Laurent series

**jbpellerin** · November 4th 2008, 10:56 PM

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

**shawsend** · November 5th 2008, 12:17 AM

Originally Posted by jbpellerin

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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