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May 2nd, 2012, 12:39 PM   #1
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contour integral with sin and e

I am somewhat stuck on this one. Zardoz?.



I rewrote as

The poles are at , but the only one inside the contour is

It was suggested to use a rectangular contour with vertices

with an indent around .

Unless I made an error, the residue is

What would be a good approach for this one?.

It should work out to .

I managed to evaluate this by obtaining a series

Then, using . This has poles at and the residues were easily found and summed up to arrive at the required result.

The contour method with the rectangular contour is what has me a little befuddled.
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May 2nd, 2012, 02:01 PM   #2
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Re: contour integral with sin and e

[color=#000000]I will try it some other time, but there is a theorem called fractional residue theorem which states the following


.[/color]
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May 6th, 2012, 05:52 AM   #3
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Re: contour integral with sin and e

If anyone is interested, I managed to find a solution to [color=red]***[/color].

Which can be written as

which is similar to the aforementioned integral and can be done in a similar fashion avoiding the poles created by .

It is rather involved and uses a contour that has a line segment AND quarter circle. The quarter circles are around the poles we're trying to avoid at 4 and 6. These are poles at 0 and i.



starting with the line segment and going counterclockwise.

As .

For where x varies from R to .



As

For , and y varies from 0 to 1

Thus, .

and y varies from .



is over the quarter circle from and we get:

.

is over the quarter circle from



Put them together: . Of course, so is omitted.

or

The limit of each improper integral does not exist, but the limit of the sum does.

Take imaginary parts on both sides:

...........[2]

Now,



.............[3]







Plug this into [3] and use [2], and we get:



Evaluate the second integral:





So, we're getting to the home stretch:







WHEW!!!!!

[color=red]***[/color]In Applied Complex Analysis with PDE's by Asnar Akhle
From what I have seen of it, this is a very nice CA text. Good for class or even self-studying.

EDIT: I managed to evaluate by using the same technique as above. Only I used a rectangle with vertices and avoided the pole at by drawing a little semicircle around it.
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May 9th, 2012, 02:05 PM   #4
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Re: contour integral with sin and e

Quote:
Originally Posted by galactus
I am somewhat stuck on this one. Zardoz?.


[color=#000000]The reason why you can't find the answer is because you use the wrong function, try , with this function is a removable singularity. When the problem with the attachments (cannot upload a figure) is resolved I will upload a solution.[/color]
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May 10th, 2012, 05:54 AM   #5
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Re: contour integral with sin and e

Thanks, Z, I would like to see your method. I managed to work it out using an indent around and a rectangle with vertices at . I would like to see your method because it sounds easier. I did not think of that function you mentioned with the removable singularity. Cool.
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May 14th, 2012, 07:53 AM   #6
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Re: contour integral with sin and e

[color=#000000][attachment=0:3mi8ld2o]gal.png[/attachment:3mi8ld2o]

Hi G. I almost forgot it. We will use the above figure and we will integrate the function . Now f is analytic in F and so .



, because using the M-L inequality





adding up .



, because .





Now,



.[/color]
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May 14th, 2012, 08:10 AM   #7
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Re: contour integral with sin and e

Thanks much, Z.

That's fantastic. May I ask what you used to generate that nice graph?. Paint?.
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May 14th, 2012, 09:04 AM   #8
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Re: contour integral with sin and e

[color=#000000]I always use geogebra, which is a free program. I use the version for linux, I think you will have to download the windows version.[/color]
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May 14th, 2012, 09:47 AM   #9
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Re: contour integral with sin and e

Thanks, I'll check it out.
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May 22nd, 2012, 03:38 AM   #10
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Re: contour integral with sin and e

[color=#000000]I received a pm asking me about the last computation of .




and so .[/color]
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