In mathematics, singularity theory studies spaces that are almost manifolds, but not quite. A string can serve as an example of a one-dimensional manifold, if one neglects its thickness. A singularity can be made by balling it up, dropping it on the floor, and flattening it. (Def: ...

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872 views

Is there a classification of isolated essential singularities?

In the thread Why do we categorize all other (iso.) singularities as "essential"?, here is one of the questions that was asked: Do we not care about essential singularities to classify ...
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3answers
445 views

How to know if a tangent bundle is trivial from its defining equations

In this question, I am considering only regular manifolds. A Trivial Bundle The circle $S^1$ is known to have a trivial tangent bundle. As a subset of $\mathbb{R}^4$, the tangent bundle of $S^1$ ...
7
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1answer
94 views

Showing a complex analytic function is unbounded

This was one of the problems on a previous year's Complex Analysis final exam. Assume $f\in \mathcal O (\mathbb H )$, non-constant, and $f(\frac {i}{\sqrt n})=0, \forall n\in \mathbb N$. Prove that ...
6
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3answers
3k views

Type of singularity of $\log z$ at $z=0$

What type of singularity is $z=0$ for $\log z$ (any branch)? What is the Laurent series for $\log z$ centered at 0, if exist? If the Laurent series has the form $\sum_{k=-\infty}^{\infty} a_kx^k$, ...
6
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1answer
100 views

Books or texts on singularity theory [closed]

So a friend is doing his PhD in maths (algebraic topology) and his advisor wants him to publish something on singularities (of which, as fas as I understand, he knows next to nothing). I want to give ...
5
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2answers
711 views

$z_0$ non-removable singularity of $f\Rightarrow z_0$ essential singularity of $\exp(f)$

Let $z_0$ be a non-removable isolated singularity of $f$. Show that $z_0$ is then an essential singularity of $\exp(f)$. Hello, unfortunately I do not know how to proof that. To my ...
5
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2answers
756 views

Singularities of $e^{z - \frac{1}{z}}$

I believe $e^{z - \frac{1}{z}}$ has essential singularities at $z = 0$ and $z = \infty$ (in both cases because of a $\frac{1}{z}$ in the exponent) but I'm having a hard time proving this. How can one ...
5
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2answers
114 views

How to get the asymptotic form of this oscilatting integral?

So the integral is like this: $$\int_1^\infty \frac{\cos xt}{(x^2-1)\left[\left(\ln\left|\frac{1-x}{1+x}\right|\right)^2+\pi^2\right]}\mathrm{d}x$$ The question is how to get the asymptotic form of ...
5
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1answer
124 views

Distribution with singularities.

I need some help to prove that $f$ defined by $\langle f,\psi\rangle:= \sum_{n=0} ^\infty \psi^{(n)}(n)$ is a distribution which has singularities of infinite order. Here $\psi$ is a test function ...
5
votes
1answer
106 views

germ finitely determined

Does anyone know any result on finitely determined germs to help me prove that the germ $f(x,y)=x^3+ xy^3$ is $4$- determined? I tried using the definition of germs finitely determined, which is:$f: ...
4
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1answer
530 views

How to analyze the asymptotic behaviour of this integral function?

Based on the asymptotic analysis of correlation functions at large distence in Physics, now I get a math question. Let the function $$f(x)=\int_{-1}^{1}\sqrt{1-k^2}e^{ikx}dk.$$ Without working out ...
4
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2answers
147 views

Do the polynomial germs generate all the ring of germs?

I'm trying to understand some equality that comes up in stability theory involving sets of germs and I think I need a result like the next one, so if anyone knows anything about this and helps me it ...
4
votes
1answer
62 views

Resolution of singularities of the determinant hypersurface

Let $$\det\nolimits_n=\sum_{\pi\in\mathfrak{S}_n} \operatorname{sgn}(\pi)\cdot x_{1,\pi(1)}\cdots x_{n,\pi(n)} \in \mathbb{C}[x_{ij}\mid 1\le i,j\le n]$$ be the determinant polynomial. It defines a ...
4
votes
1answer
152 views

How many types of surface singularities multiplicity two exist?

