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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2
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2answers
37 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 ?
-1
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0answers
30 views

How find Laurent Series for the function [closed]

about z = 0, valid for |z|<2 I do not understand this function Laurent series, which method will be used to implement the function of Laurent series
0
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1answer
18 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
votes
0answers
30 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 ...
0
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0answers
17 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 ...
0
votes
1answer
29 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?
1
vote
2answers
32 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 ...
0
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0answers
18 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 ...
0
votes
1answer
27 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 ...
1
vote
1answer
61 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
votes
1answer
53 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 $$f(z)=\frac{1}{z^4+z^2}=\frac{1}{z^2}-\frac{1}{2i(z-i)}+\frac{1}{2i(z+i)}.$$ ...
2
votes
1answer
30 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 ...
0
votes
2answers
27 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 ...
3
votes
0answers
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 + ...
0
votes
0answers
45 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 $b_n$ ...
2
votes
2answers
50 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 $w$: ...
2
votes
3answers
43 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: ...
0
votes
1answer
68 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?
2
votes
2answers
53 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} ...
0
votes
2answers
47 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 ...
0
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0answers
35 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 ...
0
votes
1answer
32 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| = ...
0
votes
0answers
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
votes
1answer
61 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
votes
1answer
75 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 ...
0
votes
1answer
38 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 ...
0
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0answers
32 views

On exponentials of formal power series

I am having a very hard time trying to understand the following paper by M. Kontsevich (http://arxiv.org/pdf/1109.2469v1.pdf), and since I cannot really find a way out by myself, I here to seek some ...
0
votes
2answers
57 views

Laurent Series Expansion computing terms

I need to compute the -5th term to the 5th term of the Laurent expansion of $(\cos(z))^2/\sin(z)$. I know that I can make this into $\csc(z)-\sin(z)$ but I wouldn't know what to do with the $\csc(z)$ ...
2
votes
3answers
83 views

The Laurent series of $(1/(z^2+1)^2$ in the annulus $0<|z-i|<2$

I can't figure it out how to solve this problem: Find the Laurent Series of the function $$f(z)=\frac{1}{(z^2+1)^2}$$ valid in $A=\{z \in \mathbb{C} : 0 < |z-i|<2\}$ I think that it is ...
2
votes
2answers
50 views

Taylor series for the function $f(z) = \frac{1}{(z-5)(z-7)}$ on a disc centered at point $z_0=3$

I started by expressing the function as sum of two fractions using partial fraction decomposition to get $\frac{-1}{2(z-5)} + \frac{1}{2(z-7)}$ However I could only then end up writing that as the ...
2
votes
2answers
47 views

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

I have to find the Laurent series of $f(z)=\frac{1}{1-z}-\frac{1}{2-z}$ on $D_1(0)=\{z\mid |z|<1\}$, on $C_{1,2}(0)=\{z\mid 1<|z|<2\}$ and on $C_{2,\infty }(0)=\{z\mid |z|>2\}$. For ...
2
votes
3answers
43 views

Does this qualify as a Laurent series?

Evaluate the Laurent series around the singularity at $ z_0 = 3$. $$ \frac{1}{z^2(z-3)} $$ I can apply the geometric series as follows: ...
1
vote
2answers
33 views

Laurent series expansion for $\lvert z\rvert >1$.

I have a simple complex function like this: $$\frac{z+1}{z-1}$$ When I expand it by its Maclaurin series: $$\frac{z+1}{z-1} = \frac{z-1+2}{z-1} = 1 - \frac{2}{1-z} = 1 - 2\sum_{k=0}^{\infty}z^{k} ...
0
votes
1answer
31 views

laurent series convergence

Find the Laurent Series representations in powers of $z$ for i) $\frac{ \cos{z}}{z}$ ii) $z^4 \cosh{\frac{1}{z^2}}$ Where do they converge? I found the Laurent Series for each of the functions, ...
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vote
3answers
28 views

Laurent series with nonzero negative infinity term

I was told that the most general case of a Laurent series has its negative index at negative infinity instead of at -m for some integer m. Can someone give an example of a function with this form? ...
3
votes
3answers
240 views

How to find Laurent series Expansion

$f(z)$ is defined like this: $$ f(z) = \frac{z}{(z-1)(z-3)} $$ I need to find a series for $f(z)$ that involves positive and negative powers of $(z-1)$, which converges to $f(z)$ when $0 \leq |z - 1| ...
1
vote
2answers
67 views

Finding the Laurent series of a given function

How do I find the Laurent series expansion for the function $\frac{z}{(z+1)(z+2)}$ which converges for $1 < |z| < 2$ and diverges elsewhere. I have done the partial fractions for $f(z) = ...
2
votes
1answer
47 views

Prove that for a holomorphic function $f$ on $\mathbb{C}$, if $-f(z) = f(\frac{1}{z})$, then the residue of $f$ at 0 is 0.

