait, that's it? Are you sure? When I looked at some videos, I found that they split the integral into several integrals along the contour path...

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

The knowledge of some things as a function of age is a delta function.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 104,366

I am not following you? What integral?

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.** **A number by itself is useful, but it is far more useful to know how accurate or certain that number is.**

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**anonimnystefy****Real Member**- From: Harlan's World
- Registered: 2011-05-23
- Posts: 16,015

The contour integral. As far as I know tthe original integral and the contour integral are not the same.

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

The knowledge of some things as a function of age is a delta function.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 104,366

I am sorry, I am still not following. The original integral?

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**anonimnystefy****Real Member**- From: Harlan's World
- Registered: 2011-05-23
- Posts: 16,015

bobbym wrote:

Please integrate that.

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

The knowledge of some things as a function of age is a delta function.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 104,366

Did we not get the right answer?

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**anonimnystefy****Real Member**- From: Harlan's World
- Registered: 2011-05-23
- Posts: 16,015

But, where did we use the contour integration?

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

The knowledge of some things as a function of age is a delta function.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 104,366

We considered the poles that were inside the contour, see the graph of the imaginary points.

Then we used the residues and then the formula. That is how you do a contour integration.

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.**

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**anonimnystefy****Real Member**- From: Harlan's World
- Registered: 2011-05-23
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Ok. And why did we take those two poles and not the other two?

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

The knowledge of some things as a function of age is a delta function.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 104,366

Good question. See rule one of my signature.

It seems that the unit circle represents the interval -∞ to ∞. The top half represents 0 to ∞, we took the poles inside there. This is the best I can understand of it.

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.**

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**anonimnystefy****Real Member**- From: Harlan's World
- Registered: 2011-05-23
- Posts: 16,015

Ok, but, why did I see some examples done with a rectangle path instead of a semicircle?

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

The knowledge of some things as a function of age is a delta function.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 104,366

That I do not know. But this I do know: One thing at a time!

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.**

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**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 7,588

As I understand it if the integral is from -∞ to + ∞ you take all the poles.

In your example you only take the ones in the right side of the Argand diagram because those are the ones enclosed by your contour. As I'm still learning this, what I just said may be rubbish.

As for the shape of the contour, I don't think it matters. It has to enclose the poles, then there's a fiddly bit where the integral simplifies to just the residues at the poles because the rest is zero.

If we get desparate for the reason, I could ask David.

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: 104,366

Hi;

Yes, if you are going from -∞ to + ∞ you would use all the poles.

The only two poles that work are the ones on top of the real axis ( looks like the x axis ) see the drawing. See post #15

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**anonimnystefy****Real Member**- From: Harlan's World
- Registered: 2011-05-23
- Posts: 16,015

Hi

It seems that rhe contour integral can be split up into several normal ones along the contour uaing the parametrization of that contour. One of those integrals will be the one we started with, but with limits from -R to R so then we should take the limit as R goes to infinity. Meanwhile, the other seems to go to zero for a reason unknown to me.

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

The knowledge of some things as a function of age is a delta function.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 104,366

Isn't that just the standard way of evaluating a limit with an infinite endpoint? It is more like a definition than a means to computation.

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.**

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**anonimnystefy****Real Member**- From: Harlan's World
- Registered: 2011-05-23
- Posts: 16,015

Not exactly. I will try posting later what I mean, when I get on my laptop.

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

The knowledge of some things as a function of age is a delta function.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 104,366

Okay, but chances are it will be one of those methods mathematicians love but does not do anything.

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.**

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**anonimnystefy****Real Member**- From: Harlan's World
- Registered: 2011-05-23
- Posts: 16,015

Here is what he did:

And, when we let R go to infinity, the second integral becomes zero and you get the value of

*Last edited by anonimnystefy (2013-03-29 00:28:35)*

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

The knowledge of some things as a function of age is a delta function.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 104,366

Those integrals look difficult the residue way does not require you to do the integral.

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.**

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**anonimnystefy****Real Member**- From: Harlan's World
- Registered: 2011-05-23
- Posts: 16,015

But, are sure it always gives a correct solution?

And, how'd you get the idea that he is calculating the integral the hard way?

*Last edited by anonimnystefy (2013-03-29 13:42:08)*

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

The knowledge of some things as a function of age is a delta function.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 104,366

The residue method? I know it does for rational functions. And how come you are not sleeping at this late hour?

After you get your rest we will do more of them.

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.**

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