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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 $${1\over{2\pi i}} \int_c {f(z)dz \over {(z-z_0)^{-j+1}}}$$.

- Type of singularity of $\log z$ at $z=0$
- Complex Taylor and Laurent expansions
- Prove Laurent Series Expansion is Unique
- Find and classify singular points of $\cot\left(\frac{1}{z}\right)$
- Principal part of Laurent expansion.
- Help with this limit?

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.

- Calculate the series expansion at $x=0$ of the integral $\int \frac{xy\arctan(xy)}{1-xy}dx$
- Examine convergence of $\sum_{n=1}^{\infty}(\sqrt{a} - \frac{\sqrt{b}+\sqrt{c}}{2})$
- Binomial Theorem Proof from Taylor Series
- Explain inequality of integrals by taylor expansion
- Exponential of powers of the derivative operator
- Conditions for Taylor formula
- Identifying $\sum\limits_{k=0}^\infty\frac{1}{k!}\frac{x^{k+6}}{k+6}$
- Removable singularity and laurent series
- To show that the limit of the sequence $\sum\limits_{k=1}^n \frac{n}{n^2+k^2}$ is $\frac{\pi}{4}$
- How many smooth functions are non-analytic?

Well the taylor series only work when your function is holomorphic, the laurent series works still for isolated singularities.

They both represent the function, but the only converges when $|z|>1$ and the other only converges when $|z|<1$.

When $f$ is holomorphic the taylor series and the laurent series are the same, and with Cauchy’s theorem you can see that. If you want to be as good as possible you have to calculate those things on your own, thats how you learn the most.

If you want some exercises come to chat, or search in google, and if you can’t solve them ask a new question.

- Adjoint functors
- Multivariate Normal Difference Distribution
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- Strong Induction Base Case
- Ring of integers is a PID but not a Euclidean domain
- Weil does not imply Cartier on variety $X$.
- if $A, B$ are open in $\mathbb R$ then so is $A+B.$
- Showing the existence of an eigenvector using groups
- Evaluation of $ \int\frac{\sqrt{\sin x}}{\sqrt{\sin x}+\sqrt{\cos x}}dx$
- Do groups of order $p^3$ have subgroups of order $p^2$?
- Equivalence of reflexive and weakly compact
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- Show that the ring of all rational numbers, which when written in simplest form has an odd denominator, is a principal ideal domain.
- the Nordhaus-Gaddum problems for chromatic number of graph and its complement