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 $$ \sum_{n=-\infty}^\infty a_n (z-z_0)^n = \sum_{i=1}^\infty {a_{-i}(z-z_0)^{-i}} + \sum_{j=0}^\infty {a_j(z-z_0)^i}$$
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 $$ \frac{1}{1-z} =1+z+z^2+...$$ 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
$$ \frac{1}{1-z} = \frac{-1}{z} \cdot \frac{1}{\left(1- \frac{1}{z} \right)} $$
Since $1<|z|$, $|1/z|<1$, so the Taylor expansion gives us $$ -\frac{1}{z} - \frac{1}{z^2} - \cdots $$ 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.
