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Questions tagged [singularity]

A singularity is a point where a mathemtical concept is not defined or well behaved, such as boundedness, differentiability, continuity.

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Is this physical model exactly solvable?

There exists a popular model in the Physics of heavy quark bound systems, called the Cornell potential model, in which the inter-quark potential is modeled to vary with radial distance $r$ as $$V(r) ...
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1answer
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Relationship between Affine definition of singular point and projective definition

Let $C : F(X,Y,Z) = 0$ be a projective curve given by a homogeneous polynomial $F \in \mathbb{C}[X,Y,Z]$, and let $P \in \mathbb{P}^2$ be a point. Prove that $P$ is a singular point of $C$ if ...
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20 views

Blow up of a planar curve at a singularity

Assume I have a planar projective curve $C\subseteq \mathbb{P}^2$. Furthermore, assume $C$ has a nodal singularity at some point $Q\in\mathbb{P}^2$. I am looking to resolve $C$'s nodal singularity by ...
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linear functions and physical constraints

I have a feeling I am missing something elementary with my question, but I can't seem to find where the problem is. In a certain physical problem, I have to evaluate an integral in order to obtain an ...
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1answer
32 views

Determining the type of singularities

Determine the type of singularities of $$f(z)=\frac{1}{(z-1)\cot(\pi/z)}\tag{1}$$ We first rewrite the function: $$f(z)=\frac{1}{(z-1)\cot(\pi/z)}=\frac{\sin(\pi/z)}{(z-1)\cos(\pi/z)} \tag{2}$$ ...
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1answer
29 views

Series solution of the second order ODE around a regular singular point

Here is the ODE I want to integrate, $$R''(y)-\frac{2}{k-y}R'(y)-\frac{l(l+1)}{(k-y)^{2}}R(y)=0$$ We see that it has a regular singular point at $y=k$ where $k<0$. Is there a way to obtain the ...
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1answer
48 views

Singularity of a surface in $\mathbb{P}(1,1,1,a)$ passing through the vertex

I would like to understand better the kind of singularities that we obtain in hypersurfaces in weighted projective spaces. For instance, let us consider the surface $S:x_0^{2a}+x_1^{2a}+x_2^{2a}=0$ of ...
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Computing singularities of a surface

Let $Y$ be the abelian variety $\mathbb{C}/\mathbb{Z}[i] \times \mathbb{C}/\mathbb{Z}[i] $. Let $X$ be the quotient of $Y$ by action of the group generated by the map $\eta(x,y)=(ix,iy)$. This group ...
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2answers
43 views

Why $\int^1_{-1}\frac{1}{\sqrt{x}}dx$ experiences no singularity at $x=0$?

Why $\dfrac{1}{\sqrt{x}}$ is not singular at $x=0$? My book says, in general, $\displaystyle\int^a_{-a}\dfrac{1}{x^n}dx$ converges for $n>1$, exists as a Cauchy principal value for $n=1$, and ...
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How to find the dimension of local algebra given an equation?

I have the equation : $ x^4+x^3 y^2+xy^5+y^7$ and I need to find the dimension of local algebra. can somebody say how should I do this?
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1answer
30 views

isolated singularity / Laurent series

I want to classify the singularities of $f(z)=\frac{\cos^2 z}{\sin^2 z}$ Maybe I can write: $$f(z)=\frac{\cos^2 z}{\sin^2 z} = \frac{1-\sin^2 z}{\sin^2 z} = \frac{1}{\sin^2 z}-1.$$ I can substitute $\...
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1answer
25 views

Laurent series/isolated singularity

I want to classify the singularities of $$ f(z)=\frac{\sin(2z)}{(z-1)^3}$$ The Taylor series is: $\sin(2z)=\sum_{k=0}^{\infty}\frac{(-1)^k}{(2k+1)!} 2^{2k+1} z^{2k+1}$ So: $ \frac{\sum_{k=0}^{\...
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1answer
44 views

