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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Largest number for which a laurent series converges

For part $(a)$ I got summation from $\sum^{\infty}_{n=0}(-1)^n\frac{z^{2n}}{(2n+1)!}$ Is this correct? Could someone explain how to do part (b) because I have no idea where to start Thanks
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1answer
14 views

Calculate laurent series in the following regions [on hold]

Can someone help me with this? Not sure where to start... I've split it up into partial fractions but not sure what to do now Thanks
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46 views

Find the largest number for which a Laurent Series converges

Not really sure where to start with this. Any help greatly appreciated
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1answer
34 views

Laurent Series expansion of $f(z)=(z-1)sin{1\over z}$

I need to find the Laurent series expansion of the function: $$f(z)=(z-1)sin{1\over z}$$ about $$A= z ∈ \Bbb C : 0<|z|<∞ $$ Any help would be appreciated!
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1answer
24 views

How to obtain the Laruent expansion of gamma function around $z=0$?

I want to prove, the laurent expansion of gamma function. \begin{align} \Gamma(z) = \frac1z-\gamma+\frac12\left(\gamma^2+\frac {\pi^2}6\right)z-\frac16\left(\gamma^3+\frac {\gamma\pi^2}2+2 ...
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60 views

How to find Laurent series expansion for $\frac{e^z}{(z+1)^2}$

Find the Laurent series expansion for $\frac{e^z}{(z+1)^2}$ for $\lvert z \rvert > 1$. I know how to find the Laurent series expansion for $\lvert z \rvert < 1$, which is ...
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64 views

$f(z)=\frac{1}{z^2-2}$ laurent series?

$f(z)=\frac{1}{z^2-2}\quad\quad \left \{ z\in \mathbb{C}:0<\left | z-1 \right |<1 \right \}$      Laurent series? My Try ...
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23 views

What is the genral form of the laurent exspansion of $\frac{1}{(z-\alpha)^n}$

This is a question from a text book (Saff and Snider, Complex analysis for matemetics science and engineering). Obtain the general formula for the laurent expansion of $$ f_n(z) = ...
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22 views

Obtain the Laurent Series expansion of these two complex functions.

I have obtained the Laurent Series expansion for the first one. I am stuck on the answer for the second one as it doesn't seem as straight forward method wise. Now, I know that $e^z$ can be ...
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22 views

$n(\gamma_1,0)=n(\gamma_2,0)$ iff $\int_{\gamma_1}f(z)dz=\int_{\gamma_2}f(z)dz$

Use Laurent decomposition to prove the next equivalence, assuming that $\gamma_1,\gamma_2$ are closed curves such that they live in the annulus $r<|z|<R$: ...
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39 views

Show that an analytic function in a strip has a complex Fourier expansion

I am a self-studier, and this is a homework problem from a course in Complex Analysis. First, let me give a plug for the course as it is outstanding. Taught by Jerry Shurman at Reed. Great lecture ...
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53 views

Find the Laurent series about $z=i$

Let $g(z)=\cfrac{3z+1}{(z-i)^3}$. Find the Laurent expansion of $g$ about $z=i$. My idea is first to find the Laurent series of $\cfrac{1}{z-i}$ about $z=i$, and then diferenciate, but I have problem ...
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1answer
27 views

Question about finding Laurent Series over closed region and classifying singularity

Represent $\sin(\pi x/(x+1))$ Laurent Series about the region $0<|x+1|<2$: Its true that $$\sin(x)=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\cdots$$ So the $$\sin(\pi x/(1+x))=\sum (-1)^{n-1} \frac{(\pi ...
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1answer
33 views

Expand $(e^{2x}-1-2x)/x^5$ into Laurent Series on 0<|x|<$\infty$ and classify its singularity

I guess I'm having difficulty with this because its not in the form of a polynomial expression, which is what I've been taught. Nevertheless here's what I did: I know that the expansion for ...
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1answer
37 views

Prove that the integral of $sin(z)/(z^2+4z+5)$ from negative to positive infinity is $-\pi sin(2)/e$

I think I've made the problem a lot nastier than it supposed to look. Here's what I have so far. First notice that $(z^2+4z+5)$ is equivalent to $(z^2+4z+4)+1$ so our singularities are -2-i and ...
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34 views

Laurent series for $\frac{1}{z(z-1)}$ in the domain $1 < |z-2| < 2$

Expand $f(z) = \cfrac{1}{z(z-1)}$ in a Laurent series valid for $1 < |z-2| < 2$ First of all I wrote the fraction in partial fraction form: $\cfrac{1}{z(z-1)} = \cfrac{-1}{z} + ...
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38 views

How should I expand $\frac{1}{z(e^z-1)}$ to find the residue and order at the pole z = 0?

