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Questions tagged [laurent-series]

This tag is for questions about finding a Laurent series of functions and their convergence. 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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Converse of bounds on the spectrum of a Toeplitz matrix

The following is from Robert M. Gray's review (https://ee.stanford.edu/~gray/toeplitz.pdf): Lemma 4.1 Let $\tau_{n,k}$ be the eigenvalues of a Toeplitz matrix $T_n(f)$. If $T_n(f)$ is Hermitian, then ...
gen's user avatar
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proof that $K((x)) = \operatorname{Frac}(K[[x]])$

I want to show $K((x)) = \operatorname{Frac}(K[[x]])$, i.e. K((x)) is the minimum field containing $K[[x]]$, where the latter is an integrity domain, since it is a commutative ring and has no zero ...
Wrloord's user avatar
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3 votes
1 answer
87 views

Expand the function $ f $ with the prescription $ f(z) = \frac{z - 9}{z^2 - 3z - 4} $ into a Laurent series

Expand the function $ f $ with the prescription $ f(z) = \frac{z - 9}{z^2 - 3z - 4} $ into a Laurent series: (a) around the point $ z = 4 $ in the region $ D = \{ z \in \mathbb{C} \mid |z - 4| > 5 \...
Markus's user avatar
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3 votes
2 answers
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$\int_{|z-2|=1} \frac{g(z) \, dz}{(z-2)^2} = 12 \int_{|z-2|=1} \frac{f(z) \, dz}{(z-2)} + 8 \int_{|z-2|=1} \frac{f(z) \, dz}{(z-2)^2}$

Let $f: \mathbb{C} \to \mathbb{C}$ be a holomorphic function and let $g(z) = z^3 f(z)$. Prove that the following holds: $$\int_{|z-2|=1} \frac{g(z) \, dz}{(z-2)^2} = 12 \int_{|z-2|=1} \frac{f(z) \, dz}...
math123's user avatar
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1 answer
81 views

Find the Laurent expansion of $f(z) = \sin{\frac{1}{z(z-1)}}$ in $0<|z-1|<1$

Here is my idea: $\sin{\frac{1}{z(z-1)}} = \sin{\left( \frac{-1}{z} + \frac{1}{z-1}\right)} = \sin{\left(\frac{1}{z-1} - \frac{1}{z}\right)} = \sin{\frac{1}{z-1}}\cos{\frac{1}{z}} - \cos{\frac{1}{z-1}}...
Irbin B.'s user avatar
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1 answer
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Could the product of two Bessel functions of the first kind be expressed in terms of infinite series $J_n(x)J_m(\alpha x)$, where $n,m\in\{0,2\}$?

It is well known that the square of the Bessel function of the first kind of order zero has the Maclaurin series expansion $$ J_{0}(x)^{2} = \frac{1}{\sqrt{\pi}}\sum_{k=0}^{\infty} \frac{(-1)^{k}}{k!}\...
Siegfriedenberghofen's user avatar
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1 answer
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Help with Analytic Function Zero in Annulus

I'm working on a problem involving an analytic function in an annulus, and I need some help with the second part of the question. Here is the problem statement: Suppose that $ f(z) $ is analytic in ...
tree tree juice's user avatar
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1 answer
64 views

Procedure to solve $\mathcal{P}\int_{-\infty}^{+\infty} \frac{\cos (\alpha x)}{x^2-1} dx$

I recently solved a complex analysis exercise that required to solve this integral using residue theory, but I don't know where to check for the correctness of the result I got, so I thought of ...
deomanu01's user avatar
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1 answer
46 views

Taylor-Laurent series expansions

I'm having some issues finding how to series expand some complex functions that my professor gave past years in exams. For example, in this exercise, it is asked to find the first two terms of the ...
deomanu01's user avatar
  • 113
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0 answers
26 views

Need Clarification on a problem about finding Laurent series

I am working on the following problem: Let $f(z) = \frac{1}{(1+z^2)(2-z)^2}$. Determine the principal part of $f$ at $z=2$ and determine the region where the Laurent series of $f$ at $z=2$ converges. ...
Koda's user avatar
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24 views

Condition for series expansion of a parametric integral

Suppose you have a function $h:(0,1)\times (0,1)\to \mathbb{R}$ and a parametric integral of the form: $$ I(x)=\int_x^1 h(x,y)\,\mathrm{d}y $$ Question: What would be the conditions on $h$ so that the ...
Hakanaou's user avatar
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1 answer
49 views

Why bounded Laurent Series is not constant?

