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
46 views

How to compute the residue of $(z^2+2z+1)\sin\left(\frac{1}{1+z}\right)$

This was an example given in my notes but all it concluded was with something about an infinite principal part. How do we compute it? we have it equal to $ \left( z + 1 \right)^2 \cdot \sin \left( ...
0
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1answer
15 views

The Laurent series expansion of 1/((z^2)(z+1))

In an example in our notes it says: Compute the Laurent series for f(z)= 1/z^2(z+1) and determine the annulus of convergence. No more information was provided. So I did it on my own by factoring it ...
3
votes
1answer
48 views

The Computation of a special kind of Laurent Series

Let $a\in\mathbb{C}$ and $k\in\mathbb{N}$, we wish to compute the Laurent Series for the function $$ f(z)=\frac{1}{(z-a)^k} $$ about $z=0$ (NOT $z=a$). So there should be two Laurent Series which are ...
0
votes
2answers
33 views

Integral along closed contour

In the Laurent series, the coefficient $$b_n = \frac{1}{2\pi i}\int_C\frac{f(z)}{(z - z_0)^{-n + 1}}dz,\qquad\left(\, n = 1,2,\ldots\,\right)$$ collapses to zero when $f(z)$ is analytic in the ...
1
vote
2answers
26 views

Compute $\int_\Gamma \frac{e^\frac{1}{z}}{z-1}dz$, where $\Gamma$ is the circle $|z-1|\le\frac{3}{2}$, positively oriented.

Compute $\int_\Gamma \frac{e^\frac{1}{z}}{z-1}dz$, where $\Gamma$ is the circle $|z-1|\le\frac{3}{2}$, positively oriented. The numerator is not analytic in $\Gamma$ so we can't use Cauchy ...
0
votes
2answers
75 views

How do I calculate the residue of $\sin(z+1/z)$?

How do I do this about $\displaystyle z=0$ ?. I tried creating a Laurent expansion and extracting it from there but I wasn't sure how to isolate the $\displaystyle 1/z$ expression. $$ \mbox{I ...
1
vote
2answers
21 views

How to figure out Laurent series expansion for $z^2 \sin(1/(z+i))$?

I am having trouble figuring out the Laurent series expansion for $z^2 \sin\dfrac{1}{z+i}$. I would really appreciate any help.
3
votes
1answer
45 views

Laurent Series of $f(z)=(z+1)\sin\frac{z^{2}+2z+5}{(z+1)^{2}}$

We are asked to find the Laurent series for the following function. $$f(z)=(z+1)\sin\frac{z^{2}+2z+5}{(z+1)^{2}}$$ Around the point $$z_{0}=-1$$ I have tried to factor the inside of sine, to no ...
0
votes
1answer
99 views

integration, laurent series, residue therorem

Evaluate the integral $\int_\gamma f(z)dz,$ where $\gamma(t)=e^{it}$, and $0\leqslant t\leqslant2\pi$. For $f(z)$ equal to: $$\dfrac{e^z}{z^3},\quad\dfrac1{z^2\sin z},\quad\tanh ...
0
votes
1answer
38 views

Residue theorem with contour integrals

I want to evaluate the integral $$ \int_{\gamma} \frac{1}{z^{2}\sin(z)} dz$$ where $\gamma(t) = e^{it}$ and $ 0 \leq t \leq 2\pi$ using the Residue theorem. I've tried expanding sin(z) with Taylor ...
2
votes
1answer
52 views

Laurent Series, region of convergence

I want to find the laurent series for $$ f(z) = \frac{z}{z^2 - (1+i)z +i} $$ in powers of $z-1$ and find the region of convergence. I am not quite sure how to do this. I know that $$ f(z) = ...
2
votes
1answer
51 views

Countour integral using residue theorem

Evaluate the integral $$ \int_{\gamma} \tanh(z) dz $$ where $\gamma(t) = e^{it}$ and $0 \leq t \leq 2\pi$. I want to do this using the residue theorem but I am unsure of how to work out the poles of ...
0
votes
1answer
66 views

Lauren expansion with different annuli

Find the Laurent expansion about $0$ of $$f(z)= \frac{1}{(z-i)(z-2)}$$ on the annuli: $0 \lt \mid z \mid \lt 1 $, $ 1 \lt \mid z \mid \lt 2$, $ 2 \lt \mid z \mid \lt \infty $. So far I have put ...
0
votes
1answer
213 views

Why are there two series representations of the natural logarithm?

