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Questions tagged [singularity]

This tag is for questions relating to singularity, which is a point where a mathematical concept is not defined or well behaved, such as boundedness, differentiability, continuity. In general, because a function behaves in an anomalous manner at singular points, singularities must be treated separately when analyzing the function, or mathematical model, in which they appear.

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Find $\displaystyle\int_\Gamma \frac{\tan z}{z^2+\pi^2} \, dz$ where $\Gamma$ is the circle $\left|z-\frac{\pi}{2}\right|=1$

Find $\displaystyle\int_\Gamma \frac{\tan z}{z^2+\pi^2} \, dz$ where $\Gamma$ is the circle $\left|z-\frac{\pi}{2}\right|=1$. This problem showed up on a qualifying exam where the usual "find the ...
Grigor Hakobyan's user avatar
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1 answer
51 views

The space of solutions for an ODE on an interval containing a regular singular point

Consider the differential operator $L(y)=x^ny^{(n)}+a_1(x)x^{n-1}y^{(n-1)}+\cdots+a_n(x)x^0y^{(0)}$, where each of $a_i(x)$ has a power series expansion at $x=0$ converging for all $|x|<r_0$ for ...
Alex Scott Johnson's user avatar
1 vote
0 answers
15 views

Does maximal contact hypersurface (m.c.h.) exist for any curve over a field of arbitrary characteristic, and if so, for what definition of m.c.h.?

Does a maximal contact hypersurface always exist for any curve over a field of arbitrary characteristic, and if so, for what definition of maximal contact hypersurface?
mathemusician's user avatar
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Sign of a complex integral

If one consider the complex value function $$ f(z)=\frac{1}{(z-1)^{3/2}(z-2)^{3/2}} $$ with branch cut chosen to be between $z=1$ and $z=2$. Consider $$ \oint f(z) dz, $$ where the contour is taken to ...
Gateau au fromage's user avatar
1 vote
1 answer
109 views

Complex integral with fractional singularities

If one consider the complex value function $$ f(z)=\frac{1}{\sqrt{z-1}\sqrt{z-2}} $$ with branch cut chosen to be between $z=1$ and $z=2$. Could someone please explain why $$ 2\int_1^2 f(x)dx=\oint f(...
Gateau au fromage's user avatar
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52 views

$\int_0^{+\infty}\frac{\cos (ax)-\cos(bx)}{x^2}\mathrm dx$ using complex analysis [duplicate]

$$ \mbox{I defined the complex function}\quad \operatorname{f}\left(z\right) = {{\rm e}^{{\rm i}z} \over z^{2}} $$ and in the end i just take the real part of my answer. I want to use Residues Theorem....
Ben Reznik's user avatar
1 vote
1 answer
104 views

Using the residue theorem to compute two integrals [closed]

Classify the singular points for the function under the integral and using the residue theorem, compute: (a) $$ \int_{|z-i|=2} \frac{z^2}{z^4 + 8z^2 + 16} \, dz, $$ (b) $$ \int_{|z|=2} \sin\left(\frac{...
GENERAL123's user avatar
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27 views

Residue of a removable singularity at inifinity

Exercise: Find all the singularities of $$\frac{z^3e^{\frac{1}{z^2}}}{(z^2+4)^2},$$ classify them, and find each residue. I found that $+2i, \ -2i$ are poles of order two. I was able to calculate ...
cor1.1.29's user avatar
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Smoothing projective nodal curve, is the general fiber smooth?

Proposition 29.9 of Hartshorne's Deformation theory states the following: A reduced curve Y in $\mathbb{P}^n$ with locally smoothable singularities and $H^1(Y,O_Y(1)) = 0$ is smoothable. In particular,...
maxo's user avatar
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Determine whether a matrix is non singular

The model of a spacecraft is the following: \begin{equation*} \dot{\sigma} = \mathbf{G}(\sigma)\omega \end{equation*} \begin{equation*}\mathbf{G}(\sigma) = \frac{1}{2}\bigg(\frac{1-||\sigma||^2}{2}\...
dodo's user avatar
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1 answer
57 views

Integral with singularities

Let $n > 1$ and $1 \le k \le (n-2)$ be integers, and set $$f(x) := (-1)^n k \left(\frac{k u \sin (k \pi u)}{n-k u}+\frac{(k+1) u \cos \left((k+1) \pi \sqrt{u}\right)}{(k+1)u-n}+\frac{n}{\pi }\...
Richard Burke-Ward's user avatar
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42 views

What is the definition of "multiple component of germ"?

