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Math Help - Laurent series

  1. #1
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    Laurent series

    Consider the identity
    0 = \frac{1}{z-1} + \frac{1}{1-z} = \frac{1}{z}\frac{1}{1-\frac{1}{z}}+\frac{1}{1-z} = \sum_{n=1}^{infinity} \frac{1}{z^n} + \sum_{n=0}^{infinity}z^n=\sum_{-infinity}^{infinity} z^n
    Does this contradict the uniqueness of the Laurent Series?

    due in 15 hours haha
    thanks ahead of time if anyone does work on this
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  2. #2
    Super Member
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    Quote Originally Posted by jbpellerin View Post
    Consider the identity
    0 = \frac{1}{z-1} + \frac{1}{1-z} = \frac{1}{z}\frac{1}{1-\frac{1}{z}}+\frac{1}{1-z} = \sum_{n=1}^{infinity} \frac{1}{z^n} + \sum_{n=0}^{infinity}z^n=\sum_{-infinity}^{infinity} z^n
    Does this contradict the uniqueness of the Laurent Series?

    due in 15 hours haha
    thanks ahead of time if anyone does work on this
    I don't think the series:

    \sum_{n=-\infty}^{\infty}z^n converges anywhere: The singular part converges outside the unit circle; the regular part, inside the unit circle and by casual inspection doesn't seem to converge anywhere on the unit circle.
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