I've only got a couple of pages of notes on this and the only example I've got is the Laurent Series of $f(x)=\frac {1}{1-z}$ for $z>|1|$. I don't really understand how to find a series or why - only it extends a Taylor expansion to negative powers. I was hoping someone could explain how to find a series, and why it works. Thanks!
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asked Apr 5, 2013 at 22:03
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1$\begingroup$ If you want other examples: \url{sym.lboro.ac.uk/resources/Handout-Laurent.pdf} $\endgroup$– mwouaApr 5, 2013 at 22:22
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1 Answer
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It looks like a geometric series doesn't it? but $|z|>1$ so it doesn't converge, but there is a trick $$\frac{1}{1-z}=\frac{1}{\frac{z}{z}(1-z)}=\frac{1}{z} \cdot \frac{1}{\frac{1}{z}-1}$$ and $|\frac{1}{z}|<1$ so here you can the geometric series again. For arbitrary function there is no general way of finding it, but partial fraction decomposition helps sometimes. But finding those series is often difficult or not possible
answered Apr 5, 2013 at 22:36
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$\begingroup$ Thank you, I needed to look at power series again. $\endgroup$ Apr 6, 2013 at 0:20