Recently I read a paper and I am confused with a word "multiple components", but I don't find its definition in this paper. I guess it is about the singularity. Here is a picture. You can ...
cbi's user avatar
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restricting a function changes its singular points and analyticity?

Let define $f(z) = \frac{1}{z-2}$ for $z\in\mathbb{C}\setminus\{2\}$. Then it is clear that, $f(z)$ has singular point at $z=2$ (Namely pole of order 1 at $z=2$). However, if I update the definition ...
General Mathematics's user avatar
1 vote
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Singularities of $\frac{1}{1-z^n}$

I'm looking to classify the singularities of $g(z) = \frac{1}{1-z^n}$ and compute the residue at each pole. Now, $g(z)$ has singularities at the roots of unity $w^k$, where $w=\exp\{\frac{2\pi \textbf ...
CatsAndDogs's user avatar
1 vote
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27 views

Singularity in Poisson's Equation

Consider an instance of Poisson's equation in spherical coordinates for the radial dimension: $$ \nabla \cdot \nabla \phi(r) = \frac{1}{r^2} \frac{d}{dr}\left(r^2 \frac{d\phi}{dr} \right) = -\sin(r). $...
SeanBrooks's user avatar
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What is the relationship between regular singular points and the Cauchy Euler equation?

In my PDE course, my professor shows me how we can take a generic homogenous polynomial coefficient 2nd order differential equation: $$P(x)y'' + Q(x)y' + R(x)y = 0$$ And transform it into the ...
jeromecho's user avatar
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Minimum singular value vs condition number to determine closeness to singularity

Which criteria should I use to determine if a matrix is close to singularity or not? My application is to try to find an optimal matrix that maximizes the minimum singular value/minimizes the ...
William Lin's user avatar
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43 views

Singularities and their nature

What are the singularities of $f(z) = e^{\frac{\sin z}{z}}$ ? It clearly has a removable singularity at $0$. A textbook says that it has essential singularities at $kπ$ for $k \in \mathbb{Z}$. How and ...
Anonymous's user avatar
1 vote
0 answers
56 views

How to tell which of several possible asymptotic forms a numerical solution to an ODE is converging to

I've previously mentioned the ordinary differential equation $$12P\left(f\left(x\right)\right)^3f''''\left(x\right)+12\left(3P-1\right)\left(f\left(x\right)\right)^2f'\left(x\right)f'''\left(x\right)+...
Daniel Hatton's user avatar
1 vote
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
  • 23
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1 answer
61 views

What is the nature of the pole of the derivative of $\frac{1}{1- \ln(x)}$ at $x=0$?

I'm interested in the function $\frac{1}{1-\ln(x)}$ on positive real line. One can experimentally see that $$ \lim_{x \rightarrow 0^+} \left[ x \frac{d}{dx} \left[ \frac{1}{1 - \ln(x)} \right] \right] ...
Sidharth Ghoshal's user avatar
1 vote
1 answer
49 views

A doubt in the proof of the spectrum of convolution operator

The following is a link showing the spectrum of convolution operator $A f(x)= \int_{-\pi}^{\pi} h(x-y) f(y) dy$ $L^2( {-\pi},{\pi})->L^2( {-\pi},{\pi})$ is range of $h$ where $h$ is continuous ...
Stack_Underflow's user avatar
8 votes
2 answers
219 views

How is the discriminant defined for $x^3+y^3+z^3+u^3+(ax+by+cz+du)^3+exyzu$?

Recently I am reading a book by Arnol'd et. al.. In this text I did not find the definition of discriminant $\Delta(a,b,c,d)$ : I wonder what is the meaning of the discriminant here ? The following ...
cbi's user avatar
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0 answers
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Proof of analytic continuation on manifold

I was reading the theorem below in this article. My question concerns the passages in bold. Specifically, why would Z be open? Is it because we can use germs to define a topology where Z would be the ...
yumika's user avatar
  • 13
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4 answers
75 views

Show that $f(\frac{1}{z})$ have a essential singularity at $0$.

We need to show that if $f(z)$ ,who is entire, periodic and non-constant , then $f(\frac{1}{z})$ have an essential singularity at 0. So we then need to show that: $f(\frac{1}{z})$ as not a pole at $z=...
Student_Number_249812341's user avatar
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1 answer
25 views

$f$ with an essential singularity at $0$ intersects $1/z^4$ in every neighborhood of $0$.