Find order of pole $\frac{e^z -1}{z^2 +4}$, about $z=2i$.

$$\frac{e^z -1}{z^2 +4},\quad\text{about $z=2i$.}$$ The textbook I'm reading isn't specific about these case, only gives basic examples. Basically to find pole I'd have to expand a Laurent series ...
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42 views

Numerical integration of long fourth order tensor components containing singularities

I need to evaluate a number of integrals over a unit circle, whereby the integrands are very long fourth order tensor components which are functions of phi but also of other tensor components, i.e. i ...
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1answer
17 views

Classification of isolated singularity /Laurent series

I want to find out, what kind of singularties does $ f(z)= \frac{1}{z^3-z^5}$ have. I would do the following steps: $ f(z)= \frac{1}{z^3-z^5} = \frac{1}{z^3(1-z)(1+z)}$ so I have $ z_1=1, z_2=-1 , ...
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25 views

Definiteness of matrix after Woodbury inversion.

Consider a real, symmetric and positive definite $n\times n$ matrix $\mathbf{K}$, and a $n\times m$ matrix $\mathbf{W}$. $\mathbf{W}$ contains $m$ columns with all zeros except a single entry in each ...
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12 views

Tangent Space of the Surface Created from a k-Determined Constraint

For a $k$-determined smooth function $f : \Bbb R^n \to \Bbb R^m ,$ I have a surface $f^{-1}(0)$. Is it true that replacing $f$ with its $k$-jet about zero defines an equivalent tangent space at zero, ...
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2answers
78 views

A counter-example for integration by parts when there are “small” singularities

I am looking for a "counter-example" to integration by parts of the following type: $\Omega \subseteq \mathbb R^n$ is an open, bounded, connected domain with smooth boundary. $u,v:\bar \Omega \to \...
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1answer
39 views

Are the singular points of harmonic function on the disk always of measure zero?

Let $f : \mathbb D^2 \to \mathbb R$ be a smooth function with no singular points, i.e. $df \neq 0$ on $\mathbb D^2$. (Here $\mathbb D^2$ is the closed unit disk in $\mathbb R^2$). Let $\omega:\mathbb ...
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1answer
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$f$ has a pole at $z=a$ implies $1/f$ has a removable singularity at $z=a$

In Section V.1 of Conway's Functions of One Complex variable, he says that if $f$ has a pole at $z=a$ implies $[f(z)]^{-1}$ has a removable singularity at $z=a$. I am confused why $[f(z)]^{-1}$ should ...
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32 views

Determination of transition from non-singular matrix to singular matrix

I have the following matrix as a biproduct of inverting a matrix sum by the Woodbury matrix identity: $$ \mathbf{A} = -(g\mathbf{G})^{-1} + \mathbf{W}^T \mathbf{K}^{-1} \mathbf{W} $$ where $g$ is ...
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1answer
54 views

Why is $z_0=0$ not an essential singularity of $\tan(1/z)$ What is this singularity?

Since for any $\epsilon>0~\exists |z|=|\frac{2}{(2k+1)\pi}|<\epsilon$ for some $k$ large enough such that $\tan(1/z)=\pm\infty$ there cannot be a Laurent series at $0$. Does there need to be a ...
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3answers
59 views

Find the residue, state the nature of the singularity, find the constant term in $1/\sin(ze^z)$ at $z=0$

Find the residue, state the nature of the singularity, find the constant term in series $1/\sin(ze^z)$ at $z=0$. We can rewrite the function $\frac{1}{\sin(ze^z)}$ as $\frac{ze^z}{\sin(ze^z)}\cdot\...
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2answers
95 views

Black Holes and the Schwarzschild Solution

From Section 9.1, in General Relativity by Woodhouse: For a normal star, the Schwarzschild radius is well inside the star itself. As it is not in the vacuum region of space-time, the Ricci tensor ...
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0answers
44 views