On the Wikipedia article of the natural logarithm one finds two different series representations for $\ln(x)$: $\ln(x)= (x - 1) - \frac{(x-1) ^ 2}{2} + \frac{(x-1)^3}{3} - \frac{(x-1)^4}{4} \cdots$ ...
1
vote
0answers
22 views

Laurent series expansion of $f(z)= \frac{\sinh(z)}{z^3}$

Find the Laurent- series expansion of $$f(z)= \frac{\sinh(z)}{z^3}$$ With small manipulations, I came up with below thing, $$\frac{\sinh(z)}{z^3}=\frac{1}{z^2}+\frac{1}{3!}+\frac{z^2}{5!}+...$$ Is ...
2
votes
1answer
45 views

Laurent series of $f(z)=\frac{1}{z(z-1)}$ given four different conditions

Expand $f(z)=\frac{1}{z(z-1)}$ in a Laurent series valid for the follwing annular domains. $a)0\lt \vert z \rvert \lt 1 \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,b)1\le\lvert z ...
0
votes
1answer
11 views

About multiplying two essential singularity containing functions

Find analytic function such that f(z) and g(z) both have essential singularity at z = 0 but when multiplied together they have a pole of order 7. I honestly do not recognize essential singularity ...
0
votes
1answer
53 views

Why does this function residue equal 0?

$$ f(z) = \frac{e^{2z}}{(z-1/2)^{2013}} $$ Why does this residue equal 0? If I expand Laurent series, the right side will have $\dfrac{a_{2013}}{(z-1/2)^{2013}}$ $$ + \frac{a_{-2012}}{(z- ...
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0answers
40 views

Finding all Laurent series of a function

I have to find all of the Laurent series for the function $$ f(z)=\frac{z}{(z+1)(z-2)} $$ about $z=0$ I'm a little confused about the regions that I'm dealing with. I started with the partial ...
0
votes
2answers
28 views

Taylor series example

Task : Write down the McLaurin series till the 4th power ( Hope the translation from german is good :) ). $f(x) = (e^{-x} - 1)^2$ What I did , because I thought that expanding $(e^{-x} - 1)^2$ till ...
2
votes
1answer
34 views

Complex integral and Laurent series

Could you help with solving this complex integral: $$\int_C z^3\exp{\left(\dfrac{-1}{z^2}\right)} dz$$ where $C$ is $|z|=5$. I am expecting that the Residue Theorem will be needed. The answer should ...
0
votes
1answer
35 views

Laurent Series Expansion for $f(z)=\dfrac{z+2}{(z+1)(z-2)}$ in $\{1<|z|<2\}$ and $\{2<|z|<\infty\}$

I'm trying to get the Laurent Series expansion of the function stated in the title in the stated regions. My approach is as follows: We can first break up $f(z)$ using partial fractions ...
0
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0answers
43 views

Showing that there exist $C \in \mathbb{C}$ such that $g(z)=C \sin(z)$ if $g$ holomorphic & $|g(z)|\leq A|\sin(z)|$ ($A\in\mathbb{R}$)? [duplicate]

I'm trying to manipulate the sine function is some complex analysis problems (I need practice) and I've encountered two slight darker points: First, I don't understand how it can be possible (I read ...
1
vote
1answer
17 views

Laurent series expansion - inequalities of series coeffiients

Let $f:U\rightarrow \mathbb{C}$ be analytic. Suppose $\overline{A}_{r,R}(z_0)\subset U$ where $r<R$ and $r,R \in (0,\infty)$. Also, $|f(z)|\leq M$ in $\overline{A}_{r,R}(z_0)$. I need to prove the ...
4
votes
1answer
80 views

Laurent series, integral over the annulus, radii

We are given $$f = \sum_{n= - \infty} ^{\infty} a_n (z-z_0)^n \in \mathcal{O} ( \text{ann} (z_0, r, R)), \ \ 0<r<R< \infty. $$ Prove that $$\frac{1}{\pi} \int _{ann (z_0, r, R)} |f(z)|^2 d ...
2
votes
2answers
34 views

