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Math Help - Series Representation

  1. #1
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    Series Representation

    Hi, I was wondering if anyone might know how to solve this problem that I've been working on.

    Find the Laurent Series for f centered at z0 = 0. Also, determine the punctured disk of maximum radius on which the series converges.

    f(z) = z^3 cosh(1/z^2)

    Thanks
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  2. #2
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    Quote Originally Posted by villa223 View Post
    Hi, I was wondering if anyone might know how to solve this problem that I've been working on.

    Find the Laurent Series for f centered at z0 = 0. Also, determine the punctured disk of maximum radius on which the series converges.

    f(z) = z^3 cosh(1/z^2)

    Thanks

    \cosh\left(\frac{1}{z^2}\right)=\frac{e^{1/ z^2}+e^{-1/z^2}}{2}= \frac{1}{2}\left[ \sum^\infty_{n=0}\frac{1}{n!z^{2n}}+\sum^\infty_{n  =0}\frac{(-1)^n}{n!z^{2n}}\right] =\sum^\infty_{n=0}\frac{1}{(2n)!z^{4n}} =1+\frac{1}{2!z^4}+\frac{1}{4!z^8}+\frac{1}{6!z^{1  2}}+\ldots\Longrightarrow

    z^3\cosh\left(\frac{1}{z^2}\right)=z^3+\frac{1}{2!  z}+\frac{1}{4!z^5}+\ldots =\sum^\infty_{n=0}\frac{1}{(2n)!z^{4n-3}}

    Tonio
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