Questions tagged [singularity]

This tag is for questions relating to singularity, which is a point where a mathemtical 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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Which type of singularity does this complex function have at z = ∞?

I'm trying to classify the type of singularity at z = ∞ (the point at infinity) of the complex function: click to see the equation Up to now, I've just been able to prove that ∞ is not a pole. So I ...
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Are the two different Forms of the damped pseudo-inverse (damped Moore–Penrose inverse) equal?

To generalize the concept of the inverse to non square matrix the pseudo inverse (also called Moore–Penrose inverse) can be used: For tall matrices with full column rank: $$A^+ = (A^T \cdot A)^{-1} A^...
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Non-isolated singularity and contour integral

I encounter contour integrals of the following form, $$ \oint_{|z| = 1} f_q(z) \frac{ dz }{ 2\pi i}\ , \qquad |q| < 1 $$ where the meromorphic function $f_q(z)$ contains a lot of simple poles of ...
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Bounding integrals using asymptotic expansions of the integrand

Im following the book "Analytic Combinatorics" by Flajolet and Sedgwick. I'm having trouble understanding the last part of the proof of the theorem regarding the standard function scale. In ...
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What about a manifold makes a velocity curve rotate about a point more than once?

I have a parameterized 2-surface $M(x,y)$. On it, I define a closed curve parameterized by $t$ using $x=r cos(t), y=r sin(t)$, for example. Then, the ‘velocity curve’ is the tangent at each point ...
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30 views

Does a slow holomorphic function exist?

Backstory. I was trying to come up with a function $f(z)$ that would have an essential singularity in $z = 0$ and would be limited by the following function with non-integer power (for some $\alpha$ ...
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1answer
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Does Associated Legendre Function of Second Kind Give Delta Function?

Associated Legendre Function of Second Kind is singular at $x=\pm 1$. So I am wondering whether it satisfies the corresponding differential equation everywhere or there is a hidden functional of delta ...
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An improper integral involving exponentials and a power function.

Let $\alpha >0$ and $2 n > \alpha$ and $n \in {\mathbb N}$. We consider a following integral: \begin{equation} {\mathcal A}_{2 n}(\alpha) := \int\limits_0^\infty \left( \frac{1-e^{-x}}{1+e^{-x}} ...
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1answer
33 views

Function computing the order of a pole

I'm wondering if there's a function (mapping into the natural numbers) that computes the order of a pole of a meromorphic function ? Put a little different, how does mathematical software finds this ...
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How to calculate the SVD of a $3×3$ diagonal only matrix without computing the eigenvalues explicitly.

Given $$A = \begin{bmatrix}1&0&0\\0&-2&0\\0&0&0\end{bmatrix} $$ What would be its SVD? Key facts we can use: a) Singular values in the Sigma matrix are on the principal ...
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Cauchy-Goursat Theorem with singularities in $\partial R_0$

Consider $\overline{R}:[-a,a]\times [-b,b]$ and $\varphi: [-a,a] \rightarrow [-b,b]$ a continous map. Let $K$ be the graphic of $\varphi$. if $f: int(\overline{R}) \rightarrow \mathbb{C}$ is continous ...
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Why does the blowup of a curve at a singular point decreases the arithmetic genus?

In Beauville's Complex Algebraic Surfaces, problem II.20 we are asked to show that an irreducible curve $C$ in a smooth surface $S$ becomes smooth after a finite number of blowups. He says that a way ...
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Behavior of $e^{ f(z)}$ if f has removable singularity at a

This particular problem was asked in my assignment which could not be discussed due to pendamic. Question : If f is holomorphic on $\Omega$/{a} prove that $e^{f(z)}$ cannot have a pole at a . What I ...
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$f$ has an isolated singularity at a point a then $e^{f} $ can't have pole at $a.$

This question was part of an assignment that couldn't be discussed due to the pandemic. Question: Let $f$ have an isolated singularity at a point $a$. Prove that $e^{f}$ cannot have a pole at point $...
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19 views

If a holomorphic function satisfies this integral condition then It must have singularity [duplicate]

This question was part of my assignment which couldn't be discussed due to pendamic. If f $\in H(0<|z|<R)$ and $\int_{0<x^2+y^2<R} {|f(x+iy)|}^2 dx dy <\infty$ prove that f has either ...
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Regular singular points of a Fuchsian equation

I was reading the book Painleve transcendents: The Riemann-Hilbert approach and it said the following: Suppose that the rational matrix function $A(\lambda)$ has only simple poles. Then all singular ...
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27 views

Behavior of $e^{ f(z)}$ if f is holomorphic on $\Omega$\ {a} [duplicate]

This particular problem was asked in my assignment which could not be discussed due to pendamic. Question : If f is holomorphic on $\Omega$/{a} prove that $e^{f(z)}$ cannot have a pole at a . What I ...
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1answer
29 views

Finding sum of residues of a rational function [duplicate]

This question is part of an assignment of an institute in which I don't study . Question: If p and q are polynomials with deg(q)>deg(p)+1 prove that sum of the residues of p/q at all its poles is ...
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1answer
29 views

Image of punctured disk around essential singularity

I am studying Picard's great theorem in complex analysis and I saw two different forms of theorem.For instance in convoy the theorem is stated as: Great Picard's Theorem: If an analytic function $f$ ...
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1answer
34 views

Desingularization of a fibered surface with smooth generic fiber - missing argument/assumption in Liu?