All varieties are over $\mathbb{C}$. Let $S$ be a reduced algebraic surface in $\mathbb{P}^3$ with a singular point $p$ of multiplicity two. The question is local so we reduce to $S \subset ...
4
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2answers
84 views

Preimage of singular points of smooth map between manifolds

Given a smooth ($C^{\infty}$) map $\phi: V \rightarrow SU(n)$ where $V$ is a (finite dim, real) vector space (of potentially very large dimension) and $SU(n)$ is the special unitary Lie group, what ...
4
votes
2answers
342 views

Singularities of $f(z)=z/\cos(z)$

Regarding complex functions (in complex variables), I was wondering why the function $g(z)= \cos(z)$ has a singularity at $z = \infty$ but $f(z)= \dfrac{z}{\cos(z)}$ does not. I am a bit confused ...
4
votes
1answer
107 views

Singularities on a weighted projective curve

Let $C$ be the curve of degree $3$ defined over $\mathbb{C}$ given by $$x(y+z)=y^3-z^3$$ which lives in the weighed projective space $\mathbb{P}(x,y,z)=\mathbb{P}(2,1,1)$. Is the curve singular ?
4
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0answers
64 views

Minimal resolutions of singularities in higher dimensions and their cobordism class.

For a given singular algebraic variety over $\mathbb{C}$, how many minimal resolutions can exist? More specifically, consider a projective quadratic cone $Q$ in $\mathbb{C}P^{4}$ (with homogeneous ...
4
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0answers
118 views

Is it true that blowing up a quasi-affine variety at a nonsingular point never introduces new singularities?

If we let $M$ be a quasi-affine variety, is it true in general that the blowup of $M$ at a non-singular point $p$ does not introduce new singularities? I came across this statement in my reading, but ...
3
votes
2answers
224 views

Types of singularities of a function

How can I determine the type of a singularity of a function $$f(z)={e^{1/z} \over z-1}+{\pi z \over 2\sin(\pi z)}$$ at $z_0=0$? I don't see an easy way to represent it using Laurent series, neither ...
3
votes
2answers
575 views

Proving that a function has a removable singularity at infinity

I'm having trouble with the following exercise from Ahlfors' text (not homework) "If $f(z)$ is analytic in a neighborhood of $\infty$ and if $z^{-1} \Re f(z) \to0$ as $z \to \infty$, show that ...
3
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1answer
105 views

Does $\frac {z^5}{\sin z^2-z^2}$ have a non-isolated singularity at $0$?

Does $\frac {z^5}{\sin z^2-z^2}$ have a non-isolated singularity at $0$? If so, is it not meaningful to discuss its residue at $0$?
3
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1answer
97 views

What is the value of this integral (using the Argument Principle),

F(z) is given by $$F(z) = e^zz^{-2}(z-1)(z^2-4)(z+8)^7$$ What is the value of the integral $$\int_0^{2\pi} \frac{F'(3e^{i\theta})}{F(3e^{i\theta})}d\theta \space \space ?$$ I think the relevant ...
3
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1answer
89 views

Meromorphic function with a simple pole and a simple zero, and satisfies an inequality. What can it be?

