Slide 2 Review of Taylor Series

Assumes function is analytic over the whole disc.

What if function not differentiable at some point?

.e.g. $f(z) = \frac{z}{z^2 + 4}$ 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: $f(z) = \Sigma_{k=-\infty}^{+\infty}a_k(z-z_0)^k = ... + \frac{a_{-2}}{(z-z_0)^2} + \frac{a_{-1}}{(z-z_0)^1} + a_{0} + {a_{1}}{(z-z_0)^1} + {a_{2}}{(z-z_0)^2} + ...$

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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