Prove that for a holomorphic function $f$ on $\mathbb{C}$, if $-f(z) = f(\frac{1}{z})$, then the residue of $f$ at 0 is 0. Note: $f$ isn't defined on 0. I'm having a bit of trouble getting this ...
0
votes
1answer
13 views

Finding a certain residue

I wish to calculate the value of \begin{equation} \text{Res}\big((z+\pi/4)^2\tan(z);z=\pi/2\big)=\frac{1}{2\pi i}\int_C (z+\pi/4)^2\tan(z)\text{d}z, \end{equation} where $C$ is any counter-clockwise ...
0
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2answers
53 views

Power Series (Laurent Series)

I need some help with this exercise: I need to obtain the power series development of this function: $$f(z)=\frac{\cos(z+1)}{(z^2-1)z}$$ Centered in $z_0=-1$ and valid in $z_1=\frac{1}{2}-i$ I know ...
2
votes
1answer
29 views

Decomposition of resolvent in projections

I am reading the book Perturbation theory for linear operators from Kato. He defines (§5 Section 3) for an operator $T : X\to X$ on a finite Banach Space the resolvent as $$ R(x) = (T- x)^{-1}.$$ ...
0
votes
1answer
48 views

Laurent Series of operator-value function

I am reading the book 'Perturbation Theory for Linear Operators' from Kato. He defines in his Book (Chapter 1 §5) the resolvent for some operator $T: X \to X$ on a finite Banach space $X$ as a ...
3
votes
1answer
51 views

How to indentify the power series I need to use?

Let $$ f(z) = \frac{1}{(z - 4)(z + 8i)} $$ a) Find the domains where f(z) is valid b) Find its power series at such domains Considering three singularities, I believe the domains are: $$ D_{1} = ...
1
vote
1answer
54 views

Problem with Laurent series

I am trying to find a Laurent series for $\cos(\frac{1}{z})z$. I know that $$ \cos(1/z) = \frac{1}{2} e^{(-i/z)}z + \frac{1}{2} e^{(i/z)}z = z \sum_{n=0}^\infty \frac{(-1)^n ...
1
vote
1answer
64 views

Proof of Laurent series co-efficients in Complex Residue

Am trying to see if there is any proof available for coefficients in Laurent series with regards to Residue in Complex Integration. The laurent series for a complex function is given by $$ f(z) = ...
0
votes
0answers
29 views

To obtain an explicit expression of f1(z) from the Laurent decomposition of f(z)

How do you do to obtain an explicit expression for f1(z) where the function is f(z)=tan z in the annulus {3<|z|<4}? Let f(z)=f0(z) + f1(z) be the laurent decomposition of f(z), so that f0(z) is ...
3
votes
0answers
64 views

Finding Laurent Series of a function

I've been assigned to write a computer program which then calculates the Laurent series of a function. Of course I'm familiar with the concept, but I've always calculated the Laurent series in an ad ...
2
votes
2answers
92 views

Laurent series expansion, can one simplify this?

I have to expand $f(z)=\frac{z-1}{(z^2+1)z}$ in an annulus $R(i,1,2)$. $$f(z)=\frac{1}{z-i}\frac{1}{z+i}-\frac{1}{z-i}\Big(\frac{i}{z+i}-\frac{i}{z}\Big)$$ ...
1
vote
5answers
85 views

Approximate $\coth(x)$ around $x = 0$

I'm trying to approximate $\coth(x)$ around $x = 0$, up to say, third order in $x$. Now obviously a simple taylor expansion doesn't work, as it diverges around $x = 0$. I'm not quite sure how to ...
1
vote
1answer
32 views

Laurent series of $z/(z-6i)^9$

I'm really confused by the Laurent series, and need a little help with this one: $ f(z) = \frac z{(z-6i)^9} $ around $z = 6i$ The partial fraction method doesn't seems adapted for this problem. I ...