$\frac{1}{z(e^z-1)}$ is the function I want to expand. I tried using the expansion for $e^z$ and got $$\frac{1}{z^2+z^3/2!+z^4/3!+...}$$ Can I put this fraction into the $b_n/(z-z_0)^n$ form, or did I ...
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1answer
19 views

Laurent series, function representation

Write the Laurent series for the function $f(z)=\frac{1}{1+z}$ $1<|z|<\infty$ I did $$\frac{1}{1-z}=\sum_{i=0}^\infty z^n\rightarrow \frac{1}{1+z}=\sum_{i=0}^\infty (-1)^nz^n$$ Is it right? ...
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30 views

Laurent series , function representation

Write the Laurent series around zero for the entire function $f(z)=z^2e^{3z}$ I'm a little confused on how to represent the complex functions by series, as I did in the calculation of real functions, ...
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16 views

Laurent series around a certain singularity

Find the Laurent series and determine the precise region of convergence of $f(z)=\dfrac{\sin z}{z-\pi/4}$ where $z_0=\pi/4$? How do I solve this?
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30 views

Laurent series and residues $f(z^2)$

If $f(z)$ is analytic in $ 0 < |z| < n$, what is the residue of the function $f(z^2)$ at $z = 0$? Attempt If $f(z)$ has a pole of order n at $z=0$ it seems like the residue of $f(z^2)$ would ...
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62 views

Laurent expansion of 1/sin(z)

I've been looking for a concise explanation of how to obtain the Laurent expansion for $$\frac{1}{\sin(z)}$$ My attempt at it has me confused by it pretty quickly. I start with the knowledge that ...
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19 views

Laurent series expansion for powers of n?

I wish to expand the function: $$\dfrac{e^z}{z^n-c^n}$$ about the point $z_0=c$, where c is a constant greater than 0 and n is greater than 2. So I have that $e^{z-c}$ expands to ...
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1answer
52 views

Expand a function into a Laurent series about a point?

Take the function $f(z)=(z^2+3z+2)e^\frac{1}{z+1}$ We want to expand this into its Laurent series about $z_0$=-1. Alright, so I'm a little confused. This converges everywhere but -1, which throws me ...
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2answers
46 views

Laurent series of $e^{e^{\frac{1}{z}}}$ around $z=0$

Actually I need only the $res(f;0)$ where $f = e^{e^{\frac{1}{z}}}$ I thought of finding the Laurent series of $e^{e^{\frac{1}{z}}}$ around $z=0$ Any other Ideas if you have ?
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33 views

Find the Laurent series for $f(z)=\frac{e^z}{(z-i)^4}$ at $z=i$.

Find the Laurent series for $f(z)=\frac{e^z}{(z-i)^4}$ at $z=i$. What I was thinking of using $e^z=\sum_{i=0}^\infty \frac{x^i}{i!}$. But from there I am not sure what to do?
2
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40 views

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 ...
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21 views

Laurent series about which point?

I have been given the question: "Find the laurent series of $1/(1-z^3)$". No other information. The question was posed in the process of finding the series' residual and in the answer I can see that ...
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1answer
33 views

laurent series expansion about $z=0$

using the Laurent expansion i got the answer to be $$-(z+1)\sum_{n=0}^\infty \frac{z^{n-1}}{2^{n+1}}$$ however, I've got a feeling I've made a mistake somewhere?
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2answers
37 views

residue theorem, laurent espansion, singularities and order of poles

for this question i have found that for part (i) at $z=0$ we have an essential singularity however, I'm not sure how to solve for the residual? (ii) at $z=0$ we have a pole of order $2$, and i ...
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22 views

laurent expansion question

I'm quiet stuck on how to solve this using the Laurent expansion as the denominator is simply $z^3$ rather than something like $1-z$ separate question! I'm not too sure how to solve this, I've ...
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1answer
31 views

Laurent series question

I'm currently trying to solve this question, I have work out the solution to be $$-\frac{1}{2} \bigg(\sum_{n=0}^{\infty} \frac{z^n}{3^{n+1}} + \sum_{n=1}^{\infty} \frac{1}{z^{n}} \bigg) $$ could ...
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1answer
35 views

Laurent series of $\frac{1}{\sin^2z}$ around $0$?

I tried to expand $\frac{z^2}{\sin(z)^{2}}$ using Taylor expansion, but the coefficient involved some limit of $\frac{0}{0}$ and was really difficult to calculate. (I tried to convince myself the ...
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1answer
70 views

Find the Laurent expansion for $f(z)=\frac{\exp{1/z^2}}{z-1}$ about $z=0$.