Suppose $f$ has an isolated singularity at $z_0$, and $f$ is analytic and bounded in an open circular neighborhood around $z_0$ (but not analytic on $z_0$). By expression of Laurent series, $$f(z)=\...
Anora's user avatar
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Steps for classifying more than one singularity

Upon doing a lot of questions where I have to classify more than one singularity, I've come u with some steps to do based on patterns that I've seen. But, I'm unsure as to whether the steps I've ...
Luke's user avatar
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Which annalus do we use when finding the Laurent series of $\frac{z}{(z+1)(z+2)}$ at $z = -2$

Find the Laurent series of $\frac{z}{(z+1)(z+2)}$ at $z = -2$ My question When a question does not provide the annalus, like this one, does it mean that we need to find all possible laurent series? I....
Luke's user avatar
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0 votes
1 answer
48 views

Laurent expansion of $\frac{1}{(z^5-1)(z-3)}$ with $1<|z|<3$

Laurent expansion of $\frac{1}{(z^5-1)(z-3)}$ with $1<|z|<3$ I'm stuck in the step which is simplify $\sum_{n=0}^{\infty}z^{5n}$ into $\sum_{n=0}^{\infty}a_nz^n$ form. The calculate process is $...
Krystal Justin's user avatar
1 vote
1 answer
58 views

Find the Laurent series of $\frac{z}{(z+1)(z+2)}$ at $z=-2$.

Here's the question: Find the Laurent series of $\frac{z}{(z+1)(z+2)}$ at $z=-2$. Here's what I've done: $\frac{z}{(z+1)(z+2)}=-\frac{1}{(z+1)}+\frac{2}{(z+2)}$ We have singularities at $z=-1$ and $...
Luke's user avatar
  • 99
2 votes
1 answer
42 views

Compute explicitly the principal part of $\sum_{n=-\infty}^{-1}a_nz^n$.

Here's the question: Let $\sum_{n=-\infty}^\infty a_nz^n$ be the Laurent expansion of $\frac{\sin(z)\cos(z)}{z^5}$ in $0<|z|<\infty$. Compute explicitly the principal part of such Laurent series,...
Luke's user avatar
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0 answers
52 views

Laurent Series Expansion of $\frac{1}{(z+1)(z+2)}$ about $z = 1$

I am working on the following exercise of computing the (1) Laurent series expansion and (2) radius of convergence. Let $f(z) = \frac{1}{(z + 1)(z + 2)}$. Consider the annular domain $3 < \lvert z ...
3m115's user avatar
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1 vote
1 answer
62 views

Is every smooth function equal to its Laurent series?

It's known that there exists non-analytic smooth functions, take for example the function $f$ defined by $f(0) = 0$ and $f(x) = \exp(-1/x^{2})$. $f$ has the zero polynomial as its Taylor series ...
conan's user avatar
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2 votes
1 answer
45 views

Understanding the Laurent expansion of a meromorphic function about $\infty$.

