# Tagged Questions

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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### Expansion of $\frac{\cos(z^2)-1}{z^2}$ about the origin [closed]

Expand the following functions about the origin and find the region of convergence in each case. $\frac{\cos(z^2)-1}{z^2}$ $\frac{e^z-z-1}{z^2}$ Please help with these functions, I'm ...
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### complex analysis singularites

These are the directions that I have for my three functions below. Please help. Determine the type of each singularity of the given function, if the singularity is removable, define the function ...
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### Laurent Series Expansion

I need to find all possible series expansions with center at $c=0$ for $f(z) = \frac{1}{z^2(z-1)(z-2)}$ I know that I need to use the fact that $\frac{1}{1-z} = z^n$ for substitution and that the ...
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### Laurent series with $e^z$

I'm trying to find the Laurent series Expansion for $$f(x) = \frac{e^z-(z-1)}{z-1}$$ on the annulus $0<|z|<\infty$. I'm aware that I am supposed to use substitution of known series. I am ...
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### Laurent series of $\frac{1}{(z-1)(z-2)}$

How can I compute the Laurent series of $$f(z)=\frac{1}{(z-1)(z-2)}$$ on the circular ring $A(0;1,2)=B(0,2)\setminus \overline{B(0,1)}$? I tried to take the Cauchy product of geometric series but ...
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### Find the Taylor-series expansion of a square of a rational function of a complex variable

I've been trying to find the Taylor-series expansion of the following function: $$f(z)=\left ( \frac{1+z}{1-z} \right )^2$$ az the origin : Z0 = 0. also I would like to find the region of ...
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### find the laurent series using z=w+1

Here is the question that my books is asking Find the same Laurent series for $f(z)=1/(z(z-1)^2)$ center at $c=1$ by using the following procedure. Set $z=w+1$, expand the resulting function in ...
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### Laurent Series expansion in sigma notation

I am trying to find the Laurent Series expansion in sigma notation of $$\frac{1}{z^3-2z^2+z}$$ where $0<|z-1|<1$. I've tried partial fractions and am still stuck on the approach necessary to ...
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### Classifying singularities of $\frac{\sin(\pi z)}{z^4+1}$

If $f(z)=\frac{\sin(\pi z)}{z^4+1}$, we have four roots of unity, which are isolated singularities of $f$: $$z=-(-1)^{1/4},z=(-1)^{1/4}, z=-(-1)^{3/4}, z=(-1)^{3/4}.$$ Do we need to find the Laurent ...
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### Laurent series of $\frac{1}{z^2(z-1)}$ when $0<\lvert z\rvert<1$

$\frac{1}{z^2(z-1)} = -\left(\frac{1}{z}+\frac{1}{z^2}+\frac{1}{1-z}\right)$. I know that $\frac{1}{1-z}=\sum\limits_{n=0}^\infty z^n$, but what about the other two terms, should they be left as they ...
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### laurent series expansion of terms like $\frac{1}{z}$, $\frac{1}{z^2}$

I have a homework question of finding the Laurent expansion of $\frac{1}{z^2(z-1)}$ on $0<|z|<1$. I've learned to decompose the function to $\frac{A}{z}$, $\frac{B}{z^2}$, and$\frac{C}{z-1}$, ...
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### Laurent's Theorem - definition of a path in an annular domain

Theorem on p. 197 of the book mentioned below: Supposed that a function $f$ is analytic throughout an annular domain $R_1<\lvert z-z_0\rvert< R_2$, cenetered at $z_0$, and let $C$ denote any ...
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### Find holomorphic function in $\mathbb{C}\setminus\{i, -i\}$ satisfying specific conditions

I'm learning complex analysis, specifically (Laurent) series and residues, and need help with the following problem: Construct a function $f(z)$ holomorphic in $\mathbb{C}\setminus\{i, -i\}$ with ...
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### Analytic continuation on the disk

Let $V = \big\{z: |z|<5,\text{Im}(z)>0 \big\}$. Let $f$ analytic in $V$, continuous in $\overline{V}$ and suppose $$\forall x \in \left[ -5,5\right]:\ f\left( x\right) \in \mathbb{R}$$ Show that ...
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### Singularities in the complex plane and expansion of Taylor/Laurent Series

The function f(z) = $\frac{\cosh(z-3i) -1}{(z-3i)^{5}}$ has one singular point in $\mathbb{C}$. I understand that the singular point is an isolated singularity at 3i, and I know there are certain ...
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### Find the Laurent Series for $\frac{z-2}{z+1}$ around $z = -1$

Find the Laurent Series for $\frac{z-2}{z+1}$ around $z = -1$ I'm not sure how to do this because it is not something with a simple numerator. If it was something like $\frac{1}{(z-2)(z+1)}$ I ...
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### Radius of convergence of the series $\sum\limits_{-\infty}^{\infty}(2^{-n}+4^{-n}) z^n$

I'm trying to find for what values of $z\in\mathbb{C}$ the series $$\sum_{n=-\infty}^{\infty}(2^{-n}+4^{-n})z^n$$ converges. My main methods are the nth root test and the ratio test. I believe it can ...
When finding the Laurent series of $$f(z):=\frac{1}{z(z-1)(z-2)}$$ valid in the region $1<|z-2|<2$ for example do we just use partial fractions to break $f(z)$ up and the just find the Laurent ...