Weird eigensystem of ODE with regular singularity

Consider the eigenvalue problem of the following 2nd-order ODE $$(x/2+a)^2y(x)-xy'(x)-x^2y''(x)=\lambda^2y(x),$$ in which $y\in(-\infty,+\infty)$ and parameter $a>0$. It has a regular singularity $...
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0answers
34 views

Bessel equation solution in $(-\infty,0)$

Let $x \in \mathbb{R}$. Homogeneous Bessel differential equation $$x^2 \frac{\mathrm{d}^2 f(x)}{\mathrm{d}x^2} + x \frac{\mathrm{d} f(x)}{\mathrm{d}x} + (x^2 - n^2) f(x) = 0 \label{a} \tag{1}$$ has ...
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2answers
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Understanding Proposition $3.14$ in Ullrich's Complex Made Simple

In Ullrich's Complex Made Simple, Proposition $3.14$ part $ii$ states If $f \in H(D'(z,r))$ (holomorphic in the punctured disk) has an essential singularity at $z$ if and only if $f(D'(z,\rho))...
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2answers
27 views

Analyzing isolated singularities using $\lim_{z\to z_0}(z-z_0)f(z)$

so I currently read about isolated singularities and residue. If we have a complex function $f(z)$ with a simple pole at $z_0$ we can use $$\operatorname{Res}(f;z_0)=\lim_{z\to z_0}(z-z_0)f(z) \tag{...
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3answers
45 views

Principal part of function at a pole

I have a function $\dfrac{e^zz}{z^2-1}$. It has isolated singularities $z=\pm 1$. To find the principal part at $z=1$, I am trying to find a Laurent series expansion around $z=1$. I have the following ...
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What mathematical quantity can characterise the extent of twisting at “twisting singular points”?

Recently, inspired by the phenomenon of frame dragging of spacetime around rotating massive bodies, I became curious on how to characterise how much a surface locally twists For example, in this ...
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1answer
46 views

Singularity $z/\sin(z)$

I'm trying to figure out the singularity of $$\frac z {\sin(z)}$$ I understand how it works with $$\frac {\sin(z)} z$$ But here I don't know what to do. I know, that the solution is that it has a ...
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1answer
39 views

Evaluate a complex function

Let $\varepsilon>0$. Let $f:B_{\varepsilon}(0)\rightarrow\mathbb{C} $ be an analytic function such that $f(0)=0$, $f(a)=a$ for some $a\in B_{\varepsilon}(0)-\{0\}$ and $\{0,a\}$ are the only fixed ...
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1answer
36 views

What are the singular points of $z/z$

I'm trying to understand what happens when you have on point that can evident can be both. Or for example $z/z^2$ or $z^2/z$. Any examples that can clarify this for me?
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1answer
25 views

Determination of entire functions given with a removable singularity. [closed]

Determine all entire functions $f(z)$ such that $0$ is a removable singularity of $f\big(\frac{1}{z}\big)$. I have no idea how to start with.
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1answer
29 views

Solving Trignometric integral with the aid of residues.

If $\alpha, \beta, \gamma$ are real numbers such that $\alpha^2> \beta^2+\gamma^2$ show that $$\int_0^{2\pi}\frac{d\theta}{\alpha+\beta \cos \theta +\gamma \sin \theta} = \frac{2 \pi}{\sqrt{\alpha^...
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1answer
21 views

Magnetic field at points on the circuit

I know magnetic field lines due to a circuit always form closed loops. Therefore $\nabla \cdot \vec{B}=0$ everywhere (even at points on the circuit). However due to singularity, magnetic fields are ...
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0answers
21 views

Find maximal ideals in semi local ring with singular

I am trying to analyze the normalization $N$ of a local ring $A_{\mathfrak{m}}$ of a variety with Singular. The normalization (integral closure in its total ring of fractions) is semi-local with its ...
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1answer
14 views