Suppose $f:\hat{\mathbb{C}}\to\hat{\mathbb{C}}$ were meromorphic, and suppose $f$ has a pole at $\infty$. I'm trying to understand the Laurent series of $f$ about $\infty$. By definition, $f$ has a ...
Ty Perkins's user avatar
1 vote
3 answers
42 views

Non-negative Maclaurin coefficients on $(0,a)$ implies convexity

I was interested in Putnam 2003, A3: find the minimum of $$|\sin(x)+\cos(x)+\tan(x)+\cot(x)+\sec(x)+\csc(x)|$$ for all real numbers $x$ (but it suffices to look over $(0,\pi/2)$, in which case $|\cdot|...
Integrand's user avatar
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0 votes
1 answer
26 views

Laurent series $f(z) =\frac{(2z-2)}{(z^2-3z+2)}$ around $0<|z-2|<1$

I've used Trinom to find roots and got $f(z) =\frac{2(z-1)}{(z-1)(z-2)}$. And for some reason I cannot remember if it would be legal to do away with the $\frac{(z-1)}{(z-1)}$ to be left with $f(z) =\...
Kaiannae's user avatar
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3 votes
0 answers
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How to find the inverse Laplace transform of function $s\csc(2s)$?

When dealing with one ODE, it happened to find the inverse Laplace transform of function $$ F(s)=\frac{s}{\sin(2s)}. $$ I suppose it exists the inverse Laplace transform, but I could not find any ...
MathArt's user avatar
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0 answers
32 views

Find the Laurent series $e^{2z} + \cos \frac{2}{z}$ in the domain $\mathbb{C} \setminus \{0\}$.

Find the Laurent series $e^{2z} + \cos \frac{2}{z}$ in the domain $\mathbb{C} \setminus \{0\}$. I just want to make sure I'm doing this right. We clearly have \begin{align} \begin{split} ...
Grigor Hakobyan's user avatar
2 votes
0 answers
34 views

Find the Laurent series of $z^2e^{1/z}$ in the neighborhood of $z=0$

Find the Laurent series of $z^2e^{1/z}$ in the neighborhood of $z=0$. Just looking for solution verification. I have: \begin{align} \begin{split} z^2 e^{1/z} &= z^2 \sum_{n=0}^\infty \...
Grigor Hakobyan's user avatar
1 vote
1 answer
60 views

Asymptotic expansion of an integral by expanding series first

Question: as $s\to\infty$ use the substitution $u = {\rm e}^{-x}\,$ to obtain the first 2 terms in asymptotic expansion in the integral: $$ \operatorname{B}\left(s,t\right) = \int_{0}^{1}u^{s - 1}\,\,\...
vegetandy's user avatar
  • 305
3 votes
3 answers
254 views

Find the principal part at poles

I am asked to find the isolated singularities of the function and determine their typing, order, and finding the principal part at each pole. So for the first function, which is $\frac{e^z-e}{z^2-1}$, ...
robert lewison's user avatar
4 votes
2 answers
88 views

Find the Laurent Series of $f(z) = \frac{1}{(z-6)(z+4)}$

Can someone help me find the Laurent Series centered at $z=6$ for $f(z) = \frac{1}{(z-6)(z+4)}?$ I was going to start by rewriting this as $f(w) = \frac{1}{w(w+10)}$ so that I can find the Laurent ...
Math Undergrad Student's user avatar
1 vote
2 answers
65 views

Find $\int_{|z|=5} \frac{\sin z}{z^2-4} \hspace{0.2cm} dz$ using Residue Theorem

Find $\int_{|z|=5} \frac{\sin z}{z^2-4} \hspace{0.2cm} dz$ using Residue Theorem We know that $\sin z = \sum_{n=0}^\infty (-1)^n \frac{z^{2n+1}}{(2n+1)!}$. But how can we manipulate $\frac{1}{z^2-4} = ...
Grigor Hakobyan's user avatar
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0 answers
47 views

which one is correct for the complex integration $C_k=\frac{1}{2 \pi i} \oint_C \frac{\sin \left(\zeta-\frac{1}{\zeta}\right)}{\zeta^{k+1}} d \zeta $

when I calculate the complex integral$$ C_k=\frac{1}{2 \pi i} \oint_C \frac{\sin \left(\zeta-\frac{1}{\zeta}\right)}{\zeta^{k+1}} d \zeta $$ where $C:z=e^{i t}, $ $t: 0 \rightarrow 2 \pi$ $$ $$I got ...
gxh gxh's user avatar
3 votes
1 answer
57 views

