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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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Divisors $D$ such that $(\mathbb{A}^{2}_{\mathbb{C}},\frac{3}{4}D)$ has strictly log canonical singularities

Let $P:= 0\in\mathbb{A}^{2}_{\mathbb{C}}$ and $D$ be a divisor in $\mathbb{A}^{2}_{\mathbb{C}}$. Let us assume that the pair $(\mathbb{A}^{2}_{\mathbb{C}},\frac{3}{4}D)$ has strictly log canonical ...
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Residue of pole at higher power in the denominator

I need to calculate the residue of $\frac{1}{z^{2017}}$ My thought process would be to use the derivative formula for a pole of higher order for the pole at $0$ of order $2017$ but I can’t be ...
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Understanding the Laurent series $\frac{1}{a_n+a_{n+1}(z-z_0)+…}=c_{-n}+c_{-n+1}(z-z_0)+…$

Let $z_0$ be a polo singularity, f(z) is analytic in the neighbourhood excluding $z_0$. Then $\phi(z)=\frac{1}{f(z)}$ which implies $\lim_{z\to z_0}\phi(z)=0$ So the $\phi(z)$ has the Laurent series:...
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Nonlinear DAE solution procedure

Nonlinear DAE solution Consider the following differential algebraic system (DAE), that are obtained by physical modeling of a system: $\dot{x}_1=k_1 x_1 \left(1-x_1^{\epsilon_1} \right)-k_2 (a+b u)(...
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Finding the singular locus of the given complex space

This problem is from Greuel et al., Introduction to Singularities and Deformations. Determine the singular locus of the complex spaces defined by the following $\mathcal{O}_{\mathbb{C}^n}$-ideals: (...
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factoring out a singularity of a 2nd order ordinary differential equation

I am looking at a 2nd order ODE: \begin{equation} \frac{d^2u}{dx^2} + p(x)\frac{du}{dx} + q(x) u = 0 . \end{equation} that has five regular singular points: say at $0$, $1$, $a$, $b$, and $\infty$...
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eigenvalue problem for a cusps model.

In applied physics we need to solve the model cusps model to get corresponding stationary solutions of the system called eigenfucnctions. These stationary solutions show the behaviour the system along ...
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The integral of a radius function 1/(r(x)*e^r(x))

I am aproximating a radius function with quadratic isoparametric line elements, like this: d1 = x1*0.5*x*(x-1)+x2*(x+1)*(x-1)+x3*0.5*x*(x+1) d2 = y1*0.5*x*(x-1)+y2*(x+1)*(x-1)+y3*0.5*x*(x+1) where ...
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Condition for the power series to have a single singularity on its circle of convergence

I'm looking to the prove of the following statement: If for the complex power series $\sum^\infty_{i=1} a_i z^i$ it holds that $$ \limsup_{n \to \infty} \sqrt[n]{\left| \frac{a_n}{a_{n+1}} -z_0 \...
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What type of singularity is $z=\infty$ for $f(z)=\frac{1}{(sin(1/z))}$?

Consider the function $$f(z)=\frac{1}{(sin(1/z))}$$ At $z=\infty$ does $f$ have an isolated singularity or not? Or is $z=\infty$ a regular point? $f(1/t)=1/(sin(t))$ has simple poles in $t=k \pi$, ...
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Why are we not using Dirac delta and ignoring the contribution to the surface integral from the point $r=0$?

Let $V'$ be the volume of dipole distribution and $S'$ be the boundary. The potential of a dipole distribution at a point $P$ is: $$\psi=-k \int_{V'} \dfrac{\vec{\nabla'}.\vec{M'}}{r}dV' +k \oint_{...
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How to prove $\displaystyle \int_V \dfrac{dx\ dy\ dz}{r^2}$ doesn't contain singularity?

Let's consider the transformation from spherical to Cartesian coordinates: $r, \theta, \phi\overset{T}{\rightarrow}x,y,z$ Let: $\vec{a}=\vec{r} (r+\Delta r,\ \theta,\ \phi)-\vec{r} (r, \theta, \phi)$...
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Clarification on some surface integrals in an MSE answer

I am trying to understand this answer. The answer is the following: I'll assume that $V'$ is bounded and that $\mathbf{M'}$ is continuously differentiable. To your primary question: Is [...
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35 views

example of algebraic variety with infinitely many singularities

Let $X$ ba an algebraic variety and $\mathrm{Sing}(X)$ be the set of all singular points. For a set $A$ , $|A|$ denotes the cardinality of $A$ . I konw examples of algebraic variety with finite ...
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Defined and Undefined Vector Fields

Given a vector field which has circulation but no curl resulting from an undefined point, for example, $\begin{bmatrix}-\frac{y}{x^2+y^2}\\\frac{x}{x^2+y^2}\end{bmatrix}$, does there exist, or can ...
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Determining the order of the poles of the function $\frac{1}{\sin z-\sin 2z}$