Finding the Laurent series of $\frac{1}{z^{2}(1-z)}, 1<|z|<\infty $

Finding the Laurent Series of $\frac{1}{z^{2}(1-z)}, 1<|z|<\infty $ I tried to divide both numerator and denominator by $z^2$, so that there is a term of $\frac{1}{(1-z)}$. I am tempted to ...
2
votes
2answers
62 views

Using Residue theorem

I read Book by Egorychev in Russian. I don't understand the following identity $$S_m=\frac{1}{2\pi i m}\int_{|w|=\frac12}(2+w)(1+w)^m(1+w+w^2)^{-1}w^{-m-1}dw=-\frac1m ...
1
vote
1answer
38 views

Laurent series for $f(z)= \frac{1}{ (z-i)(z+2i)}$

I'm struggling with this question. I tried to break $f(z)$ using partial fractions and modify each equation so it looks like $\dfrac{1}{1-z}$ series but that's where I get stuck. Any help would be ...
0
votes
0answers
36 views

What are the possible expansions for $f(z)$ at $0$ for disks and annuli?

For the expression $f(z)$ what are its all possible expansions (I am considering disks and annuli) around the origin and where do they converge? $$ f(z) = z + 2z^2 + 3z^3 + \ldots + nz^n + \ldots = ...
5
votes
2answers
63 views

Prove that $\zeta(4)=\pi^4/90$

I am asked to "use the calculus of residues" to prove that $$\displaystyle\sum\limits_{n=1}^{\infty} \frac{1}{n^4}=\frac{\pi^4}{90}$$ I think I can do this given the Laurent series for $\cot z$ ...
0
votes
1answer
24 views

Find Laurent series for $\frac{(z+1)}{z(z-4)^3}$ in $0<|z-4|<4$

Find Laurent series for $\frac{(z+1)}{z(z-4)^3}$ in $0<|z-4|<4$. First we perform partial fraction and we get: $\frac{A}{z}+\frac{B}{z-4}+\frac{C}{(z-4)^2}+\frac{D}{(z-4)^3}$. My first ...
0
votes
1answer
26 views

Singularities of $\frac{1}{e^{\frac{1}{z}}+2}$ and classification?

I'm considering the function $$\frac{1}{e^{\frac{1}{z}}+2}$$ Clearly, it has a unique singularity at $z=0$. I feel like its an essential but I can't find the Laurent expansion or any other way of ...
1
vote
0answers
23 views

Calculating the Laurent Series of $\tan z$

I need help calculating the laurent series of $\tan z$ around the points $z=0$, $z=\pi/2$, and $z=\pi$. How would one go about doing this? I solved an almost identical question that was "Derive the ...
2
votes
2answers
31 views

Calculate Laurent Series for $\frac{\ln z}{(z-1)^3}$ about $z=1$

Calculate the Laurent series of the function $g(z)= \frac{\ln z}{(z-1)^3}$ about the point $z=1$. Well since the singularity and the centre of the circle we are expanding about collide, I can just ...
2
votes
2answers
56 views

Complex Analysis- Finding Laurent Series

I'm a bit confused about how to calculate all the laurent series about a given point in the complex plane. I have the complex function $$ f(z)=\frac{1}{z^2(z-3)}$$ I need to find all the Laurent ...
0
votes
0answers
18 views

Residue (Laurent Series) of $\exp(\frac{1}{z}-1)$ at $z=0$

I am trying to calculate the residue of $\exp(\frac{1}{z}-1)$ at $z=0$. My professor suggested me to use $\exp(\frac{1}{z}-1)=e^{-1}\exp(\frac{1}{z})$ I don't know how this relation comes from.
2
votes
2answers
51 views

Computing the Laurent series of $\frac{1}{z^2 + z - 6}$ in the region $2 < |z| < 3$

Make a series expansion of $f(z)=\dfrac{1}{z^2+z-6}$ valid in the region $2<|z|<3$. By partial fractions, $$f(z) = \frac{1}{(z-2)(z+3)} = \frac{1}{5(z-2)}-\frac{1}{5(z+3)}.$$ From here, ...
0
votes
1answer
45 views

complex analysis: Laurent series of $\frac{1}{z^2(z - 3)^3}$

Find the Laurent series about the point $z = 3$ of $$ \frac{1}{z^2(z - 3)^3}. $$ Since we are finding the Laurent series about $z = 3$, we have $0 < |z - 3| < 3$. Let $u = z - 3$. Then ...
0
votes
0answers
27 views