Finding residue of $f(z) = \frac{1}{z^2-1} \sin\frac{1}{z^2 + z^4}$ at $z = 0$

Trying to find the residue for $f(z) = \frac{1}{z^2-1} \sin\frac{1}{z^2 + z^4}$ at $z = 0$. I am concentrating on the sin term in particular. What is a good strategy to approach this? One thing I ...
giorgio's user avatar
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3 votes
1 answer
148 views

Can you remove the poles from $\frac{f'(x)}{f(x) - y}$ with $L^\infty_y L^1_x$ error?

I believe that the following below holds, which I need for some estimates. I have been struggling to prove it though! Let $f \in C^1(\mathbb{R}; \mathbb{R})$ have finitely many critical points and be ...
Robert Wegner's user avatar
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0 answers
47 views

Taylor series, Laurent series and a simple pole

Let $f$ be holomorphic in $|z|<2$ except for $z=1$, in which it has a simple pole. Let $f(z)=\sum_{n=0}^{\infty} a_n z^n$ be the Taylor series of $f$ around $z=0$. Prove or find a counterexample: (...
Dr. John's user avatar
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1 vote
0 answers
37 views

Laurent Series Partial Fractions

I have $\frac{1}{z+z^2}$ defined on the annulus $|z+1|>1$. I write: $f(z) = \frac{1}{z} - \frac{1}{z+1} = -\frac{1}{z+1} + \frac{1}{(z+1)-1} = -\frac{1}{z+1} + \frac{1}{z+1} \cdot \frac{1}{1-\frac{...
adisnjo's user avatar
  • 247
2 votes
0 answers
34 views

Laurent Series question for Exponentials

I must find the Laurent series for $f(z) = \frac{e^z}{z^2}$ in powers of $z$ for the annulus $ |z| > 0$. I wrote $f(z) = \frac{1}{z^2} \sum_{n=0}^{\infty} \frac{z^n}{n!} = \sum_{n=0}^{\infty} \frac{...
adisnjo's user avatar
  • 247
1 vote
0 answers
60 views

Laurent series expansion of $𝑓(𝑧)=\tan(𝑧/(𝑧-1))$ around $𝑧 = 1$

I am tasked with finding whether the function $𝑓(𝑧)=\tan(𝑧/(𝑧-1))$ can be developed into a Laurent series around $𝑧=1$, and if so what is the radius of convergence and what is the residue? My ...
giorgio's user avatar
  • 583
1 vote
1 answer
78 views

Laurent series expansion of $f(z) = \cos(z/(1-z))$ around $z = 1$

I am tasked with finding whether the function $f(z) = \cos(z/(1-z))$ can be developed into a Laurent series around $z = 1$, and if so what is the radius of convergence and what is the residue? So far ...
giorgio's user avatar
  • 583
1 vote
1 answer
41 views

Find the laurent expansion

Find the Laurent expansion of $\dfrac{e^\frac{1}{z}}{z+1}$ in the domain $|z|>0$. My attempt: Since we have to work on domain $|z|>0$, let $z=\dfrac{1}{t}$, so I have to find the Laurent ...
user avatar
0 votes
1 answer
44 views

Commutant of a polynomial

Let $Q$ a polynomial in $\mathbb{C}[X]$ (or in $\mathbb{K}[X]$ with $\mathbb{K}$ a field of characteristic zero). I wonder what is the commutant of $Q$, i.e. the set of polynomials $P$ such that $P \...
Dlem's user avatar
  • 898
2 votes
0 answers
40 views

How to use stationary phase method when the zero point is at infinity?

Thank you for reading my questions. It is known that the stationary phase method has this form: $$ \int_a^bg(t)e^{jf(t)dt}\approx\sum\limits_{t_0\in\Sigma}g(t_0)e^{jf(t_0)+j*\text{sign}(f''(t_0))\frac{...
Xiangyu Cui's user avatar
-2 votes
2 answers
89 views

How to use Laurent expansion of a function into this form?

