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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Invertible matrices over ring of formal Laurent series

Let $A$ be a commutative ring, let $A[[t]]$ be the ring of formal power series and consider the ring of formal Laurent series $A((t)) = A[[t]][t^{-1}]$. I would like to know: What is $Gl_n(A((t)))$,...
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Generator of inertia group in function field extension

Can Someone help me solve the following problem? Let C((T)) be the field of formal Laurent series in the variable T over an algebraically closed field C of characteristic $0$. (1) Prove that ...
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Finding Laurent Series of a function

I've been assigned to write a computer program which then calculates the Laurent series of a function. Of course I'm familiar with the concept, but I've always calculated the Laurent series in an ad ...
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How to find the constant $c_0$ term in $\displaystyle F(z) = \sum_{n=-Large1}^{Large2} {c_nz^{2n}}$

How to determine if $c_0 = 0$ ? Where $$F(z) = \left( \prod_{k = 0}^{N} (1+z^{2a_k}) \right) \left( \prod_{j = 0}^{M} (z^{-2b_j} + z^{-2d_j}) \right) = \sum_{n=-Large1}^{Large2} {c_nz^{2n}}$$ ...
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Find the Laurent series for $p(4/z)$

Find the Laurent series for $p(4/z)$ given that $p(z)=(z-3)^3$ My attempt: if the Taylor series for $p(z)$ looks like $$\frac{-27}{0!}+ \frac{27z}{1!}-\frac{18z^2}{2!}+ \frac{6z^3}{3!}+0+0+0...$$ ...
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Expanding $f\left ( z \right )=\frac{e^{az}}{1+e^{z}}$ about $z= i\pi$

$$f\left ( z \right )=\frac{e^{az}}{1+e^{z}} ,\left ( a\in\left ( 0,1 \right ) \right )$$ The point $z=i\pi$ is one of the nonremovable singularities of this function. In order to expand it about that ...
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Limit of $\frac{1}{2^n}\sum_{n=0}^{2^n-1}f(e^\frac{2k\pi}{2^n}i)$ for a complex-analytic function

Let consider a Laurent series $\displaystyle{ \sum_{k\in\mathbb Z}a_kz^k }$ with complex coefficients and converging inside the annulus $A=\{\ z\in\mathbb C\ |\ r<|z|<R\ \}$, with $r<1<R$....
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How come the definition of analytic continuation doesn't require the smaller and the bigger open subsets to be connected?

The reason that is making me think that these subsets should be connected / simpled connected is because I think that the Taylor disks of convergence of f and F, which is the continuation of f to the ...
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Laurent polynomial regression?

Polynomial regression is a common way of doing curvilinear regression. It is common to also use the inverse transform x^-1 (http://pareonline.net/getvn.asp?v=8&n=6). One can extend the concept ...
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Finding the Laurent series given the poles and residues

I am working on the following problem, suppose that $f$ has a simple pole at $-1$ with $Res(f,-1) = 1$. A double pole at $2$ with $Res(f, 2) = 2$. Also $f(0) = 7/4$ and $f(1) = 5/2$. I am supposed ...
Using the Laurent-expansion of $f(z)$ around $0$ $$f(z) = \sum_{n=0}^{+\infty} a_n z^n + \sum_{n=1}^{+\infty} b_n z^{-n} \tag{1}$$ Theorem The coefficients of the Laurent series are given by ...
Consider the function $$f(z) = {1 \over (z-i)(z+i)}$$ with a Laurent series expansion at $z_0=i$ on a domain $\;\Omega=\left\{z\in \mathbb{C}:2\lt\left|z-i\right|\right\}$ \begin{eqnarray}f(z)={1 \...