# 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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### Converse of bounds on the spectrum of a Toeplitz matrix

The following is from Robert M. Gray's review (https://ee.stanford.edu/~gray/toeplitz.pdf): Lemma 4.1 Let $\tau_{n,k}$ be the eigenvalues of a Toeplitz matrix $T_n(f)$. If $T_n(f)$ is Hermitian, then ...
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### proof that $K((x)) = \operatorname{Frac}(K[[x]])$

I want to show $K((x)) = \operatorname{Frac}(K[[x]])$, i.e. K((x)) is the minimum field containing $K[[x]]$, where the latter is an integrity domain, since it is a commutative ring and has no zero ...
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### which one is correct for the complex integration $C_k=\frac{1}{2 \pi i} \oint_C \frac{\sin \left(\zeta-\frac{1}{\zeta}\right)}{\zeta^{k+1}} d \zeta$

when I calculate the complex integral$$C_k=\frac{1}{2 \pi i} \oint_C \frac{\sin \left(\zeta-\frac{1}{\zeta}\right)}{\zeta^{k+1}} d \zeta$$ where $C:z=e^{i t},$ $t: 0 \rightarrow 2 \pi$ I got ...
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### Finding residue of $f(z) = \frac{1}{z^2-1} \sin\frac{1}{z^2 + z^4}$ at $z = 0$

Trying to find the residue for $f(z) = \frac{1}{z^2-1} \sin\frac{1}{z^2 + z^4}$ at $z = 0$. I am concentrating on the sin term in particular. What is a good strategy to approach this? One thing I ...
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### Can you remove the poles from $\frac{f'(x)}{f(x) - y}$ with $L^\infty_y L^1_x$ error?

I believe that the following below holds, which I need for some estimates. I have been struggling to prove it though! Let $f \in C^1(\mathbb{R}; \mathbb{R})$ have finitely many critical points and be ...
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### Taylor series, Laurent series and a simple pole

Let $f$ be holomorphic in $|z|<2$ except for $z=1$, in which it has a simple pole. Let $f(z)=\sum_{n=0}^{\infty} a_n z^n$ be the Taylor series of $f$ around $z=0$. Prove or find a counterexample: (...
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### Laurent series expansion of $𝑓(𝑧)=\tan(𝑧/(𝑧-1))$ around $𝑧 = 1$

I am tasked with finding whether the function $𝑓(𝑧)=\tan(𝑧/(𝑧-1))$ can be developed into a Laurent series around $𝑧=1$, and if so what is the radius of convergence and what is the residue? My ...
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### Laurent series expansion of $f(z) = \cos(z/(1-z))$ around $z = 1$

I am tasked with finding whether the function $f(z) = \cos(z/(1-z))$ can be developed into a Laurent series around $z = 1$, and if so what is the radius of convergence and what is the residue? So far ...
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1 vote
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### Find the laurent expansion

Find the Laurent expansion of $\dfrac{e^\frac{1}{z}}{z+1}$ in the domain $|z|>0$. My attempt: Since we have to work on domain $|z|>0$, let $z=\dfrac{1}{t}$, so I have to find the Laurent ...
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