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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Products of Laurent Series

I'm trying to find the Laurent expansion for $$\frac{e^{1/z^2}}{z - 1}$$ about $z_0 = 0$. Writing the series for $e^{1/z^2}$ and $1/(z-1)$ individually gives $$\frac{e^{1/z^2}}{z - 1} = -\left(\...
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2answers
38 views

Laurent expansions of $\frac{1}{z-1}$

I want to calculate Laurent expansion of $\frac{1}{z-1}$ thtat are valid in the annuli $\begin{align} (a) & \;\;1<|z|<3\\ (b) & \;\;0<|z-3|<2 \end{align}$ For part $(a)$ since $|...
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1answer
23 views

The difference between Taylor and Laurent expansions for Holomorphic functions

I have encountered 2 similar but different theorems on expansions of holomorphic functions to power series, but am not sure how exactly do they differ. Is correct that any $f: U \rightarrow \mathbb{C}...
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1answer
38 views

Laurent expansion - Faster technique

I'm currently preparing for an exam in complex analysis. There is a type of exercise, where I need to compute Laurent expansions about different places. However, my ...
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1answer
38 views

How Do We Know How To Expand Function As Laurent Series

I'm a little confused about a one part of a specific example of a Laurent series that was given by a prof. It seems like it should be pretty straightforward. I want to expand $f(z)=\frac{1}{z^2-z}$ as ...
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2answers
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Laurent series of $z\mapsto\frac{z^4}{z^2-1}$ in $\infty$

I want to calculate the Laurent series and the main part in $\infty$ of $$f:z\mapsto\frac{z^4}{z^2-1}.$$ The Taylor series of $f(\frac{1}{z})$ in $0$ is $$\sum_{n=0}^{\infty}z^{2(n-1)}=1/z^2+1+z^2+z^...
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1answer
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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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32 views

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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1answer
17 views

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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1answer
58 views

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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3answers
72 views

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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2answers
28 views

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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1answer
21 views

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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2answers
31 views

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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1answer
31 views

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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2answers
25 views

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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1answer
23 views

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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1answer
29 views

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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1answer
40 views

Finding Laurent and Taylor series

I need to find both a Laurent and a Taylor expansion. $$f(z)=\frac{z}{(z-1)(z-2)} = \frac{-1}{(z-1)}+\frac{2}{(z-2)}$$ If I choose $z_0=0$ $$f(z)=\frac{1}{(1 + z)} - \frac{4}{\left(1 - \frac{z}{4}...
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2answers
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Singularities of $f(z) = \frac{e^z - e}{(z^2 - 2z + 1)} + z^3\sin(\frac1z).$

In the solution section of my textbook it is said that the function $$f(z) = \frac{e^z - e}{(z^2 - 2z + 1)} + z^3\sin(\frac1z).$$ has a pole of order $1$ at $z_0 = 1$. I don't understand why this is ...
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Singularities of $f(z) = \frac{z\cos(z) - z}{\sin^3(z)}$

I'm having some difficulties classifying the singularities of $$f(z) = \frac{z\cos(z) - z}{\sin^3(z)}.$$ Here's my work so far: Using the trigonometric identity $\sin^2(z) = (1-\cos^2(z))$ is is ...
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1answer
29 views

Singularities of $f(z)=\frac{z^4}{\cos(z^2)-1}$

I'm having difficulties classifying the isolated singularities of the function $$f(z)=\frac{z^4}{\cos(z^2)-1}.$$ The function $f$ is undefined when the denominator equal $0$, that is $$\cos(z^2)-1=...
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1answer
29 views

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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1answer
42 views

Application of the Casorati-Weiertrass theorem

I'm learning complex analysis, specifically series applications, and need help with the following problem: Let $f(z)$ be an entire function which is not a polynomial. Show that for every $c \in \...
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1answer
34 views

If $g(z) = z^kf(\frac{1}{z})$ is bounded in some annular region, then $f(z)$ is a polynomial

I'm learning complex analysis, specifically Laurent series, and need help with the following exercise: If $f(z)$ is entire and for some $k \in \mathbb{N}$ the function $g(z) = z^kf(\frac{1}{z})$ ...
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1answer
31 views

How to rigorously deduce the Laurent series of $\log\frac{z-p}{z-q}$?

Of course, the logarithm here is defined on the ring region $|z|>R\ge\max\{|p|,|q|\}$ as $$\log\frac{z-p}{z-q}=\int_{z_0}^z \left(\frac1{w-p}-\frac1{w-q}\right)\mathrm d w. $$ Here the integral is ...
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1answer
26 views

Isolated singularity of $\frac{\sin(\pi z)}{z - 1}$

I'm learning complex analysis, specifically Laurent series and isolated singularities, and need help to understand the solution to the following exercise: Find and determine the nature of the ...
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3answers
141 views

Which function of $x$, other than $x +c$, and Integral of ($\cos x)^2+(\sin x)^2$, and Integral of $e^{iPi}$ has derivative = 1. [closed]

It is a simple question: Which function of x, other than x +c, and Integral of (cosx)^2+/(sinx)^2, and the integral of $-e^{i*Pi}$, has derivative =1.
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1answer
27 views

Finding the Laurent series of a function

I'm trying to work through the following example: Find the Laurent series of: $$ f(z) = \frac{1}{z(z-2)^3}, $$ about the singularities $z = 0$ and $z = 2$ (separately). Hence ...
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3answers
46 views

How to determine Laurent series associated to $f(z)$ [closed]

The function is $$f(z)= \frac{1}{(e^z -1)},$$ $z$ belong to $\mathbb{C}$ and $0<|z|<1$. I need a general expression in term of a sum from 0 to infinity
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1answer
38 views

