Questions tagged [laurent-series]

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 of $f(z)=\frac{2}{(z+2)^2}-\frac{5}{z-4}$ that converges at $z=1$ in powers of $z-2$

I am trying to find the Laurent series of, $$f(z)=\frac{2}{(z+2)^2}-\frac{5}{z-4},$$ in powers of $z-2$ that converges at $z=1$. My attempt: I think our radius for convergence is $|z-2|<2$. Now, \...
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How can I evaluate this complex integral $\int_{|z|=1}e^{\frac{1}{z}}\cos{\frac{1}{z}}dz$?

I'm trying to evaluate the following complex integral using the residue method. $$\int_{|z|=1}e^{\frac{1}{z}}\cos{\frac{1}{z}}dz$$ The point $z_0=0$ seems to be a singularity. I'm not sure but I ...
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How can I find the residue of this removable singularity?

We have the following function : $$f(z)=\frac{z^2}{1-cosz}$$ where $z_0=0$ is a removable singularity since the limit as z goes to 0 is 2. In such cases, in order to find the residue I proceed by ...
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Principal part of Laurent Series for $\frac{1}{(1-\cosh(z))^2}$

in this exercise I am asked to provide the principal part of the Laurent series of $$\frac{1}{(1-\cosh(z))^2}$$ And i am kinda struggling with fonding a solution or even a pattern towards one Thanks ...
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$\frac {1}{(x^2+1)^2}$ as a Laurent series

How can I write this function $\frac {1}{(z^2+1)^2}$ as a Laurent series around the point $z =i$? I am struggling with the partial fractions part. It's unreducible. The only thing that came to my ...
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Laurent series of $f(z)= \frac {z^2-2z+5}{(z-2)(z^2+1)}$

I want to find the Laurent series of the following function around the point $z=2$, $z$ is a complex number. $f(z)= \frac {z^2-2z+5}{(z-2)(z^2+1)}$ After the fraction decomposition we received ...
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Poles of alternating series

Consider the function $f(\cdot)$ defined as follows, $$f(x) = \sum_{k=0}^{\infty} a_k \left(\frac{-1}{x}\right)^k$$ where $a_0 = 1$ and $a_k > 0$ for all $i$. Assume the series converges in ...
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Integral of $e^\left(1/z^2\right)$ around $|z|=1$ in the complex plane [closed]

$e^\left(1/z^2\right)$ has an essential singularity at $0.$ Don't know how to do this integral.
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Laurent Expansion of $\frac{1}{f(z)^2}$, for $f$ a holomorphic function on a nbd of $a \in C$ with $f(a) = 0$ $f'(a) \ne 0$ and $f''(a) = 0$.

First, we are asked: For $f$ a holomorphic function on a nbd of $a \in C$ with $f(a) = 0$ $f'(a) \ne 0$, find the residue of $\frac{1}{f(z)}$. I used $f'(a) \ne 0$ to conclude the $f$ is not constant,...
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Show that the field of fractions of $\mathbb{Z}[[x]]$ is properly contained in $\Bbb Q((x))$

I've been working on this for a while but I don't know how to proceed. Here it's what I've done: Clearly, my goal is to show that every element of the field of fractions is in $\Bbb Q((x))$ and to ...
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Laurent series expansion of a function with removable singularities

Let $$f(z)=\frac{\sin^2z}{z}, \quad z\neq0$$ with $z_0=0$ being an removable singularity of $f$ since $$\lim_{z \to 0}\, zf(z)=\lim_{z \to 0}\,\sin^2z=0$$ To find its Laurent series expansion within ...
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Doubt on Laurent series of $f(z) = 1/z(z-2)^2$

I've a doubt about this exercise Find the Laurent series of the function $$f(z) = \frac{1}{z(z-2)^2}$$ addend $z=0$ in the annular regions of interest The problem it's not totally about the how ...
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Confusion with the intuitive understanding of Laurent's series and it's reduction to Taylor's

I know that Laurent's series is used for an annulus with the function $f(z)$ being analytic in the region R defined by the annulus ($r<|z|<R$, where $R > r$). Now my question is that ...
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Uniform convergence and series of expansion

Assume the following sum: $$s(x) = \sum_{n=1}^{\infty} f_n(x) = \sum_{n=1}^{\infty}\frac{1}{n \sqrt{n^2+x^2} \left(\sqrt{n^2+x^2}+n\right)},$$ where $x \in \mathbb{R}$. Since $f_n(x) = f_n(-x)$, it ...
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Find the Laurent expansion $f(z)=\frac{1}{z(1-z)^2}$.

Find the Laurent Series expansion of $$f(z)=\frac{1}{z(1-z)^2}$$ at $z=1$. How do I do this when I am not given any region?
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Find the two Laurent series for $f(z)=\frac{1}{z^3-z^4}$ that involve powers of z, and state their domains of convergence.

I'm working on the following problem: Find the two Laurent series for $f(z)=\frac{1}{z^3-z^4}$ that involve powers of z, and state their domains of convergence. I'm not sure how to go about it, any ...
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An apparent contradiction based on considering the series of $f(z)$ and relating it to $f(1/z)$

I have a function with a relationship between $f(z)$ and $f(1/z)$ that, when interpreted naively using the series for $f$, gives wrong results. I'm going to give this naive argument in the hopes that ...
Laurent series for $f(z) = \frac{\sinh(z + 3i)}{z(z + 3i)^3}$ at $-3i$ (not leaving as a product of series)
I got this problem in my complex analysis class: Find the Laurent series of $$f(z) = \frac{\sinh(z + 3i)}{z(z + 3i)^3}$$ to calculate the residue at $z=-3i$. Is there an easy way to calculate the ...