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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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complex analysis Laurent series around no singularity point?

hi Guys I am trying to determine a series for the following function: $$f(x)=\frac{2z}{(z+i)(z-1)}$$ about $|z-1|<\frac{1}{2}$ I already developed partial fractions: $$\frac{1 + i}{z + i} + \...
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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 ...
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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 ...
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Finding Residues Using Partial Fractions

I'm calculating some countour integrals of complex functions by residues, but I can't understand an certain way to calculate these residues in some cases. We have a theorem that states that the ...
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Laurent series around z=-i for $\frac{1}{z(z-1)}$ and $1<|z+i|<\sqrt2$

I'm given the function : $$f(z)=\frac{1}{z(z-1)}$$ I'm interesting in finding the Laurent series around $z=-i$ for the one finite circular ring corresponding to this function given its ...
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Show that the Laurent series expansion of $f(z)$ in the region $0 < |z-z_0| < r < R$ does not involve negative power of $(z-z_0)$.

Suppose that a function $f$ is analytic and bounded in $B(z_0;R)-\{z_0\}$ of a point $z_0$. Show that the Laurent series expansion of $f(z)$ in the region $0 < |z-z_0| < r < R$ does not ...
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Laurent series for $e^\frac{1}{z}$ about $z=0$

What I've currently done is Taylor expansion for $e^z$ $$e^z = \sum_{k=0}^\infty \frac{z^k}{k!}$$ And $\frac{1}{z}$ substitution $$e^\frac{1}{z} = \sum_{k=0}^\infty \frac{1}{k!} \frac{1}{z^k}$$ But ...
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Laurent series expansion of complex analysis

Suppose $f:\Omega \mapsto \mathbb{C}$ is analytic expect ${z_1,z_2,...z_r}$, but defined on full set $\Omega$, if in Laurent series expansion of f around these points ${z_1,z_2,...z_r}$ has no ...
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Why is the composition of convergent Laurent series convergent?

In Rick Miranda's book Algebraic Curves and Riemann Surfaces, the author proves the order of a meromorphic function $f \colon X \to \mathbf{C}$ at a point $p$ is independent of the complex chart that ...
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Algebraic elements in a bijection between $p$-adic numbers and formal Laurent series over $\mathbb F_p$

Let $p$ be a prime number. We have a bijection $$\begin{align*}\sigma : \mathbb Q_p & \to \mathbb F_p(\!(t)\!) \\ \sum_{i \geq -n} a_i p^i & \mapsto \sum_{i \geq -n} a_i t^i \end{align*}$$ ...
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Let $f$ be holomorphic in $\mathbb{C}\setminus\{i,2i\}$.

Let $f$ be holomorphic in $\mathbb{C}\setminus\{i,2i\}$. Show that if $f$ has an non avoidable singularity in $z = i$ and $z = 2i$, then, the Laurent series of $f$ in $\{1 <|z| < 2 \}$ has ...
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Multiple annuli of laurent series expansion

Let $A_z(R_1,R_2)$ denote the open annulus about $z$ with inner radius $R_1$ and outer radius $R_2$. According to the basic theorem of Laurent series, if $f\in Hol(A_z(R_1,R_2))$ then $f$ has a ...
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Multiplying a Laurent Series and a Maclaurin Series

Find the Laurent expansion for $$\frac{1}{z-1}\frac{1}{z+3}$$ inside the annulus $1<\vert z\vert<3$. Since $|z|>1$, $$\frac{1}{z-1}=\frac{-1}{1-z}=\frac{-1}{(1-1/z)(-z)}=\frac{1}{z}\sum_{n=0}...
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What will be the Laurent's series expansion of $e^{f(z)}$ about $z=0$?

Let $f$ be a meromorphic function having a pole at $z=0$. Then what can we say about the Laurent's series expansion of $e^f$ about $z=0$? My confusion is in the following lines "Can we first write $...
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Laurent series of $f(z) = \frac{5}{z^2 + (2i - 1)z - 2i}$

Let $f: \mathbb{C} \setminus \left\{ 1, -2i \right\} \rightarrow \mathbb{C}$ with $$f(z) = \frac{5}{z^2 + (2i - 1)z - 2i}$$ Determine the Laurent series of $f$ in the annuli $\left\{ z \in \mathbb{C} ...
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What is the Laurent series expansion of $e^{-e^{\frac {1} {z}}}$ about $z=0$?