Laurent Series 1/(z-3i)

Apologies if this is a simple question. I've just read a chapter on Laurent series which seems to indicate that in practice we don't calculate the coefficients of a Laurent series using the integral, ...
0
votes
1answer
15 views

Problem from complex analysis regarding series representation

The question asks Represent the function $f(z) = \frac{z+1}{z-1}$ by its Maclaurin series for $|z|<1$ and its Laurent series for $1<|z|<\infty$. The answers the book gives are: Maclaurin ...
4
votes
0answers
40 views

Manipulation of Taylor/Laurent series

I have a question regarding how to expand a given rational function into its Taylor/Laurent series representation. Suppose we are given the function $$f(z) = \frac{z}{(z-1)(z-3)},$$ and are asked to ...
2
votes
2answers
124 views

Laurent-series expansion of $1/(e^z-1)$

Find the Laurent series for the given function about the indicated point. Also, give the residue of the function at the point. $$ \frac{1}{e^z - 1} $$ at $z_0=0$(four terms of laurent series). I ...
1
vote
1answer
35 views

Expanding $(x+yi)^c$ to series

I need to evaluate a complex expression $f(x,y)=(x+yi)^c$, where $x,y,c\in\mathbb{R}$, in double-precision arithmetic on the GPU. It is done in a usual way, i.e., computing $\exp(c \log(x + yi))$. ...
1
vote
1answer
31 views

What is the Laurent series of $\exp(\frac{1}{z})\exp (2z)$?

What is the Laurent series of $\exp(\frac{1}{z})\exp (2z)$ ? I know how to do the Laurent series of $ \dfrac{1}{z(z+5)}$ (I make use of the geometric series of $\dfrac{1}{1-z}$) but I don't know how ...
0
votes
1answer
40 views

Uniqueness of Laurent series expansion of $f(z) = \frac{1}{z-1} + \frac{1}{1-z}$.

We first have $$ \frac{1}{z-1} = \frac{1}{z}\frac{1}{1-1/z} = \sum_{n=-\infty}^{-1} z^n.$$ We also have $$\frac{1}{1-z} = \sum_{n=0}^\infty z^n.$$ Now here is the perceived issue. Since $f(z)$ is ...
0
votes
1answer
38 views

Negative index coefficients of Laurent series for 1/sin(z)

Given $f(z) = \dfrac{1}{\sin(z)}$ a) Give singularities b) Determine coefficients $a_{-1}$ and $a_{-3}$ of the Laurent series So I thought: a) $n \pi$, where $n$ is an integer b) ...
1
vote
0answers
42 views

Difference between expanding $\frac{1}{1-x}$ around $x=1$ and $\frac{1}{1-e^x}$ around $x=0$

The title pretty much gives it away already, I'm trying to find the difference between Laurent expanding $\frac{1}{1-x}$ around $x=1$ and $\frac{1}{1-e^x}$ around $x=0$. The first expansion does not ...
1
vote
1answer
41 views

Question about laurent series

Find laurent series for $f(z)=\dfrac{1}{z^2-1}+\dfrac{1}{z(z-1)};z_0=0$ that converges in $0<|z|<1$. I tried to find the solution for the first fraction like this. \begin{align} ...
0
votes
1answer
29 views

Laurent series at z = 0

I want to determine the Laurent series around z = 0 (so a Maclaurin series I think) of the following function $f(z) = 4/(z^2+2z-3)$ which converges in $z = 1 + i$ I can rewrite the function as ...
1
vote
3answers
52 views

Find the Laurent series of $f(z) = \frac{1}{z-2} + \frac{1}{z-3}$ for $2 < |z| < 3$ and for $|z| > 3$

Find the Laurent series of $f(z) = \frac{1}{z-2} + \frac{1}{z-3}$ for $2 < |z| < 3$ and for $|z| > 3$. Is the first step here to notice that $$ \frac{1}{z-2} + \frac{1}{z-3} = ...