How can I compute the Laurent expansion of the function $$f(z)=\frac{1}{(2z+1)(z-2)}$$ when $\frac{1}{2}<|z|<2$ with this $\frac{1}{z^2}$ term?
WangSan's user avatar
  • 17
0 votes
0 answers
26 views

Leading divergent term in integral before integration

Consider, for example, the Beta function: $$ B(\alpha,1+\alpha) = \frac{\Gamma(\alpha)\Gamma(1+\alpha)}{\Gamma(1+2\alpha)} = \int_0^1 dt\ t^\alpha (1-t)^{\alpha-1}. $$ Expanding the integrated result ...
Marcosko's user avatar
  • 175
0 votes
0 answers
37 views

Confused about a proof in Rational Solutions of the Fifth Painleve Equation by Kitaev et al.

In the article Rational Solutions of the Fifth Painleve Equation (see here) proposition 2.3 derives the orders and residues of singularities of rational solutions to a particular differential equation ...
H. de Gracht's user avatar
1 vote
0 answers
74 views

Laurent series for $\frac{1}{z+1}+\frac{1}{z-3}$.

Find the Laurent series expansion for $$f(z)=\frac{1}{z+1}+\frac{1}{z-3},$$ in the domains (a) $0<\vert z \vert <1$, (b) $1<\vert z \vert<3$, and (c) $3<\vert z \vert<\infty.$ ...
homosapien's user avatar
  • 4,225
1 vote
1 answer
28 views

Fast way to find the __regular__ part of the Laurent expansion of a function at a pole?

To evaluate a certain limit, I need to calculate the first term in the regular part of the Laurent expansion of the function $$\frac{\pi}{\sin \frac{\pi(s+1)}{2}}$$ around -1 (should be the same thing ...
Daigaku no Baku's user avatar
1 vote
0 answers
50 views

Laurent Series around $z_0=0$ and $z_0=1$

I tried to expand three functions in Laurent series, but I don't have any given answer for them and I'm not very confident that what I'm doing is correct. The first one is $f(z)=\frac{z^3e^{1/z}}{z+1}$...
poxipollepi1's user avatar
2 votes
1 answer
106 views

How to calculate the residue of $\frac{\sin(z)}{\cos(z^3)-1}$ at $z=0$ using Laurent series?

My sheet propose the following exercise: Consider the function $$ f(z)=\frac{\sin(z)}{\cos(z^3)-1} $$ Classify the singularities and, at z = 0, calculate the residue. It is therefore clear that ...
Foxy's user avatar
  • 255
1 vote
0 answers
61 views

Name of Laurent series with countably many poles

Is there a name for series of the type $S=\sum_{k=-\infty}^{\infty}a_k(z-b_k)^k$ for some complex sequences $(a_n)_{n\in\mathbb{Z}},(b_n)_{n\in\mathbb{Z}}$ such that $S$ converges at least in some ...
Jfischer's user avatar
  • 1,271
2 votes
1 answer
184 views

Contour integral of $\sqrt{\frac{z}{z-1}}$

Which Laurent series could be used to solve $$\oint_{|z|=2}\sqrt{\dfrac{z}{z-1}}dz$$ if it has a branch cut at $y=0$, $x\in(0,1)$? I thought maybe it couldn't be solved by residue theorem because ...
Conreu's user avatar
  • 2,568
1 vote
0 answers
31 views

Radius of convergence at $2+i$ of $f_i(z)=\frac{1}{\phi_i-2^{1/4}}$ with $2^{1/4}$ being the positive real root

Let $\phi_k(z), k=0,1,2,3$ the branch cuts of $z^{1/4}$. Consider $$f_k(z)=\frac{1}{\phi_k-2^{1/4}}, \quad 2^{1/4}=|2|^{1/4}e^{i0}>0$$ Find the radius of convergence of the series expansion at $2+i$...
Mateo's user avatar
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