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Math Help - 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; September 29th 2009 at 02:17 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 \frac{z^2}{\sin^2(z)}k.

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

    If it has a pole of order one, then \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 \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 \pi. What is:

    \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:

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

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

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

    That's infinity right?

    How about then:

    \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 \pm i, well, e^{f(z)} has an essential singularity anywhere 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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