Describe all meromorphic functions f(z) in the complex plane with a simple pole at z=1, a simple zero at z=-1, and for which $$|f(z)|\le M|z|,$$ for $|z|\ge 2$ for some $M>0$. I know that, ...
3
votes
1answer
140 views

Classify Singularities

So, I'm trying to classify the singular points of the following function: $$ f(z)=e^{\cot(\frac {1}{z})} $$ Obviously, when z is zero, the function tends to approach infinity, so that must be a ...
3
votes
1answer
53 views

Why is $P$ singular

This is from Shafarevich's 'Basic Algebraic Geometry 1': Let $P=(\alpha,\beta)\in X$, and suppose that the equation of $X$ is written in the form $f(x,y)=a(x-\alpha)+b(x-\beta)+g$, where $g$ is a ...
3
votes
1answer
57 views

Help with understanding the notation $\mathbb{C}\{f\}$

I am reading an article "Relative Cohomology and volume forms" of J. P. Francoise. Here the author considers the germ of a function $f\colon(\mathbb{C}^n,0) \to (\mathbb{C},0)$, and he speaks of the ...
3
votes
1answer
78 views

Can the Milnor number be used to resolve curve singularities?

Let $f(x,y)\in \mathbb{C}[x,y]$ define a curve $C$ which is singular at the origin. By successively blowing-up the origin, we can resolve the singularities of $C$. Of course to make sure this process ...
3
votes
1answer
120 views

Singularities in (Elementary) Real Algebraic Geometry

I've taken an introductory course in algebraic geometry, and am currently studying Sumio Watanabe's book, "Algebraic Geometry and Statistical Learning Theory". In this book Watanabe gives a, possibly ...
3
votes
1answer
124 views

Discrepancy non log canonical singularities

Suppose that $Y$ is a normal variety such that its canonical class $K_Y$ is $\mathbb{Q}$-Cartier. , and let $f:X \to Y$ be a resolution of the singularities of $Y$. Then $$ K_X= f^*(K_Y)+\sum_i ...
3
votes
0answers
41 views

General Fiber in Positive Characteristic

It is well-known that a complex polynomial, considered as a function $f:\mathbb{C}^n\to\mathbb{C}$, is a fiber bundle over a cofinite set of "atypical values" which include the singular values of $f$. ...
3
votes
1answer
85 views

Laurent series of an analytic function divided by $z$

This is a probably basic question about Laurent series. Say $g(z)$ is an analytic function, that $g(0) = 0$, and $f(z) = g(z)/z$. My textbook says $z = 0$ is a removable singularity of $f(z)$. A ...
3
votes
2answers
211 views

The notion of a germ in singularity theory

I quote from my lecture: Let $X$ be a topological space (think of $X=\mathbb{C}^n$ with the classical topology), $p\in X$, $A,B\subseteq X$. Then $A\sim B$ if there exists an open subset ...
2
votes
2answers
444 views

What is $\lim\limits_{z \to 0} |z \cdot \sin(\frac{1}{z})|$ for $z \in \mathbb C$?

What is $\lim\limits_{z \to 0} |z \cdot \sin(\frac{1}{z})|$ for $z \in \mathbb C^*$? I need it to determine the type of the singularity at $z = 0$.
2
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1answer
1k views

Invertibility of a square matrix with zero diagonal elements and positive non-diagonal elements

$M$ is square and $$M(i,j)=0, i=j$$ $$M(i,j)>0, i\ne j$$ Is $M$ full-rank or invertible? Actually the $M$ I am studying has much stronger properties but I guess the simple conditions above might ...
2
votes
2answers
912 views

Contour integral $\int_{|z|=1}\exp(1/z)\sin(1/z)dz$

Evaluate the contour integral $$\int_{|z|=1}\exp(1/z)\sin(1/z)\,dz$$ along the circle $|z|=1$ counterclockwise once. The singularities are $\dfrac1{\pi k},k\in\mathbb{Z}$ plus the limit point $0$. So ...
2
votes
1answer
35 views

Question about finding Laurent Series over closed region and classifying singularity

Represent $\sin(\pi x/(x+1))$ Laurent Series about the region $0<|x+1|<2$: Its true that $$\sin(x)=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\cdots$$ So the $$\sin(\pi x/(1+x))=\sum (-1)^{n-1} \frac{(\pi ...
2
votes
2answers
62 views

Why does the function $f(z) = 1/\sin(\pi/z)$ have isolated singular points?

