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Math Help - Isolated singularities (Complex Analysis)

  1. #1
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    Isolated singularities (Complex Analysis)

    Locate each of the isolated singularities of the given function and tell whether it is a removable singularity, a pole, or an essential singularity. If the singularity is removable, give the vlaue of the function at the point; if the singularity is a pole, give the order of the pole.

    (1) \frac{e^z-1}{z}

    (2) \frac{z^4-2z^2+1}{(z-1)^2}

    (3) \frac{2z+1}{z+2}

    If anyone could show me how to do any of these, I would appreciate it. I don't understand it..Thanks!
    Last edited by shadow_2145; October 9th 2008 at 02:52 PM.
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    (1) If z_0 is a removable singularity, then \lim_{z\to z_0} (z-z_0)f(z)=0 right?

    So \lim_{z\to 0}z\frac{e^z-1}{z}=0.

    (2) If f(z) has a pole of order m then f(z)=\frac{\phi(z)}{(z-z_0)^m} with \phi(z_0)\ne 0. The numerator of the second one is zero at z=1. So it's not as simple as it looks. Need to factor the numerator:

    \frac{z^4-2z^2+1}{(z-1)^2}=\frac{(z+1)^2(z-1)^2}{(z-1)^2}. That just looks like a removable singularity right but we could check that by (1) above:

    \lim_{z\to 1}(z-1)\frac{(z+1)^2(z-1)^2}{(z-1)^2}=0

    The third one has a pole of order 1 at z=-2 by (2).
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  3. #3
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    Quote Originally Posted by shadow_2145 View Post
    Locate each of the isolated singularities of the given function and tell whether it is a removable singularity, a pole, or an essential singularity. If the singularity is removable, give the vlaue of the function at the point; if the singularity is a pole, give the order of the pole.

    (1) \frac{e^z-1}{z}

    Mr F says: Removable singularity at z = 0. (Expand e^z, simplify numerator and note that z is a common factor of numerator and denominator .....) Define f(0) = 1.

    (2) \frac{z^4-2z^2+1}{(z-1)^2}

    Mr F says: Removable singularity at z = 1. (Factorise the numerator .....) Define f(1) = 4.

    (3) \frac{2z+1}{z+2}

    Mr F says: Simple pole at z = -2 because (z + 2) f(z) is differentiable.

    If anyone could show me how to do any of these, I would appreciate it. I don't understand it..Thanks!
    In each case it's a simple matter of applying the basic definitions (both theortical and practical .....)
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    I am reviewing for an exam and I have a problem I don't know how to do.

     \pi cot\pi z

    Any help would be appreciated, thanks!
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    Quote Originally Posted by shadow_2145 View Post
    I am reviewing for an exam and I have a problem I don't know how to do.

     \pi cot\pi z

    Any help would be appreciated, thanks!
    You have not said what you don't know how to do here.

    \frac{\pi \, \cos(\pi z)}{\sin (\pi z)} obviously has isolated singular points at z = n.

    Note that \lim_{z \rightarrow n} \frac{\pi \, \cos(\pi z) \, (z - n)}{\sin (\pi z)} = 1.

    Therefore \frac{\pi \, \cos(\pi z)}{\sin (\pi z)} has simple poles at z = n \, ....

    By the way, new questions should be posted in a new thread.
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