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Thread: Please help with 3 complex analysis questions. Need help asap

  1. #1
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    Please help with 3 complex analysis questions. Need help asap

    Hi I have attached 3 questions which I am very stuck with. I have looked up the definitions and such for Q1 and dont know where else to start with any of them. Thanks for looking!


    Edit: Have successfully Solved Q1 with the kind help of shawsend.

    Q2 here I come
    Attached Thumbnails Attached Thumbnails Please help with 3 complex analysis questions. Need help asap-ca.gif  
    Last edited by Niall101; Sep 29th 2009 at 01:17 PM.
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  2. #2
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    For 1:

    You need to look at these in terms of limits and Laurent series and focus on just the term that has the singular component. For example, at z=0, the exp term is a constant so when looking at the function at z=0, you could just as well consider $\displaystyle \frac{z^2}{\sin^2(z)}k$.

    If the pole is removable, then $\displaystyle \lim_{z\to z_0} f(z)$ exists. So what is $\displaystyle \lim_{z\to 0}\frac{z^2}{\sin^2(z)} $?

    If it has a pole of order one, then $\displaystyle \lim_{z\to z_0} (z-z_0) f(z)$ exists and is not zero. And if it has a pole of order k, then $\displaystyle \lim_{z\to z_0} (z-z_0)^k f(z)$ exists and is not zero.

    Now, what about all the pi's? They are all the same in terms of singular order right so just consider the one at $\displaystyle \pi$. What is:

    $\displaystyle \lim_{z\to\pi} (z-\pi)^k \frac{z^2}{\sin^2(z)}$ for k=1, k=2, . . . until it exists and is not zero?

    Well consider:

    $\displaystyle \lim_{z\to\pi} (z-\pi) \frac{z^2}{\sin^2(z)}$

    The $\displaystyle z^2$ term just goes to $\displaystyle \pi^2$ so can consider:

    $\displaystyle \lim_{z\to\pi} (z-\pi) \frac{1}{\sin^2(z)}$

    That's infinity right?

    How about then:

    $\displaystyle \lim_{z\to\pi} (z-\pi)^2 \frac{1}{\sin^2(z)}$

    Do a double L'Hospital on it and get one. So it's a pole of order 2.

    As far as the $\displaystyle \pm i$, well, $\displaystyle e^{f(z)}$ has an essential singularity anywhere $\displaystyle f(z)$ is singular. You can prove this by writing it's Laurent series.
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  3. #3
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    Thank you very much! I had a couple of steps correct so not as bad as I thought. I am going to work more on this now. Thanks again!!
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  4. #4
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    Hello Again. I have worked out all of question 1! Woo thanks again for your help and not just giving me the answer as I have learned a lot working out the rest!
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  5. #5
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    Thanks solved the rest. Used residue Thm on Q 3.
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