What is the Laurent series expansion of $e^{-e^{\frac {1} {z}}}$ about $z=0$? I know that if the Laurent series expansion of a function $f$ about $z=0$ is $$f(z)= \cdots + \frac {b_2} {z^2} + \frac ...
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Show that $e^{\frac{1}{2}\lambda(z+\frac{1}{z})}= a_0 + \sum_{n=1}^\infty a_n(z^n+\frac{1}{z^n})$

Here $\lambda \in \mathbb{C}$ and $$a_n = \frac{1}{\pi}\int_0^\pi e^{\lambda \cos(t)} \cos(nt)\,dt$$ I'm clueless about how to reach that expression for $a_n$. The things that came to my mind were to ...
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Find the coefficient of $z$ in the Laurent series expansion of $\frac{e^z}{z-1}$ in ${|z| > 1}$

I've found this duplicate from '15: Find the coefficient of $z$ in the Laureant Series expansion of $\frac{e^z}{z-1}$, but I think it's wrong, since it looks for the Laurent expansion in ${|z-1|>1}$...
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Order of Integration in Divergent Integrals

I'm looking to compute the integral $$I = \int_0^\infty dy \, \sinh^{d-2 \epsilon -1} y $$ as a Laurent series in $\epsilon$. (For those who are familiar with it, this is related to dimensional ...
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Evaluating $\oint_{|z|=2} \frac{\log{(z-\alpha)}}{z}-\frac{\log z}z \, \mathrm dz$

Let $\alpha \in (-1,0)$ and define $$f(z)= \frac{\log{(z-\alpha)}}{z}-\frac{\log z}z$$ where the logarithms take their principal values, i.e. the arguments of $z$ and $z-\alpha$ are between $-\pi$ and ...
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Finding last part of Laurent series in annulus

Find the Laurent series in given annulus: I have function in annulus $1<|z-1|<2$, $${1 \over {z(z-3)^2}}$$ After splitting function into partial fractions, I got the first two sums, but ...
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Find the Laurent series in $1<|z|<2$ ring

I just started Laurant series, I have this function $$f(z) = {z^4 + 1 \over {(z-1)(z+2)}}$$ in $1<|z|<2$ ring, I know the theorems,but still cant figure it out, Any help with idea or ...
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How to determine the number of zeros?

Question: given $f(z)= e^z-2i$ then determine the number of zeros of $f$. My attempt: $e^z -2i=0$ ⇔$e^z=2i$ Now by taking natural log on both side we can see, $f$ has countably infinite number of ...
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A step by step process for how I think Laurent series should be obtained.

So I'm trying to calculate the Laurent series of $f(z)=\frac{1}{2z^3}-\frac{2}{z^3+i}$. Here are the steps I beieve I should take; Step.1) Well first I noted that it has two singularities $z_0=0$ ...
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A minor question on whether I've expressed my Laurent series with the correct exponent.

Say we have the function $f(z)=\frac{1}{z^2+1}$ and we want to calculate it's Laurent series about $z=i$. Then I know that we can decompose this function using partial fractions into $f(z)=\frac{\...
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Laurent series of $f(z)=\frac{z-1}{z(z^3-1)}$ for $0<|z|<1$ and $|z|>1$.

I need to find the Laurent series of $f(z)=\frac{z-1}{z(z^3-1)}$ for $0<|z|<1$ and $|z|>1$. I believe I am nearly there but I have a confusion. Notice $f(z)=\frac{z^2-1}{1-z^3}+\frac{1}{z}$ ...
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Power series (Laurent series), in a disk $B_1(0)$ , of $\frac{1}{(1-z)^m}$ where $m\in N$

I'm trying to find a Power series (Laurent series), in a disk $B_1(0)$ , of $\frac{1}{(1-z)^m}$ where $m\in N$
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Laurent series expansion in powers of $z$

Find the Laurent series expansion in powers of $z$ of $f(z) = \frac{\cos(z^2)}{z^3}$ valid in the region $|z| > 0$ My Instinct is to make use of the fact that $\cos(z^2) = \frac{1}{2}(e^{z^2 i\...
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Proof of Laurent’s Expansion

Question: Prove Laurent’s Expansion$$f(z)=\sum\limits_{i\geq0}a_n(z-z_0)^n+\sum\limits_{j\geq0}\frac {b_j}{(z-z_0)^j}$$where$$a_i=\frac 1{2\pi i}\oint\limits_{C_1}dw\,\frac {f(w)}{(w-z)^{i+1}}$$$$b_j=\...
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Laurent series expansion in powers of z

$f(z) = \frac {e^{z^{2}}}{z^3}$, valid for $|z| >0$ The way I am thinking about this problem is to factor out the $e^{z^{2}}$ term as I am use to Laurent series of the form $\frac{1}{1-w}$. I am ...
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why there isn't laurent series at a non isolated singularity

I am studying the laurent series expansion around singularities. I don't understand why at a non-isolated singularity the laurent series expansion doesn't exist, whereas we define the laurent series ...
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Finding the singularities of a complex function.

