I have never taken complex analysis, but I am preparing for a GRE for this week end and I am trying to learn a bit about Laurent Series.
So far what I get is that the Laurent Series are of form $$\Sigma_{i=1}^\infty {a_{-i}(z-z_0)^{-i}} + \Sigma_{j=1}^\infty {a_j(z-z_0)^i}$$
where $a_i$ is the usual Taylor coefficients and $a_{-j}$ is given by $${1\over{2\pi i}} \int_c {f(z)dz \over {(z-z_0)^{-j+1}}}$$.
I have absolutely no idea how this works, but I saw that in practice, we just manipulate the Taylor Series to get the Laurent Series some how.
For example, the Taylor expansion of $1\over 1-z$ is $1+z+z^2+...$ for $|z|<1$.
So this is what I would really like to understand. Supposedly in order to "avoid" the singularity at Z = 1, now $1 \over 1-z$ must be expanded in a Laurent Series in the region $1<|z|<+\infty$. To do so the series will be manipulated as such $${1 \over 1-z} = {1 \over {z({1\over z}-1})} = {-1 \over z } ・{1\over {1-{1\over z}}} $$
Since $1<|z|$, $|1/z|<1$, so the Taylor expansion gives us $${-{1 \over z}}-({1 \over z})^2- ・・・$$ for $1<|z|$.
I only have a couple of example problems in the GRE practice book, and I failed to understand all of them. I recon that I am not getting the motivation of when to use the Taylor Series and when the Laurent.
I would also like to claim that since they both represent $1 \over {1-z}$, why do they look different and HOW are they different ?
My book also did not generalize how to manipulate the Taylor Series to make it into a Laurent series, so can someone guide me to where I could learn this a little bit more with concrete examples and details or explain this to me?
I know I am asking a lot, but mathematics means the life to me and I want to do as good as possible on the GRE.