Theorem 8.3.50 in Liu's Algebraic Geometry and Arithmetic Curves states the following (this is only the relevant part of the statement): Let $S$ be a Dedekind scheme of dimension $1$. Let $\pi:X\to S$...
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Question about poles of the Lerch Transcendent

The Lerch Transcendent is defined as the analytic continutation of the sum $$ \Phi(z,s,a)=\sum_{k=0}^\infty(k+a)^{-s}z^k. $$ According to Wolfram functions, for fixed $s$, $a$, the function $\Phi(z,s,...
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1answer
51 views

Riemann's theorem for singularity(Removable)

Let the $f(z)$ be analytic and $\vert f(z) \vert \leq \vert \sin({1\over z})\vert$ on $\mathbb{C}^\# (= \mathbb{C} \setminus \{0\})$[There are no more information aobut the $f$] Put $f(\frac{1}{z}) =g(...
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Is there any good way to deal with singularities(when x=y) in numerical integration ?

The integral equation with a singularity is as follows: $$\Omega=\int^\pi_0\frac{\cos(\theta)}{\sqrt{1-\cos(\theta)}}\,d\theta$$ I want to calculate this integral numerically. Is there an analytic ...
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Monodromy element: Why that name?

Let $(H,R)$ be a quasitriangular Hopf algebra, i.e. $R$ is a choice of an universal $R$-matrix for the Hopf algebra H. (You can find a definition of the term quasitriangular Hopf algebra on wikipedia.)...
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2answers
48 views

If $g(z) = f(1/z)$ and $g$ has a pole at $0$ and $f$ entire.Then there is a $z_0$ such that $f(z_0) = 0$.

I figured I can somehow use Liouville's theorem. I don't really know what the method is for solving or where to begin. Suppose that $f: \mathbb{C} \to \mathbb{C}$ is an entire function. Let $g(z) = f(\...
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50 views

How to calculate $\int_C \cos \big( \cos \frac 1z \big) dz$?

The question says: If $C$ is a closed curve enclosing origin in the positive sense. Then $\int_C \cos \big( \cos \frac 1z \big) dz=$ ? $(1)\quad 0$ $(2)\quad 2\pi i$ $(3) \quad \pi i$ $(4)\quad -\pi i$...
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26 views

How to show the singularity is isolated when the following condition holds?

Definition: $f$ has an isolated singularity about $z=0$ if there exists a punctured disc about $z=0$ such that it is holomorphic in that punctured disc. I read about the characterisation of a pole in ...
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1answer
42 views

$f$ has a pole order $n$ then there exists a positive constant $C$ such that $C|z|^{-n}\leq |f(z)|$

Question: Suppose $f$ has a pole order $n$, $n\in \mathbb{Z}_{>0}, $at $z=0$ then there exists a positive constant $C$ such that $C|z|^{-n}\leq |f(z)|$ on sufficiently small punctured disk about $0$...
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2answers
71 views

How bad an $L_1$ function can get in terms of having $\operatorname{esssup}$ as $\infty$ at many places?

Let $f:[0,1]\to \mathbb{R}$ be a $L_1$ function (i.e., $\int_0^1 |f(x)|dx<\infty$) with $\operatorname{esssup} f=\infty$. Think of functions like $\frac1{\sqrt{x}}$ or $\frac{1}{\sqrt{x(1-x)}}$ for ...
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2answers
50 views

Are all strictly positive semi-definite matrices singular?

If I have some matrix A with an eigenvalue of 0, what makes this matrix singular? and I am assuming All positive definite matrices are non singular so all strictly positive-semidefinite matrices would ...
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1answer
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Can the closest non singular matrix be seen as the closest positive/negative definite matrix? [closed]

Now I understand that positive definite matrices are non-singular as the determinant is non-0, could the closest non singular matrix be treated as the closest positive definite matrix? I think I am ...
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37 views

Does the integral of a function, which has no singularities, also have no singularities?

I'm evaluating an complex integral and this is a part of it: $$\lim_{\substack{\epsilon\to 0 \\ \delta\to 0}}\,\int_{-\delta}^{\delta}\frac{\left(\epsilon e^{\mathrm{i}\varphi}\right)^{(s-1)}e^{\...
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1answer
35 views

Existence of a complex sequence with given property

How to show the existence of a complex sequence $(z_n)$ with $z_n\ne 1, \forall n$ but $\lim_{n\to\infty}z_n=1$ such that $\lim_{n\to \infty}\sin(\frac{1}{1-z_n})=100$? Can such a sequence be ...
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1answer
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Understanding when to use residue theorem and when Cauchy's formula to solve integrals

This integral made me wonder, what should be used: $ \underset{|z-3 \pi|=4}{\int} \frac{1}{z \sin{z}} dz$ Here $0$ is not a relevant pol since it's not in the circle. so the 3 relevant pols are: $z_0 =...
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2answers
101 views

Derivatives of $ \frac{1}{r} $ and Dirac delta function

I am trying to understand the formula \begin{equation} \nabla^2\left(\frac{1}{|{\bf r}-{\bf r}'|}\right) = - 4 \pi \delta(\bf{r}-\bf{r}'), \qquad\qquad {\rm (I)} \end{equation} where ${\bf r}=(x,y,z)$....
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Find asymptotics given equation satisfied by generating function.

