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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3
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1answer
42 views

Singularity type of $\frac{1}{z} e^{-\frac{1}{z^2}} $

I've been asked to compute the singularity type of $f(z) := \frac{1}{z}e^{-\frac{1}{z^2}} $. Here's my reasoning: $$ \frac{e^{-\frac{1}{z^2}}}{z} = z^{-1} \sum_{n=0}^\infty \big( -z^{-2} \big)^n ...
0
votes
2answers
70 views

The Laurent Series of $\dfrac{e^z}{z^2-1}$

The Laurent Series of $\dfrac{e^z}{z^2-1}$ At $z=1$ As we seek for powers of $z-1$, note that: $$e^z=e\cdot e^{z-1}=e(1+(z-1)+\dfrac{(z-1)^2}{2!}+\dfrac{(z-1)^3}{3!}+...)$$ So: ...
2
votes
2answers
45 views

Asymptotic Expansion of $\ f(x)=(1-\beta \frac{ log(log(x))}{log(x)})^{\beta}$

So I got this function and I'm looking for an asymptotic expansion for different values of$\ \beta > 1 $ $\ f(x)=\left(1- \beta \frac{\log \left( \log(x) \right)}{\log(x)} \right)^{\beta}$ as $\ x ...
-1
votes
1answer
16 views

Find the Laurent's series [closed]

Find the Laurent's series of $$f (z) = \frac1{z(1-z)^2}$$ $0<|z|<1$ $|z-1|<1$ Please help me.
0
votes
1answer
24 views

How do I write a terminating series representation of $_2 F_1(p, n+1, n+2, x)$

How do I find a terminating series representation of the hypergeometric function $_2 F_1(p, n+1, n+2, x)$, for real $p \in \mathbb{R}$ but $n \in \mathbb{Z}$, $n\geq0$? Mathematica gives (...
3
votes
2answers
50 views

Is there a simpler way to compute the residue of a function at a pole of order 3?

The function $$\frac {1}{z^2(e^{i2\pi z}-1)}$$ has a triple pole at z = 0. To compute the residue of f at z = 0, I can compute the Laurent expansion of f about z = 0, and then read off the ...
0
votes
1answer
37 views

Expanding a function into a series

I am trying to follow a proof in QFT notes, however I am unable to follow this step - it's basically Laurent/Taylor expansion but I have very little experience with it. It's claimed that: ...
2
votes
3answers
132 views

Help with this limit?

I am trying to focus on the limits of functions with similar series expansions and I stumbled on this. ...
0
votes
2answers
14 views

Lower bound on Indexing set of Formal Laurent Series

For a formal Laurent series defined over a ring R, we require that the indexing set is finitely bounded in the negative direction, or equivalently that the sequence of coefficients of R terminates at ...
0
votes
1answer
40 views

Finding the Laurent series of $f(z)=\frac{1}{(z−1)(z−2)}$ for $R =\{z∣0<|z|<1\}$ [closed]

Let $f(z)=\frac{1}{(z−1)(z−2)}$ and let $R =\{z∣0<|z|<1\}$.
1
vote
1answer
28 views

The Laurent series around $z=0$ of the function $f(z) = \frac{z}{(z-i)(z-2)}$ in the annulus $A(0,1,2)$

What I got so far: $$ \frac{z}{(z-i)(z-2)} = \frac{z}{(2-i)(z-i)} + \frac{z}{(2-i)(z-2)} $$ which is equal to $$ \frac{z}{(2-i)(z-i)} + \frac{z}{(2-i)(z-2)} = \frac{z}{(2-i)z + 1-2i} + ...
1
vote
2answers
88 views

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

I am having difficulties finding Laurent series of the above function, around these two domains: $$0<|z-1|<1$$ and $$|z-1|>1$$ The function $f(z)$ takes the form ...
0
votes
1answer
19 views

Determine a meromorphic function satisfying certain conditions

Suppose $f$ is meromorphic on the Riemann sphere, and suppose also that $f(0) = 0$, $f(-1) = 2$, $f(3) = 3$, $f$ has a simple pole at $1$ with residue $1$, and $f$ has a triple pole at $2$ with ...
3
votes
1answer
57 views

Find the Laurent series about $z=0$

Let $f(z)=\cfrac{e^{-3z}}{z^2(z-2)^2}$, find the Laurent series about $z=0$. On the region $0<|z|<2$, I get $\cfrac{1}{(z-2)^2}=\displaystyle\sum_{n=1}^{\infty}\cfrac{nz^{n-1}}{2^{n+1}}$, ...
6
votes
1answer
68 views

When proving that f(z) is a polynomial, is it enough to consider just one point instead of keeping z arbitrary?