I encounter a question in my problem sheets, which asks to identify the type of isolated singularities of the following function: $$\frac{1}{\sin z-\sin 2z}$$ Firstly, by trig identities, I can ...
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The fundamental solution for Laplace's equation in cylindrical coordinates

I am working my way through the excellent textbook of Garabedian on partial differential equations and have two questions related to the topic of the fundamental solution in chapter 5: Garabedian ...
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Principal Part of Laurent Series Converges in Punctured Disc

I'm trying to work through the following problem: Prove that if the holomorphic function $f$ has an isolated singularity at $z_{0}$, then the principal part of the Laurent series of $f$ at $z_{0}$ ...
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Extreme values of $\frac{1}{x^2 + y^2 -1 }$

Let $f(x,y)= \frac{1}{x^2 + y^2 -1 }$ . I want to find its extreme values. Its first partial derivatives are $f_x(x,y) = \frac{-2x}{(x^2 + y^2 -1)^2}$ and $f_y(x,y) = \frac{-2y}{(x^2 + y^2 -1)^2}$ ...
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Can singularities appear on cartesian planes as vertical lines?

The equation $y(x-1)=x^2-1$ can be graphed on Cartesian plane by inserting values in for $y$ and then solving for $x$ For example, if $y=3$ then: $3(x-1)=x^2-1$ $3x-3=x^2-1$ $3x=x^2+2$ $0=x^2-...
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local and regular integral closure

Let $(A,\mathfrak{m})$ be a local domain with integral closure $B$ which is a regular local domain with maximal ideal $\mathfrak{M}$. Assume in addition that $A/\mathfrak{m} = B/\mathfrak{M}$ and $B\...
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Show that if $f(z)=w$ has at most $n$ solutions for each $w \in \mathbb{C}$ then the pole $z_0$ has order at most $n$

Let $D \subset \mathbb{C}$ be open and connected, $z_0 \in D$ and $n \in \mathbb{N}$. Suppose that $f : D \ \backslash \{z_0\} \to \mathbb{C}$ is holomorphic in $D \ \backslash \{z_0\}$ with the ...
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Construction and properties of flat family of elliptic curves

I have the following situation: let $k$ be algebraically closed of characteristic 0 (one can assume $k=\mathbf{C}$ if this simplifies the discussion) and let $\varphi : \mathfrak{X}\longrightarrow \...
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1answer
133 views

Showing volume and surface integration is unaffected by the singularity at $\mathbf{r'}=\mathbf{r}$

This question is not entirely similar to the question here. Please read this question and the reader will see it is obviously not the same. $\mathbf{M'}$ is a continuous vector field in volume $V'$ ...
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Piecewise holomorphic map and complex submanifolds

Let $X$ be a complex manifold and $Y\subset X$ a compact complex submanifold of codimension $1$. Let $f:X\to Z$ be a continuous map such that $f|_{X\setminus Y}$ is biholomorphic and $f|_Y$ is ...
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Application of Casorati Weierstrass

I have the following Theorem in my lecture notes: If $f:D \to \mathbb{C}$ is holomorphic on $D-\{z_0\}$ where $D$ is open and connected and $z_0$ is an isolated singularity of f, then: $f$ has a pole ...
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Inverse of a matrix which is difference of a singular matrix with a small diagonal matrix?

If $A$ is a real symmetric singular matrix (similar to a Laplacian matrix, which comes from $M^{T}GM$, where M is incidence matrix and G is a diagonal matrix). G has large values compared to $B$, ...
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46 views

Singular points of non-linear ODE

EDIT: Sorry i messed up, I forgot a minus sign in front of the left hand side. I added it now. I am not sure how to proceed with this. Given this non-linear ODE$$\partial_{t}u(t,x)=-\cot(t)\left[\...
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Why is the singularity ignored?

In this article "Reflections on Maxwell’s Treatise", Section 4.2, it says: He replaces $\mathbf{m}$ with a volume element of magnetization $\mathbf{M}\ dV$ , integrates over $V$ , and lets the same ...
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Monodromies of complex differential equation

Let $A:\mathbb{C}^*\to\mathrm{M}_n(\mathbb{C})$ be a holomorphic map. Consider the system of first order differential equations \begin{equation} \begin{cases} \frac{dY}{dz} = A Y\\ Y(1)= I, \end{cases}...
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Classifying of isolated singularities

I want to find out, which singualrities $f(z)=\frac{z}{e^z+1}$ have? $e^z+1=0 \Leftrightarrow z_k=(2k+1)i \pi $ But how can I find out, of which type these singualrities are?
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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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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
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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
34 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
50 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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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
34 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
30 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
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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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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
20 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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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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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
45 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
31 views

$f$ 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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0answers
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 ...