I'm interested in a sequence of numbers whose ordinary generating function obeys the equation: $$F(z) = 1-z^2+z(F(z))^3.$$ Is there some (relatively simple) way to get a good upper bound on the ...
2
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1answer
28 views

Calculate residues at all isolated singularities of $f(z)=\frac{z^2+4}{(z+2)(z^2+1)^2}$.

Calculate residues at all isolated singularities of $f(z)=\frac{z^2+4}{(z+2)(z^2+1)^2}$. So I found the isolated singularities to be $z=-2$ and $z=i$. Then I found the residue at $z=-2$. $\...
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Removing coordinate singularities

Consider the Riemannian metric given in this picture. Wikipedia claims that this Riemannian manifold has the topology $\mathbb{R}^2\times S^2$, but the coordinate expression given in the picture seems ...
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Resolution of branch point singularity

Let $\pi:Y\to\mathbb{P}^2$ be a Galois cover of the projective plane which is branched along $r$ lines $L_1, L_2,...,L_r$ in $\mathbb{P}^2$. Suppose the lines $L_1, L_2,...,L_r$ all pass through the ...
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1answer
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How to compute the residue of $\frac{1}{e^{\frac{1}{z}}-1}$ around $z=0$?

Here is a problem in my complex analysis notes, i.e. compute the residue of $\frac{1}{e^{\frac{1}{z}}-1}$ around $z=0$. I think that this isolated singularity is an essential singularity. The problem ...
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1answer
80 views

Prove that if the only singularities of a function are poles then the function must be rational.

Prove that if the only singularities of a function are poles then the function must be rational. So originally this was an iff statement, but I have solved the other direction of the proof. I'm just ...
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2answers
42 views

Find the singularities of $f(z) =\frac{1}{(2\sin z - 1)^2}$.

Find the singularities of $f(z) =\frac{1}{(2\sin z - 1)^2}$. I am just learning about singularities and I was wondering if someone could give me feedback on my work. So I think, for this function, ...
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1answer
55 views

Are the singularities of $f(z) = \frac{z^2+1}{z^2(z+1)}$ removable?

Looking at the function $f(z) = \frac{z^2+1}{z^2(z+1)}$, I have found the singularities to be at $z=0$ and $z=-1$. My question is if they are removable. I expanded this into the Laurent series $\frac{...
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1answer
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Question about proof of Riemann removable singularity theorem.

Theorem: Let $f:D^*(z_0,r)=D(z_0,r)-\{z_0\}\to \Bbb C$ be holomorphic and bounded. Then $\lim_{z\to z_0}f(z)$ exists and the function $\hat{f}:D(z_0,r)\to \Bbb C$ defined by $$\hat{f}(z) = \begin{...
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60 views

Milnor number/Mapping Degree

I am reading in John Milnor's Book Singular Points of Complex Hypersurfaces and struggle to do a similar computation he did. Namely, how can I compute explicitly the Milnor number $\mu(f)$ on the ...
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4answers
66 views

Why is the pole of $\frac{x}{\sinh(x)}$ a simple pole and not a removable singularity?

I'm doing my homework for my Complex Analysis class and I'm asked to solve a definite integral of $\frac{xdx}{\sinh(x)}$. This is not a problem for me, however the problem tells us directly that there ...
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1answer
20 views

Invertible $X^TX$ - what happens when you clone rows of $X$?

My question is inspired by https://stats.stackexchange.com/questions/70899/what-correlation-makes-a-matrix-singular-and-what-are-implications-of-singularit, in particular ttnphns's answer where they ...
2
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1answer
80 views

Unexpected singularities in an integral

I am doing an indefinite integral, $$f=\int\frac{r^2}{(r^2 + d^2 -2rd\cos{\theta})^2}dr$$ where $0\leq r< \infty$, $0\leq d< \infty$ and $0\leq \theta \leq \pi$. The integral that I am getting (...
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0answers
32 views

Residue theorem, where did I make a mistake?

I have to compute $$\int_0^{2\pi}\frac{dx}{7+6\cos(x)}$$ First, I was going to find a contour, which is the unit circle. $z(\theta) = e^{i\theta}$, $0 \leq \theta \leq 2\pi$. Now, $\cos(\theta) = \...
2
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1answer
28 views

Find residues at the singularities

I have a function $f(z) = \frac{\cos(z)}{z^6}$. I have to find the singularities and the corresponding residues. I think there is a single pole at $z=0$, which has order 6. For the residue, I did this:...

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