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Math Help - Laurent series expansion

  1. #1
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    Laurent series expansion

    Give the Laurent series expansion for the function f(z) centered at z_0. State the possible domains.

    f(z) = e^z\sin z z_0=0


    Here, I see I will get two series multiplied together. I do not see how to get one Laurent series expansion for this one though. Any hints would be nice. Thanks in advance.
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  2. #2
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    Quote Originally Posted by pascal4542 View Post
    Give the Laurent series expansion for the function f(z) centered at z_0. State the possible domains.

    f(z) = e^z\sin z z_0=0


    Here, I see I will get two series multiplied together. I do not see how to get one Laurent series expansion for this one though. Any hints would be nice. Thanks in advance.
    What you will get is a Maclaurin series.

    Perhaps the question meant f(z) = \frac{e^z}{\sin z} z_0=0 ....
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  3. #3
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    Quote Originally Posted by mr fantastic View Post
    What you will get is a Maclaurin series.

    Perhaps the question meant f(z) = \frac{e^z}{\sin z} z_0=0 ....

    Maybe the question I have was what I typed before. I will have to ask for sure though. So, I will assume for now that it is the fraction so that I will get the Laurent series it is asking for.
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  4. #4
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    I was wondering if someone can check this.

    From what I understand this is the Taylor series expansion. However, the problem is asking for the Laurent series and the domain. So far I have


    e^z \sin(z) =

    (1+z+z^2/2+z^3/6+z^4/24+\cdots)(z-z^3/6+z^5/120-z^7/5040+\cdots)
    <br />
=z+z^2+z^5/120-z^7/5040+\cdots. However, this is not in series notation and I do not know the domain either. I am stuck here. I need some direction on how to get the Laurent series. Thank you.
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