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Math Help - Cauchy's Residue Theorem

  1. #1
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    Cauchy's Residue Theorem

    Hey guys, I was hoping you'd be able to help me with a question. I got the right answer but my method of doing it was a little different to the solution to the answer in my textbook. The question was this:


    What I did was use notice that the above integral was the same as:

    I then went onto the complex plane and made a semicircle closed contour with infinite radius as shown below:


    I found the singularities of the function, with two of them being inside the contour (e^(3pi/4)i and e^(pi/4)i) and then applied Cauchy's Residue Theorem:


    Worked out the values of the residues of the two poles inside the contour and came to a final answer of:

    What my maths lecturer did was way more in depth and used a lot of weird things that did not make too much sense to me and are too long to present here. Is there anything wrong with the way I solved that integral? Surely if you have a closed contour with a pole in it you can straight away use Cauchy's Residue Theorem to find the value of the residue at the pole and multiply it by 2(pi)i?
    Last edited by Alexrey; May 12th 2011 at 08:09 AM.
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  2. #2
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    Hi,

    I'm trying to do a similar question http://www.mathhelpforum.com/math-he...ng-180323.html but getting quite confused. Can you explain how you got the singularities and how then you apply the residue theorem. I'm thinking the singularities for my problem are (2a^2)e^3iPi/4 and (2a^2)e^iPi/4 which is like yours, I don't know how to go about wirking out the residue etc after this! Any help advice you could offer would be appreciated!
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  3. #3
    MHF Contributor FernandoRevilla's Avatar
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    Quote Originally Posted by Alexrey View Post
    What my maths lecturer did was way more in depth and used a lot of weird things that did not make too much sense to me and are too long to present here. Is there anything wrong with the way I solved that integral?

    For example, we need to prove that the given integral is convergent, otherwise we are only finding the Cauchy Principal Value.

    Also, we need to prove that

    \lim_{R\to +\infty}\int_{\Gamma}f(z)\;dz=0

    on the semicircle (usually, using a well known lemma of Jordan).
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  4. #4
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    Sorry if this may seem trivial, but surely, since I made a closed contour I can straight away apply Cauchy's Residue Theorem without having to prove anything. Or can I only apply it like that when the integral given is already in it's complex form (i.e. do I only have to do those above-mentioned proofs when I complexify a real integral and solve it in the complex plane)?
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  5. #5
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    Yes you can apply the residue theorem here as you have, however the original integral that you want to evaluate is on the real line, you therefore need to show that the integral around the semi-circular part of your contour becomes zero as R tends to infinity, thus leaving you with just the part on the real line as you require.
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  6. #6
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    Ahhhh I see now! Makes so much more sense, thanks so much! Regarding part of my previous question though, I'm guessing I won't have to do this for an integral along the same closed contour that was originally in its complex form right?
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