I think so - but I'd rather ask the MSE community too. Say I am given the bound |f(z)| < $|z|^3$, and that f is entire. Show f must be a polynomial. I used Cauchy's Integral Formula for ...
2
votes
0answers
23 views

How come the definition of analytic continuation doesn't require the smaller and the bigger open subsets to be connected?

The reason that is making me think that these subsets should be connected / simpled connected is because I think that the Taylor disks of convergence of f and F, which is the continuation of f to the ...
3
votes
1answer
61 views

How to solve this limit using laurent series?

$$\lim_{x\to\infty}\left(\left(\frac{x^2+5}{x+5}\right)^{3.7}+\left(\frac{x^3+5}{x+5}\right)^{1.6}\right)^{20/37}-\left(\left(x-5\right)^{3.7}+(x^2-5x+25)^{1.6}\right)^{20/37}=60$$ It is possible to ...
0
votes
0answers
17 views

Laurent series of a logarithmic derivative

Actually, the problem I have to solve is to find the number of zeroes of $f(z)=z^2-2+\frac{z-8}{z^2+2}$ within $D=\{z\in\mathbb C : 2<\left| z \right| <3 \}$. As $f(z)$ is analytic in $D$, I ...
1
vote
1answer
45 views

How to find Laurent Series for $z/(z-1)(z+4)$

How do I find the Laurent series for $$\frac{z}{(z-1)(z-4)}?$$ on: i) $0<|z-1|<5$; ii) $5<|z-1|$. I broke it up into $$\frac{1}{5}\left(\frac{4}{z+4}+\frac{1}{z-1}\right)$$ but now I am ...
1
vote
0answers
13 views

Laurent polynomial regression?

Polynomial regression is a common way of doing curvilinear regression. It is common to also use the inverse transform x^-1 (http://pareonline.net/getvn.asp?v=8&n=6). One can extend the concept ...
13
votes
3answers
581 views

Limit approach to finding $1+2+3+4+\ldots$

When exploring the divergent series consisting of the sum of all natural numbers $$\sum_{k=1}^\infty k=1+2+3+4+\ldots$$ I came across the following identity involving a one-sided limit: ...
1
vote
4answers
124 views

How does one calculate: $\left(\frac{z}{2!}-\frac{z^3}{4!}+\frac{z^5}{6!}-\cdots\right)^2$

How does one calculate: $$\left(\frac{z}{2!}-\frac{z^3}{4!}+\frac{z^5}{6!}-\cdots\right)^2$$ Is the best way to just take the first term times the following two, and the second two times the next two ...
2
votes
2answers
48 views

What is the power series expansion for Riemann-Zeta at $0$?

What are the first few terms of the Laurent series expansion of $\zeta(0)$? It gets mentioned here but they only show the first term and I am kind of confused on how they got $-1/2$.
1
vote
1answer
45 views

How to determine the Laurent expansion of $\tan{z}$ around $z=0$ that is convergent in $z=\pi$

I want to determine the Laurent expansion of $\tan{z}$ around $z=0$ that is convergent in $z=\pi$ (only the first couple of terms). Now I know that if $\sum_{n=-\infty}^{\infty}c_nz^n$ then ...
1
vote
1answer
31 views

Laurent series calculation(they seem to calculate it without Laurent series?)

Laurent sreies expansion of the function $f(z)=z^{-1}\sinh(z^{-1})$ about the point $0$. I thought I was meant to use this: $$f(z)=\sum_{n=0}^\infty a_n(z-z_0)^n + \sum_{n=1}^\infty ...
1
vote
1answer
27 views

How to practically classify singularities in complex analysis?

I am having trouble developing an intuition around the different types of singularity in complex analysis. The types of singularity that I am aware of are: Poles - These arise at $a_{0}$ when ...
0
votes
1answer
46 views

Laurent Serie of the function $f(z)=e^{z+1/z}$ around zero and prove: $\dfrac{1}{2\pi}\int_0^{2\pi}{e^{2\cos\theta}\cos{n\theta}}$

How to find the Laurent Serie of the function $f(z)=e^{z+1/z}$ around zero. Then, show that: $$\dfrac{1}{2\pi}\int_0^{2\pi}{e^{2\cos\theta}\cos{n\theta}}=\sum_{j=0}^{\infty}{\dfrac{1}{(n+j)!j!}}$$ ...
0
votes
1answer
27 views

Laurent Series of $\frac{z+1}{z(z-4)^3}$ in $0<|z-4|<4$

Find the Laurent Series of $\displaystyle \frac{z+1}{z(z-4)^3}$ in $0<|z-4|<4$ I thought about doing partial fraction decomposition first, so I'd have $\displaystyle ...
1
vote
1answer
45 views