This question is from a qual in complex analysis at my Uni, and I feel like I'm losing my mind. My solution seems far too simple for it to be a qual question, and it makes me feel like I must be wrong....
Ty Perkins's user avatar
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1 answer
64 views

How to find the matrix $A$ from $C_0=A \times B$ or $C_1=B \times A$, given the singluar matrix B.

Kindly help me in the following: Let $C_0=A \times B$ and $C_1=B \times A$, where $A$ is a full rank square matrix, $B$ is a non-zero singular square matrix. Entries in $A,B,C$ are from finite ...
X.H. Yue's user avatar
0 votes
2 answers
50 views

Differentiability of $\frac{(x - y) f'(x)}{f(x) - f(y)}$ away from poles when $y$ is not a critical point

Let $f: \mathbb{R} \longrightarrow \mathbb{R}$ be three times differentiable and let $y \in \mathbb{R}$ such that $f'(y) \neq 0$. With a lengthy manual calculation I could show the following: $$ \frac{...
Robert Wegner's user avatar
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0 answers
18 views

If a singular square matrix is multiplied by a non-singular square matrix, the null space of the result is what?

If a non-singular square matrix $B$ is multiplied by a singular square matrix $A$ of the same order, the nullspace of the resulting matrix $C=B\times A$ or $C'=A \times B$ remains unchanged from that ...
X.H. Yue's user avatar
0 votes
1 answer
72 views

Great Picard theorem for $e^{\frac 1 z}$

In accordance with the Great Picard theorem, the function $f(z) = e^{\frac 1 z}$ assumes every complex value except $0$ in every neighborhood of the origin. I would like to know an elementary ...
AlpinistKitten's user avatar
4 votes
0 answers
53 views

intersection homology and naturality

I'm learning about intersection homology, and I'm trying to write a proof of the following statement: Take X,Y two filtered spaces with perversities $p$ and $q$ resp. , a continuous function $f:X\to ...
bml64's user avatar
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How to deal with this 3-dimensional global optimization problem with singular points?

$$\max_{C_1,C_2,C_3} \frac{2C_1+C_2-\sqrt{C_2^2+4C_1C_3}}{2(C_1+C_2-C_3)}:=G(C_1,C_2,C_3)$$ s.t. $$C_1\in(0,2],C_2\in(0,1],C_3\in(0,2]$$ $$D\in(0,1]$$ $$C^2D^4-2BCD^3>0 \Longleftrightarrow C_1+C_2-...
chloe's user avatar
  • 1,052
0 votes
0 answers
39 views

Quadratic cone and paths (elementary algebraic geometry).

Let $J\subset \mathbb{C}[x_1,\dots,x_n]$ be an ideal such that $J\subset (x_1,\dots, x_n)$ we can define the Zarisky tangent space of the Scheme $X = Spm(\mathbb{C}[x_1,\dots,x_n]/ J)$ at $0$ by $T^{\...
Clément Legrand's user avatar
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0 answers
30 views

Show that $f$ has a simple pole at $0, -1, -2, \ldots$.

I am given the function $f:\mathbb{C}\setminus \{0, -1, -2, \ldots \}\to\mathbb{C}$ that is holomorphic on its domain, such that $f(1)=1$ and $f(z+1)=zf(z)$. I want to show that $0, -1, -2, \ldots$ ...
Aadi Rane's user avatar
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0 answers
52 views

Blow up of Limao $x^2 - y^3z^3$ in $\mathbb{A}^3$

Let me take $X:=x^2-z^3y^3=0$ to be our surface over $\mathbb{A}^3$. Then we see we have a singular locus on the y and z axis. So, I will blow up initial on the Z-axis $V = Z(<x,y>)$. Then $\...
ben huni's user avatar
  • 173
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0 answers
111 views

Singularity extraction: $\int_{-1}^{1}\int_{-1}^{1}\frac{dxdy}{\sqrt{(x-.3)^2+y^2}}$

$\int_{-1}^{1}\int_{-1}^{1}\frac{dxdy}{\sqrt{(x-.3)^2+y^2}}$ has the following physical meaning: it is the potential of the uniform surface source distributed on a square $|x|,|y|<1$ observed at a ...
Aria's user avatar
  • 422
1 vote
0 answers
31 views

When Principal Value Can Diverge

This question is motived by the following observation: Consider the principal value integral $$ P.V.\int_{-\eta_1}^{\eta_2}\frac{1}{a_1 x + a_2 x^2 + a_3 x^3 + \cdots} $$ Here I choose $\eta_1$ and $\...
Lawrence's user avatar
1 vote
0 answers
50 views

Can a non-holomorphic function have a pole?

