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Math Help - Laurent Series

  1. #1
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    Laurent Series

    Find the laurent series on 2<|z|<3
    f(z)=1/[(z^2)*(z-2)*(z+3)]
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  2. #2
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    Quote Originally Posted by scubasteve123 View Post
    Find the laurent series on 2<|z|<3
    f(z)=1/[(z^2)*(z-2)*(z+3)]
    Around what value of z is the series to be found?
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  3. #3
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    There isnt an indicated center. All the question says is "Find Laurent series expansion(in powers of z) that represents the function in the region 2<|z|<3
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    Quote Originally Posted by scubasteve123 View Post
    There isnt an indicated center. All the question says is "Find Laurent series expansion(in powers of z) that represents the function in the region 2<|z|<3
    "(in powers of z)" means you find it around z = 0. So the centre is given. It helps if the whole question is posted.

    I suggest writing the given function as \frac{1}{z^2} \left( \frac{A}{z-2} + \frac{B}{z + 3}\right) where the partial fraction decomposition is left for you to finish.

    Then expand each of \frac{1}{z - 2} and \frac{1}{z + 3} around z = 0. Then substitute etc.

    eg. \frac{1}{z - 2} = \frac{1}{1 - \frac{2}{z}} = 1 + r + r^2 + r^3 + ..... where you should recognise an infinite geometric series with r = \frac{2}{z} (note that \left| \frac{2}{z} \right| < 1 since 2 < |z| < 3.
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