I need to find and classify the singularities of the function: $ f(z) = \frac{z^2+1} {z^4-2}$. I'm aware that I'm going to have to first find the Laurent series corresponding to this function. I ...
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131 views

Evaluating the Laurent series of $\frac{1}{\exp (z) -1 }$

I am trying to find the Laurent series of the function $ f(z) =\frac{1}{\exp (z) -1 }$ above but I don't know how to find the term $$a_{-1} = \frac{1}{2\pi i}\int f(\xi ) \,d\xi $$ I tried ...
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Laurent series for $\frac{1}{z(z+3)(z-1)^2}$

Find the Laurent Series for $$\frac{1}{z(z+3)(z-1)^2}$$ in $1 < |z-1| < 4$ So I did the partial fraction decomposition which yields: $$\frac{1}{4(-1+z)^2} - \frac{5}{16(-1+z)} + \frac{1}{3z} ...
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Laurent Series of $\frac{1}{(1+z^2)^2}$

Expanded to the Laurent Series at the deleted neighbourhood of $$z=i$$ and try to give the Convergence range I'm trying to make $$\frac{1}{(1+z^2)^2}=\bigg(\frac{-1}{2z}\bigg)\bigg(\frac{1}{1+ z^2 }\...
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Laurent Series of $\frac{e^z}{z(1+z^2)}$ [at z=0]

How to get the Laurent Series of $$\frac{e^z}{z(1+z^2)}$$ at $z=0$? I know the answer is $$\frac{1}{z} + 1 - \frac{1}{2}z - \frac{5}{6}z^2 + \frac{13}{24}z^3 + \frac{101}{120}z^4 + O(z^5)$$ But I ...
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Show that the singular point of the following functions is a pole and verify using Laurent’s expansion

Given that I have $\frac{1-e^{2z}}{z^3}$ and $\frac{e^{2z}}{(1-z)^2}$, how would I go about verifying it in each case using Laurent's expansion? I can find the residues and I know the singular points ...
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Clasifying singularities.

I would like to clasify the singularities of the function $$ f(z) = \frac{1}{z^2+2z+1} $$ I tried to make the Laurent expansion on $ z = -1$, but at this point the function is not holomorphic and I ...
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Obtaining Laurent series of 1/f(z) given the laurent series of f(z).

Suppose I have a complex function ($f_{}^{}(z_{}^{})$), analytic in the annular region $\mathcal{C} = \{z : 0\leq r_{}^{}<|z_{}^{}-z_{0}^{}|<R_{}^{} \leq \infty\}$ with the Laurent series ...
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Calculate residue in every pole of $\frac{z^2}{(z-2)^2(\cos{z}-1)^3}$

The problem is as simple as the title suggests. Of course, the formula for the residue of an order $n$ pole involving $n-1$ derivatives could be applied, but the computation will be extremely long, as ...
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Finding the Laurent series of the following function

I need to find the laurent series and the residue of the following complex function $$f(z)=(z+1)^2e^{3/z^2}$$ at $z=0$. Since $e^z=\sum z^n/n!$, then $$e^{3/z^2}=\sum_{n=0}^\infty \frac{3^n/n!}{z^{...
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What am I doing wrong while finding the Laurent series of $\frac{1}{\sin(z)}$? [duplicate]

I'm trying to compute the Laurent series of $\frac{1}{\sin(z)}$ at $z_0=0$. From what I've seen on the internet this is given as $f(z)=\frac{1}{z}+\frac{z}{3!}+\frac{7z^3}{360}+...$ My attempt: $f(...
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Brauer group of the field of Laurent series with coefficients in a finite field

In a course I attended at university, we calculated the Brauer group of $\mathbb{F}_q((t))$ with $q=p^n$ , $p$ prime number and we proved it was $\dfrac {\mathbb{Q}}{\mathbb{Z}}=Br(\bar{\mathbb{F}_q}((...
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How much of a function to consider , when classifying its singularities?

If we are given a function such as $e^{z+\frac{1}{z}}=e^ze^{\frac{1}{z}},$ and we are trying to classify it's singularities. Is it correct to say, that because $e^{\frac{1}{z}}$ has a singularity at $...
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How to compute Laurent series?

when dealing with Laurent series I am confused about how to compute negative coefficients. Laurents theorem states (in my course at least) that if f is differentiable on $D(z_0, R)/ \{z_0\}$ R>0. ...
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Find the Taylor series about $z=0$ and the Laurent series about $z=-3$

Let $f(z)=\frac{10z}{z^2+z-6}$, find the coefficient of $z$ in the Taylor series of $f$ expanded about $z=0$ and state the open set in $\mathbb C$ where the series converges. Find the Laurent series ...
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Is $G(k(t))$ dense in $G(k((t)))$ for $G$ a linear algebraic group?

Let $G$ be a linear algebraic group. Let $k$ be a field, let $k(t)$ be the field of fractions of the polynomial algebra $k[t]$ and let $k((t))$ be the field of Laurent series. I'm interested in ...
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Laurent series for $f(z)=\frac{1}{\sin z}$

Since the isolated singularities are $z=k\pi, k \in \mathbb Z$, so we divided the complex plane into the disjoint annulus, i.e. $\{z: n\pi <|z|<(n+1)\pi\}, n \in \mathbb N \cup \{0\}$. On these ...
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Complex isolated singularities

I require some help with identifying and classifying the isolated singularities of complex-valued functions. For example: $f(z)=\frac{(z^3+1)}{z^2(z+1)}$, here I understand that there is a pole of ...