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Math Help - coefficients of a laurent series

  1. #1
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    coefficients of a laurent series

    A function f(z) does have a laurent series about the origin, with real a_n coefficients. Show that \bar{f}(z) = f(\bar{z})

    Regards
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  2. #2
    Behold, the power of SARDINES!
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    Quote Originally Posted by hurz View Post
    A function f(z) does have a laurent series about the origin, with real a_n coefficients. Show that \bar{f}(z) = f(\bar{z})

    Regards
    So that means that f can be written as

    f(z)=\sum_{n=-\infty}^{\infty}a_n z^n,a_n \in \mathbb{R}

    Now write out

    \overline{f(z)}=\overline{\sum_{n=-\infty}^{\infty}a_n z^n}=...

    f(\bar{z})=\sum_{n=-\infty}^{\infty}a_n \bar{z}^n=...

    Use some properties of complex numbers and simplify
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  3. #3
    Super Member girdav's Avatar
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    So, what did you try ? Write the Laurent series and use the conjugate to compute \overline{f(z)}. You can put the conjugate into the sum because z\mapsto \overline z is continuous.
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  4. #4
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    Done. Thanks!
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