## Coursera Complex Analysis 7.1 Laurent Series

### Aug 14, 2016

## Slide 2 Review of Taylor Series

Assumes function is analytic over the whole disc.

What if function not differentiable at some point?

.e.g. is not differentiable (or even defined) at +/- 2i.

e.g. f(z) = Log(z) not continuous, so not differentiable, on(along) $late (-\infty,0]$

## Slide 3 Laurent Series Expansion

f is analytic on some complex region U (note: U may contain some points or small regions where f is not defined or *would not be analytic*, so these have been *excluded from U*} and there is some annulus in U, {r <| z – z_0|<R } around those regions which have *already been excluded from U*,

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then f has a **Laurent series expansion:**

Notice similar to Taylor series but limits of the sigma run between **minus and plus infinity**, therefore series runs infinitely in “both directions”. There are **negative powers** of z-z_0 as well as positive powers.

This series converges at each point in the annulus …

## Slide 4 & 5 Partial series example

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