# Questions tagged [laurent-series]

This tag is for questions about finding a Laurent series of functions and their convergence. The Laurent series is a generalisation of the power series which allows negative indices and is essential for investigating the behaviour of functions near poles.

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### Laurent series for a polynomial in the numerator

I have been given the following function for which I am supposed to construct the laurent series around the singular point -1: $$f(z)=\frac{z^2+4z+4}{z+1}$$ From the general formula I know that this ...
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### Did I do this Laurent series correctly?

I am wondering whether I did the Laurent series for the following function correctly since I am pretty unsure as to what I am doing there: $$f(z)=-\frac{6}{(z-2)(z-4)}$$ for $2<|z|<4$. Through ...
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### Identical Functions in a Domain

Problem statement: If f1(z) and f2(z) are analytic in a domain D and equal at a sequence of points zn in D that converges in D, show that f1(z)=f2(z). I'm not clear on this - I understand that since ...
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### Laurent series of $f(z)=\frac {\exp (1/z)}{1-\sin(1/z)}$ around 0.

as you can see i would like to expand in a Taylor-Laurent around 0 the function in the title. I can kinda see that it should be an essential singularity, because it should be for ${\exp (1/z)}$ and ...
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### Laurent series expansion in 2 complex variables

I'm reading about the $j$-function from Washington's book. We define $S_N$ to be the set of matrices of the form $\begin{bmatrix}a & b\\ 0 & d\end{bmatrix}$ where $a,b,d\in\mathbb{Z}, ad=N$ ...
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### Laurent series of $\ln(1-\frac{2i}{z+i})$

How to find Laurent series of function $ln\left(1-\frac{2i}{z+i}\right)$ if it is known $1 \leq |z| \leq 2$. I am confused how to find this, because I tried to prove that $|\frac{2i}{z+i}|<1$ and ...
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