In the complex analysis text book "Complex Variables and Applications 8th edition", it states the function $1/sin (\pi/z)$ has singular points $z = 0$ and $z = 1/m; (m = \pm 1,2,3,4,\dots.) $ I sort ...
2
votes
1answer
124 views

Singularity models of the Ricci flow

I faced this sentence in my studies on Ricci flow: The Bryant soliton is a singularity model for the degenerate neckpinch. Q1: What is the definition and meaning of singularity model? Can one model ...
2
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1answer
67 views

Do singularities always appear on all Riemann sheets?

Consider a function $$f(z) = \frac{\ln z}{z^2+1}.$$ Besides the branching point $z=0$, the function also has singularities at $z = \pm i$. This singularities should appear on all Riemann sheets. Is ...
2
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1answer
82 views

Local rings and classifying singularities

My query is a little vague, but I'll try to be as concrete as possible. Is there some sense in which the local ring of an algebraic variety (or more general complex space) at a point depends only on ...
2
votes
1answer
5k views

Singular points of ODE

My friend and I have conflicting answers and since his phone is off, I can't get his full solution and I don't understand his argument. Consider this ODE $$(x+1)y''+\frac{1}{x}y' + (x+3)y= 0$$ ...
2
votes
1answer
123 views

Lipschitz manifold and semi-algebraic is Lipschitz graph?

It is known that there are Lipschitz manifolds that are not Lipschitz graphs and $C^1$ manifold is locally $C^1$ graph. If $M\subset \mathbb{R}^m$ is a Lipschitz manifold (with the outer distance) ...
2
votes
1answer
85 views

Removable singularities for continuous functions

Let $f: D - K \rightarrow \mathbb{C}$ be holomorphic, where $D$ is a planar domain and $K$ is a compact subset of $D$. Suppose that $f$ extends continuously to all of $D$. On which conditions on $K$ ...
2
votes
1answer
41 views

Conditions for the number of degenerated fibers of a morphism to be finite.

I'm having trouble to find a theorem about this: Let's $F_z$ be a family of curves defined by $$F_z : F(x,y) + z = 0$$ where $F$ is an irreducible polynomial and $z \in \mathbb{C}$. My question is: ...
2
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1answer
60 views

Growth rate of Taylor convergents near pole

For any fixed $z_0\in\mathbb{C}\setminus \{0\}$ and $\beta\in\mathbb{R}^{+}$, prove that $$\left.T_n\left(\log^{\beta}z;z_0\right)\right|_{z=0}\sim\log^{\beta} n$$ Note: I observed that this holds ...
2
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1answer
240 views

How to describe the singularities of a function

This is a question of an old exam of my complex analysis course and although I think I understand what a singularity is, I oftentimes have troubles 'finding and describing' them properly. I looked at ...
2
votes
1answer
33 views

Decomposition of resolvent in projections

I am reading the book Perturbation theory for linear operators from Kato. He defines (§5 Section 3) for an operator $T : X\to X$ on a finite Banach Space the resolvent as $$ R(x) = (T- x)^{-1}.$$ ...
2
votes
0answers
87 views

Manifolds, isolated singularities, open map

I can't figure out how to solve the following question: 1) Let $M$ and $N$ be $C^k$-manifolds, such that $\dim M = \dim N = n>1$, and let $f:M\rightarrow N$ be a $C^k$-function. Show that if the ...
2
votes
0answers
48 views

Requesting information on constructed discontinuous functions (from any perspective)

Suppose $f:\mathbb{R}\rightarrow \mathbb{R}$ is a continuous function. Define the function $F:\mathbb{R}\rightarrow \mathbb{R}$ as $$F(x)=f(x)\prod_{n=1}^\infty\frac{x-\frac{1}{n}}{x-\frac{1}{n}}$$ I ...