Overview
Taylors Theorem
Given a function {% f(z) %} that is analytic throughout a disc {% |z-z_0| < R %}, then {% f(z) %} has a series representation
{% f(z) = \sum_{n=0}^\infty a_n(z-z_0)^n %}
Laurent Series
Given a function {% f(z) %} that is analytic throughout an annular disc {% R_1 < |z-z_0| < R_2 %}, and let C be contour around {% z_0 %} and lying in the disc, then {% f %} has the series representation
{% f(z) = \sum_{n=0}^\infty a_n(z-z_0)^n + \sum_{n=1}^\infty \frac{b_n}{(z-z_0)^n} %}
{% a_n = \frac{1}{2 \pi i} \int_C \frac{f(z)dz}{(z-z_0)^{n+1}} %}
{% b_n = \frac{1}{2 \pi i} \int_C \frac{f(z)dz}{(z-z_0)^{-n+1}} %}