As far as my studies have brought me, I've only see so far the definition of a "pole" for a complex valued function $f:\Omega \rightarrow \mathbb C$ if we assume that the function is ...
Andreas Compagnoni's user avatar
-1 votes
1 answer
85 views

Evaluation of Integral via the Residue Theorem

I was trying to solve some complex integrals via Contour Integration and found myself stuck with the following exercise: $$ I = \oint_{\gamma}z\sin\left(\frac{1+z}{1-z}\right)\mathrm{d}z,\,\gamma = ...
Claudio's user avatar
  • 432
1 vote
0 answers
99 views

if $x^TAx = 0$ for every $x$, then $A$ is singular?

Let $A$ be some matrix, If $x^TAx = 0$ for every vector $x$, then $A$ is singular. How can I prove it? Here is what I thought: first of all, for every matrice $M$, $$ x^T(M + M^T)x = x^TMx + x^TM^Tx =...
Butterwix6O1's user avatar
1 vote
1 answer
91 views

Problem in understanding Riemann's removable singularity theorem.

Theorem $:$ Suppose that $f$ is holomorphic in the region $\Omega' \subseteq \Omega$ which is obtained from $\Omega$ by a removing a single point $a \in \Omega.$ Then $f$ has a unique holomorphic ...
Anacardium's user avatar
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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
0 votes
1 answer
56 views

Is $\frac{1}{\sqrt{\mid x \mid}}$ integrable in interval -1 to 1?

I know that I can't integrate some unlimited functions while my interval of integration contains the singularity point. For example, $\int_{-1}^{1}\frac{1}{x}dx$ is undefined, it makes sense to me ...
TY FIRE's user avatar
  • 17
1 vote
0 answers
69 views

How pathological can a Laplace transform be?

Almost every treatment of the Laplace transform that I come across talks about "the poles" of the Laplace transform function $F(s)$, thereby seeming to implicitly assume that $F(s)$ is ...
tparker's user avatar
  • 6,280
0 votes
0 answers
60 views

Type of singularity at $z=\infty$ in $\frac{1+e^z}{1-e^z}$

In my book, the poles of $f(z)$ have been found out as $z=(2n+1)πi$. So, the singularity is non-isolated essential as $z=\infty$ is a limit point of these poles. I understand this. However, if I ...
Yash Aggarwal's user avatar
4 votes
0 answers
122 views

What is a numerical method to solve IVP at irregular singular point?

I have tried to search for numerical method to solve IVP with irregular singular point but didn't find any. What is the proper method I can use to solve such equations? $$y''+\frac{y'}{x^2}+y=x^3+6x+3$...
Mohamed Mostafa's user avatar
1 vote
5 answers
156 views

How to compute limit to infinity of an exponential in complex analysis?

My question originates from trying to find the following limit: $$ \lim_{z\to\infty} e^{-z^2} $$ Disclaimer: before getting into details, I want to point out that English isn't my first language, so I ...
propriofede's user avatar
2 votes
1 answer
125 views

Integrating a Modified Bessel function of the second kind with a singularity

Does someone know how to handle the integral $$\int_{-\infty}^{\infty} \frac{K_0\!\left(\lvert \tau \rvert \sqrt{q^2 \alpha ^2}\right)}{q^2}\cos (q x)\,\mathrm{d}q $$ $\alpha$ is a real number and $\...
Roeland van den Wildenberg's user avatar
2 votes
1 answer
94 views

Meromorphic function with infinitely many poles must have essential singularity at point of infinity?

Question: Suppose $f: \mathbb{C}\setminus P \to \mathbb{C}$ is meromorphic on the complex plane, where $P:=\{z_n: n\in\mathbb{N}\}$ is a set of infinitely many isolated poles with $\infty$ as their ...
user760's user avatar
  • 1,670
2 votes
1 answer
123 views

How to force my differential equations give a bounded solution?

I have modeled the interaction of two physical quantities, $S$ and $B$, by the following differential equations (the second one is a delay differential equation): $$S'(t) = 0.31 S(t) \Big( 1 - \frac{S(...
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