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May 2nd, 2012, 01:39 PM  #1 
Senior Member Joined: May 2011 Posts: 501 Thanks: 5  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. 
May 2nd, 2012, 03:01 PM  #2 
Math Team Joined: Nov 2010 From: Greece, Thessaloniki Posts: 1,989 Thanks: 133 Math Focus: pre pre pre pre pre pre pre pre pre pre pre pre calculus  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] 
May 6th, 2012, 06:52 AM  #3 
Senior Member Joined: May 2011 Posts: 501 Thanks: 5  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 selfstudying. 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. 
May 9th, 2012, 03:05 PM  #4  
Math Team Joined: Nov 2010 From: Greece, Thessaloniki Posts: 1,989 Thanks: 133 Math Focus: pre pre pre pre pre pre pre pre pre pre pre pre calculus  Re: contour integral with sin and e Quote:
[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]  
May 10th, 2012, 06:54 AM  #5 
Senior Member Joined: May 2011 Posts: 501 Thanks: 5  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.

May 14th, 2012, 08:53 AM  #6 
Math Team Joined: Nov 2010 From: Greece, Thessaloniki Posts: 1,989 Thanks: 133 Math Focus: pre pre pre pre pre pre pre pre pre pre pre pre calculus  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 ML inequality adding up . , because . Now, .[/color] 
May 14th, 2012, 09:10 AM  #7 
Senior Member Joined: May 2011 Posts: 501 Thanks: 5  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?. 
May 14th, 2012, 10:04 AM  #8 
Math Team Joined: Nov 2010 From: Greece, Thessaloniki Posts: 1,989 Thanks: 133 Math Focus: pre pre pre pre pre pre pre pre pre pre pre pre calculus  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] 
May 14th, 2012, 10:47 AM  #9 
Senior Member Joined: May 2011 Posts: 501 Thanks: 5  Re: contour integral with sin and e
Thanks, I'll check it out.

May 22nd, 2012, 04:38 AM  #10 
Math Team Joined: Nov 2010 From: Greece, Thessaloniki Posts: 1,989 Thanks: 133 Math Focus: pre pre pre pre pre pre pre pre pre pre pre pre calculus  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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