Calculate the residues of this complex function

Calculate the residues of this complex function $$\frac{1}{z^2\sin(z)}$$ I can notice that we have singularities at $z=n\pi$, where $n=0,1,2,3,\dots$ But, how to find the residues?
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Laurent-Series on an annulus

I solved an exercise and I want to know if it is correct. I'm trying to find the Laurent-Series for $$g(w)=\frac{w}{1+w^2}$$ On the annulus $D_{1,2}(-i)$ What I did so far: We have 2 poles in $w=i$ ...
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42 views

Laurent expansion of $f(z)=\frac{z}{z^2+1}$

The exercise: Find the laurent expansion of $f(z)=\frac{z}{z^2+1}$ in $K_{1,2}(-i)$. My thoughts: $K_{1,2}(-i)$ denotes the annulus. 1 and 2 the radiuses. First thing I did is decompose in partial ...
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1answer
34 views

Radius of convergence and the existence of antiderivative

I think I have some misunderstandings regarding some basic concepts. First, the question I'm dealing with is the following: Let $f$ be analytic in $\{z ;|z|>1 \}$, and $\int_{|z|=2}f(z)dz=0$. ...
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1answer
44 views

Find and classify isolated singularities of $f(z) = \frac{z}{1-e^{z^2}}$ and calculate residues on them

Obviously, all isolated singularities will be of the form $z=\sqrt{2 \pi i k}$ for $k \in \mathbb{Z}$ but I don't know how to classify. I tried expanding $\frac{1}{f} = \frac{1-e^{z^2}}{z}$ to $- \...
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2answers
134 views

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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1answer
48 views

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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Series representation for Bessel functions

Let $\lambda\in\mathbb{C}$. I want to know if the following is correct: $$\exp\left(\frac{\lambda}{2}(z+z^{-1})\right)=\exp\left(\frac{\lambda}{2}z\right)\exp\left(\frac{\lambda}{2}z^{-1}\right)=\sum_{...
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Series representation of $1/\pi\int_0^\pi e^{\lambda \cos(t)}\cos(nt)dt$

Are there representations of $a_n=1/\pi\int_0^\pi e^{\lambda \cos(t)}\cos(nt)dt,~n\in\mathbb{N},~\lambda\in\mathbb{C}$ as a series? These $a_n$ are the coefficients of the Laurent series $a_0+\sum_{n=...
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1answer
36 views

Laurent series of $\exp({\frac{\lambda}{2}(z+z^{-1}))}$

Let $\lambda\in\mathbb{C}$. I want to get the laurent series of $f(z)=\exp({\frac{\lambda}{2}(z+z^{-1}))}$ on $C(0;0,\infty):=\mathbb{C}\setminus\{0\}$ as $$a_0+\sum_{n=1}^{\infty}a_n(z^n+z^{-n})$$ ...
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1answer
36 views

Find a Laurent expansion

I need to find Laurent expansion for function $$f(z) = \frac{iz^2 + 4iz + 4 +12 i}{(z^2+4)(z+2-i)}$$ for $2 < |z-i| < 3$. I start with division: $$\frac{iz^2 + 4iz + 4 +12 i}{(z^2+4)(z+2-i)} = \...
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1answer
25 views

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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2answers
81 views

Proving that two branch cuts can cancel out

Define the following functions $\mathbb{C}\to\mathbb{C}:$ $$u(z)=\frac{\log \left(z+\frac{1}{2}\right)}{z}\quad \left[-\pi\leqslant\arg \left(z+\tfrac12\right)<\pi\right];\quad v(z)=\frac{\log z}{z}...
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1answer
74 views

Laurent series of $\frac{e^z}{z^2+1}$

I cant figure out the laurent series of the following function. Let $f(z)= \frac{e^z}{z^2+1} $ and $|z|\gt 1$ $$\frac{1}{z^2+1}=\sum_{n=0}^{\infty}(-1)^nz^{-2n-2}$$ and $$e^z = \sum_{n=0}^{\infty}...
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37 views

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 ...
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35 views

When finding Laurent series when to use partial fractions?

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 ...
2
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1answer
49 views

How to find the Laurent series of $\frac{1}{z^4(1-z)^2}$ for |z|>1?

A hint is given that $$\frac{1}{(1-\frac{1}{z})^2} = \frac{z^2}{(1-z)^2}$$ and we know that $$\frac{1}{1-w} = \sum_{n=0}^{\infty} w^n$$ for $|w|<1$. I don't know how to make $\frac{1}{z^4(1-z)^...
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1answer
38 views

Verification on finding the radius of convergence of a Laurent series, “the largest R”.

Question: Determine the largest number $R$ such that the Laurent series of $$f(z)= \dfrac{2}{z^2-1} + \dfrac{3}{2z-i}$$ about $z=1$ converges for $0<|z-1|<R$? Attempt: The radius of ...
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34 views

Laurent series of $f(z)=\frac{1}{z(z-1)(z-2)}$

I am trying to find the Laurent series of the function $$f(z)=\frac{1}{z(z-1)(z-2)}$$in the rings: 1) $0<|z-1|<1$, 2) $1<|z-1|$, 3) $1<|z-2|<2 $ First I expressed $f$ as $$f(z)=\...
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1answer
51 views

Determineing the largest number such that the Laurent series of converges for a trig function.

Question How to determine the largest number $R$ such that the Laurent series of $$f(z)= \dfrac{2sin(z)}{z^2-4} + \dfrac{cos(z)}{z-3i}$$ about $z=-2$ converges for $0<|z+2|<R$? Attempt : Its ...