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Math Help - Complex Power Series

  1. #1
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    Complex Power Series

    Can anyone help with this question?

    Find the power series expansions of
    (a) log z about z = 1;
    (b) 1/(1 + z) about z = −5.
    What is the radius of convergence in each case?

    Thanks for any help
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  2. #2
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    Quote Originally Posted by alawrie View Post
    Can anyone help with this question?

    Find the power series expansions of
    (a) log z about z = 1;
    (b) 1/(1 + z) about z = −5.
    What is the radius of convergence in each case?

    Thanks for any help
    For (a), where are you stuck? What have you tried?

    (b) Note that \frac{1}{1 + z} = \frac{1}{(z + 5) -4}.

    Region I: 0 < \frac{|z + 5|}{4} < 1.

    \frac{1}{(z + 5) -4} = \frac{-1}{4 - (z + 5)} = -\frac{1}{4} \left( \frac{1}{1 - \left(\frac{z+5}{4}\right)} \right) and you can get a series using the sum of an infinite geometric series, noting that r = \frac{z + 5}{4}.

    Region II: \frac{|z + 5|}{4} >1.

    \frac{1}{(z + 5) -4} = \frac{1}{z + 5} \left( \frac{1}{1 - \left[ \frac{4}{z + 5}\right]} \right) and you can get a series using the sum of an infinite geometric series, noting that r = \frac{4}{z + 5}.
    Last edited by mr fantastic; February 10th 2010 at 05:03 PM. Reason: Fixed a couple of bad typos.
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