Find the Laurent expansion for $f(z)=\frac{\exp{(1/z^2)}}{z-1}$ about $z=0$. I was able to determine the series for each of the factors. We have ...
3
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1answer
71 views

Find the Laurent Expansion of $f(z)$

Find the Laurent Expansion for $$f(z)=\frac{1}{z^4+z^2}$$ about $z=0$. I have found the partial fraction decomposition $$f(z)=\frac{1}{z^4+z^2}=\frac{1}{z^2}-\frac{1}{2i(z-i)}+\frac{1}{2i(z+i)}.$$ ...
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1answer
32 views

Prove $f$ analytic on $D(z_0;R)\setminus\{z_0\}$ implies $\exists M, f(D(z_0;r)\setminus\{z_0\})\supset\{z\in\mathbb{C}:|z|>M\}$

Suppose $f$ is analytic on $D(z_0;R)\setminus\{z_0\}$, and $z_0$ is a pole of $f$. Prove that for any $r\in(0,R)$, there is $M\in(0,\infty)$ such that ...
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2answers
29 views

Prove $f|_{U_0}$ is $m$-to-$1$ except at $z_0$.

Let $f$ be analytic on a domain $U$, $z_0\in U$, and $w_0=f(z_0)$. Suppose that $\mbox{ord}_{z_0}(f-w_0)=m\in\mathbb N$. Prove that there is an open set $U_0$ with $z_0\in U_0\subset U$ such that ...
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46 views

Finding the coefficients of the Weirestrass $\wp$ function.

I am trying to find the coefficients of the $\wp$-function. Right now I have the Laurent series about the pole $ z = 0$: $$\wp(z) = \frac{c_{-n}}{z^n} + \cdots + \frac{c_{-1}}{z} + c_0 + c_1 z + ...
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50 views

Laurent Series Expansion for $\sin(\frac{1}{z})$

I am having trouble with Laurent series expansions. I am supposed to find the Laurent Series Expansion for $\sin(\frac{1}{z})$ around $0<|z| < \infty$. I know the definitions of $a_n$ and $b_n$ ...
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2answers
53 views

Laurent series of $f(z)$ with squared linear denomimator

I want to find the Laurent expansion of $$f(z)=\frac{z-1}{z^2}, \qquad |z-1|>1.$$ Since the annulus is described for $z-1$, I substitute $w=z-1$ to use a geometric series in terms of $w$: ...
3
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3answers
50 views

Find the Laurent series of an indeterminate function

$$ f(z) = \frac{z}{(\sin z)^2} $$ at $z_0 = 0$ (for the first four terms). So I thought I knew what to do, but I don't. Since it appears to be an indeterminte form, could I by L'hopital turn it into: ...
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1answer
70 views

How to find Laurent series of exp(1/z+z)

My approach was to expand both $e^{1/z}$ and $e^{z}$, and multiply them together, but it seems like a lot of work, is there any shortcut that I can use in solving this problem?
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2answers
55 views

What method was used here to expand $\ln(z)$?

On Wikipedia's entry for bilinear transform, there is this formula: \begin{align} s &= \frac{1}{T} \ln(z) \\[6pt] &= \frac{2}{T} \left[\frac{z-1}{z+1} + \frac{1}{3} \left( \frac{z-1}{z+1} ...
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50 views

Laurent Series Expansion about $z=1$

If $p(z)=(z-1)^3(z-2)^4$ then how would I find the Laurent Expansion about $z=1$ for $1/p(z)$? I am having trouble computing expansions when it's not about $z=0$ so could someone walk me through the ...
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36 views

Residue of complex exponential

I need to find the residue of $e^{\frac 1{1-z}}$ using Laurent series. How would I manipulate the function to make it easier? I need to find all singularities and the corresponding residues. I believe ...
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1answer
33 views

A question about Laurent series

In deriving Laurent series centered at, say $0$, there are two contours, -- concentric circles with radii $R_{1}, R_{2}$ s.t. $0 < R_{1} < R_{2} < \infty$. The integral on $\{z: |z| = ...
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25 views

Singularities and Residues [Demonstration]

How could I solve the following problem: "Consider $F$, an analytic function, so that $$f(z)=F(\frac{1}{z-1})$$ has a pole. Demonstrate that F(z) is a polynomial." I know that an analytic function ...
3
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1answer
83 views

Prove Laurent Series Expansion is Unique

Suppose that $f$ is holomorphic on $A=\{r<|z|<R\}$, where $0\le r<R\le \infty$. Suppose that there are two series of complex numbers $(a_n)_{n\in{\mathbb Z}}$ and $(b_n)_{n\in\mathbb Z}$ such ...
2
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1answer
84 views

Laurent series of $f(z) =\frac1{(z−1)(z−2)}$

Find the Laurent series of the function $$f(z) =\frac1{(z−1)(z−2)}$$ in each of the following domains: (a) $\{z ∈C : 1 < |z| < 2\} $ (b) $\{z ∈C : 0 < |z−1| < 1\}$ I am very new to ...
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1answer
47 views

Is there real analogue for Laurent series?

For example, $$\frac{3}{z^2-5z+4}$$ has a Laurent series expansion on the angular region $1<|z|<4$. Does the real function $$\dfrac{3}{x^2-5x+4}$$ have some sort of Laurent series on ...