Determination of the order of a pole

In the function $$f(z) =\frac{sin(\frac{\pi}{2}(z+1))}{(z^2+2z+4)(z+1)^3}$$ the order of the pole in $z=-1$ is correctly two? Or maybe it is an eliminable singularity? I have a problem because often ...
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1answer
56 views

Calculating the residue of $\frac{1}{z^2 \sin z}$ at $z = 0$

I am having some difficulty calculating the residue of $\frac{1}{z^2 \sin z}$ at $z = 0$. From what I can tell, we are dealing with an essential singularity here and so the problem becomes that of ...
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2answers
21 views

Singularity of $\frac{z+2}{e^{\frac{1}{(z+2)^2}}}$

I have a singularity in $z=-2$, now I wolud like to find the kind of singularity, so I have to compute the limit: $$\lim_{z\to-2}\frac{z+2}{e^{\frac{1}{(z+2)^2}}}$$ for me this limit is $0$, because ...
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1answer
76 views

Isolated and non isolated essential singularity at same point?

I need to find the singularities of $$f(z) = \frac{1-e^z}{2+e^z}$$ My effort: Poles of function are given by $$2+e^z=0\implies e^z = -2 \implies z = \log 2+i(2k+1)\pi$$ for k integer. All these are ...
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2answers
43 views

If $f : D(0;1) \setminus \{ 0 \}$ holomorphic such that $f(1/n) = 0$ then either $f=0$ or $f$ has an essential singularity at $0$

I'm trying to solve this question: Let $f$ be a holomorphic function on $D(0; 1)\setminus\{0\}$ with the property that $f(1/n) = 0$ for every positive integer $n$. Show that $f$ is either ...
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0answers
24 views

limit of the ratio of two divergent integrals

I have to calculate two ratios of two pairs of proper integrals, for a range of parameters. I am stuck when it comes to calculating these ratios when the parameters are such that the integrands start ...
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1answer
74 views

Identify and classify the singularities of $\frac{1}{\exp(\frac{1}{z}) + 2}$

The title pretty much explains it. I'm trying to answer a question where I'm given a few complex functions and it asks me to identify their singularities, and then to classify any that are isolated. ...
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2answers
38 views

Why is $i$ a removable singularity of $\frac{\sin(z-i)}{z^2+1}$?

Why is $i$ a removable singularity of $f(z)=\frac{\sin(z-i)}{z^2+1}$? We can find the Taylor expansions of the $\sin(z-i)$ and $\frac{1}{z+i}$ to get the Laurent series (actually Taylor series since $...
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1answer
31 views

How to show that f / f' has a removable singularity at 0

I'm new to complex analysis and am not sure where to start with this. The question states: Let the origin be a pole of order m > 0 of an otherwise analytic function f of a complex variable. Show ...
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1answer
34 views

Contradiction: Portion of our area is greater than our full area

I was reading an answer to a stack exchange post titled Is the electric field of a volume charge distribution well defined? . That answer is shown in the image below: Now I make a comparison ...
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0answers
25 views

Extension of holomorphic function to exceptional divisors

Let $(X,0) \subset \mathbb{C}^n$ be complex analytic set with an isolated singularity. Let $f:(X,0) \to (\mathbb{C},0)$ be a germ of a reduced holomorphic function germ defined on it. Let $\pi:E \to \...
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4answers
40 views

How to find the orders of the poles of $\frac{z^2}{1-\cos(z)}$?

The poles are $z=2k\pi$ for $k\in \mathbb Z$. I think the order of $z=0$ is one, but how to show it? How to show the orders of other poles are also $1$?
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0answers
27 views

Classifying the Singular points of $\frac{z\sin{(\pi z)}}{(z+1)(z-1)^2}$

I am trying to classify the singular points of the function $$g(z)=\frac{z\sin{(\pi z)}}{(z+1)(z-1)^2}.$$ The singular points are $z=\pm 1$. For $z=1$, I noticed that this is a simple zero of $z\...