Taylor expansion of fraction

I am trying to Taylor expand the function $f(x) = \frac{x}{x+3}\frac{1}{x-2}$ aound the point $x_0 = 2$. Clearly, the last factor explodes around this point, so I will try and expand that term. ...
2
votes
1answer
45 views

Find principal part of Laurent series

Find principal part of Laurent series: $$f(z)= \frac{1}{\sin z + \sinh z - 2z}$$ I calculate it and I have something like this: $$\frac{1}{\dfrac{2z^5}{5!}+\dfrac{2z^9}{9!}+\cdots}$$ and don't know ...
2
votes
2answers
61 views

Laurent series of cotangent

Compute the principal part of the Laurent series of $\cot(\pi z)$ on $1<|z|<2$. EDIT: After using either of the approaches below, we get that the principal part is equal to $$ \frac 1 \pi ...
1
vote
1answer
51 views

Finding the Laurent representation of a complex function

How can i find the Laurent representation fot the function: $$f(z)=\dfrac{1}{1-z^2}+\dfrac{1}{3-z}$$ In the region of: a) $\{z\in\mathbb C:1<|z|<3\}$ b) $\{z\in\mathbb C:1<|z-2|<3\}$
0
votes
1answer
40 views

Laurent series: how to join the 2 sums for $f(z)= \frac{1}{(z-1)(z+1)}$ about z = 1 for $0 < |z − 1| < 2$

We are to find the Laurent series for f(z) about $z = 1$ for $0 < |z − 1| < 2$: $f(z)= \frac{1}{(z-1)(z+1)}$ Assumptions: $|\frac{z−2}{1}| < 1 ⇔ |z − 1| < 2$ For $\frac{1}{(z-1)}$ we ...
1
vote
1answer
28 views

Laurent series to converge in $0<|z-1|<R$

Question: Determine the largest number $R$ so that the Laurent series of $$f(z)=\frac2{z^2-1} + \frac3{2z-i}$$ about $z=1$ converges for $0<|z-1|<R$. Attempt: I really don't understand this ...
1
vote
1answer
44 views

Classify singularities of $\frac{e^z \sin(3z)}{(z-\sqrt2)(z+\sqrt2)z^2}$

They are $0, \pm \sqrt2$. With the zero, $f(0)$ makes the numerator vanish and I have no idea how you would expand the whole function at $0$ because of the denominator. So what do you do to classify ...
0
votes
1answer
36 views

Largest $R$ value in domian $0<|z-1|<R$

Determine the largest real number $R>0$ such that the Laurent series of $$f(z)=\frac1{z-1} +\frac2{z-i}$$ about $z=1$ converges for $0<|z-1|<R$. The singularities are $1$ and $i$. But in the ...
0
votes
1answer
30 views

Laurent series in domain $|z|>0$

Find Laurent series, in powers of $z$, of $$f(z)=\frac{\sin(2z)}{z}$$ valid in the region $|z|>0$. The singularity is $0$ but $0$ isn't inside the region of the domain so what do you exactly ...
0
votes
2answers
38 views

Finding Laurent series with imaginary numbers

$$f(z)=\frac{2z}{z^2+1}=\frac1{z-i} +\frac1{z+i}$$ Find Laurent series in powers of $z$ in the domain $|z|<1$. So I got to find two Taylor series of the two terms in the function but how do you do ...
0
votes
0answers
40 views

Laurent series confusion

I've split it up into partial fractions and got $1/z$ - $2/(z-1)$ + $1/(z-2)$ but I'm unsure sure what to do now. I think I have done part $(i)$. I get $$z^{-1} + \sum_{n=0}^\infty ...
2
votes
1answer
46 views

Laurent series of $1/({z^3-z})$

Question: Find the Laurent series of the function $$f(z) = \frac{1}{z^3 - z}$$ at the domain $|z-1|>2$. Attempt: So we have $$\frac{1}{z(z-1)(z+1)}$$ and we only have to find a Laurent ...
0
votes
0answers
45 views

Laurent series for $\frac{2}{(z)(z-1)(z-2)}$

! So I think I am getting the hang of Laurent Series, but having a bit of trouble with one of the fractions for part a). So I split this up in to partial fractions: $\frac{1}{z} - ...
1
vote
0answers
71 views

Laurent series $\frac{-1}{z}+\frac{1}{2(z-1)} +\frac{1}{2(z+1)}$ at |z-1| > 2

This is what I tried to do. Is this correct?
0
votes
1answer
42 views

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
0
votes
1answer
43 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!
0
votes
1answer
33 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 ...
1
vote
1answer
86 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 ...
0
votes
0answers
67 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 ...
1
vote
0answers
33 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) = ...
0
votes
0answers
28 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 ...
0
votes
1answer
27 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 $r<|z|<R$: ...