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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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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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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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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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Show that if $f$ is analytic then $ \frac{1}{f}$ has a pole using given definitions

Definition 1: $f$ is analytic at $z=0$ if $f$ has a power series expansion at $0$ i.e. $f$ is analytic at $0$ $\iff f(z)=a_0+a_1z+a_2z^2+\cdots +a_nz^n+\cdots$ Definition 2: $f$ has a pole at $z=...
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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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35 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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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
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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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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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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
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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
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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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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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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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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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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
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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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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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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
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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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Order of the equations defining the Zariski tangent space

Consider a function $F: \Bbb R^n \to \Bbb R^m$. We assume that the Taylor series of this function converges to the function itself. The Zariski tangent space of $F^{-1}(0)$ at a singular point $p$ can ...
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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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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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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\...
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1answer
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Singular points on complex projective-algebraic curve vs affine curves

I am trying to understand singular points on a complex projective -algebraic curve. I remember that singular points on a affine algebraic curve are determined by taking the partial derivatives and ...
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1answer
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Divergence of Series vs Dense Singularities

I am preparing for a Complex Analysis exam and have a question where I have to show that $$\sum_{n=1}^\infty nz^n=\frac{z}{(1-z)^2}$$ which I did using a geometric series of $\frac{1}{1-z}$. The next ...
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Leaving out equations from a singular linear system

Consider a solvable linear system $\mathbf{Ax=b}$, where $\mathbf{A}$ is a sparse, large, and square matrix (of typical size 20,000×20,000) with one zero eigenvalue, $\mathbf{x}$ is an unknown vector, ...
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35 views

Resolution of plane curve singularity

Given a plane curve $C$ over an algebraically closed field of characteristic 0 and $p\in C$ a singular point, if $f:\tilde{C}\to C$ is a resolution of the singularity and $f^{-1}(p)$ consists of one ...
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1answer
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What is the nature of the singularity of this function?

If I've been asked to determine the nature of the singularity of the function $\frac{1}{z^2+a^2}$, where $z$ and $a$ have not been defined but from the context I assume $z$ is a complex variable and $...
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Suppose $g(z)$ is analytic at $z_0$. Proof $g(z)$ has zero of order $m$ at $z_0$ iff $ lim_{z\rightarrow z_0} \frac{g(z)}{(z-z_0)^m} \neq 0$.

Suppose $g(z)$ is analytic at $z_0$. Proof $g(z)$ has zero of order $m$ at $z_0$ iff $ lim_{z\rightarrow z_0} \frac{g(z)}{(z-z_0)^m} \neq 0$. Sketch: We have that $g(z)=\sum_{n\rightarrow\infty} a_n(...
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Classify the points at 0 and $\infty$ of $x^7~\frac{d^4y}{dx^4}=y'$

$$\displaystyle x^7~\frac{d^4y}{dx^4}=y'$$ I know that 0 is an irregular singular point. At $\infty$, I'm using the change of variable $x = \frac{1}{t}$ and I don't understand how to differentiate ...
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1answer
26 views

Classification of singularities of complex multivalued function

I have some problems dealing with multivalued functions when it comes to handling singularities. I'll give an example and try to ask questions based on it. I want to classify the singularities of $f(...
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Classifying the singular point $z=\frac{\pi}{2}$ for $f(z)=\frac{z-\frac{\pi}{2}}{1-\sin(z)}$

I am trying to find determine the type of singular point $z=\frac{\pi}{2}$ is for the function, $$f(z)=\frac{z-\frac{\pi}{2}}{1-\sin(z)}.$$ My attempt: I originally thought that $$\lim_{z\to\frac{\...
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1answer
15 views

Prove that the improper integral is independent of the intermediate value

Suppose the function $ f:[a,b] \rightarrow \mathbb{R} $ only has singularities at $ a $ and $ b $, and the integral $$ \int_{a+\epsilon}^{b-\epsilon} f $$ exists for all $ \epsilon > 0 $. With the ...
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2answers
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Classifying singular points of $\frac{\sin(z^2)}{z^3-\frac{\pi}{4}z^2}$

I am trying to classify the singular points of the function $$f(z)=\frac{\sin(z^2)}{z^3-\frac{\pi}{4}z^2}.$$ My attempt: $$f(z)=\frac{\sin(z^2)}{z^3-\frac{\pi}{4}z^2}=\frac{\sin(z^2)}{z^2\left(z-\...
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1answer
31 views

Confusion about the definition of analytic and singularity.

In my textbook the definition of analyticity is given as A function is said to be analytic in a domain D if f(z) is defined and differentiable at all points of D. The function f(z) is said to be ...
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Problem in Hythothesis of a given problem

Let $f$ be a meromorphic function in a neighborhood of the closed unit disk $\bar{\mathbb{D}}$. Suppose that $f$ is holomorphic in $\mathbb{D}$ and $$ f(z) = \sum_{n=0}^\infty a_n z^n $$ for $z \...
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Why does subtracting function with the same singularities make it analytic

When estimating asymptotics of a series from its generating function, we look for singularities (this makes sense to me) and then try to remove them (this also makes sense) by subtracting a function ...
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Is the Schwarzschild singularity stretched in space as a straight line?

I am trying to visualize the Schwarzschild geometry and would appreciate a help of the experts. The geometrized radial ($\theta=\phi=0$) Schwarzschild metric outside the horizon is $$ d\tau^2 = \left(...
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Integral $\int\frac{\sin(2zj)}{z(z^{2}+\frac{\pi^{2}}{4})^{2}}dz = 0$ (residues)

Hi guys I'm solving this integral : $$\int_{+\partial D}\frac{\sin(2zj)}{z(z^{2}+\frac{\pi^{2}}{4})^{2}}dz\,,$$ where $D=\{z\in\mathbb{C}:|z|<\pi \}$ I have found that for $z=0$ the residue is $0$ ...
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Determining type of singularity

Let $f$ be holomorphic in $\mathbb{C} \setminus \{0\}$. Prove or disprove: If $|f(\frac{i}{n})| \ge n$ for all $n \in \mathbb{N}$, then $f$ has a pole in $0$. The equation yields that $f$ is not ...
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1answer
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To find $f(\{z\in \mathbb{C}:0<|z|<\varepsilon\})$ if $f(z)=\cos\left(\frac{1}{z}\right)$.

Let $\varepsilon>0$. I want to find the image of $\{z\in\mathbb{C}:0<|z|<\varepsilon\}$ through of $f(z)=\cos\left(\frac{1}{z}\right)$. I know that $f\left(\{z\in\mathbb{C}:0<|z|<\...
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In $C_{\infty}$ , find all singularities for $\frac{1}{z(e^z-1)}$?

I know it has pole of order 2 at z = 0 but I'm not sure it has essential singularity at z = $\infty$. In $C_{\infty}$ , $\pm\infty , \pm i\infty$ are same point. They should give same value if it ...
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In $C_{\infty}$ , does function $f(z) = \frac{1}{z}$ have removable singularity at $z=\infty$?

In $C_{\infty}$ , does function $f(z) = \frac{1}{z}$ have removable singularity at $z=\infty$ ? Because $f(\frac{1}{w}) = \frac{1}{\frac{1}{w}}$ , $\frac{1}{w}$ isn't define at w = 0 and $lim_{w\to0}\...
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1answer
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f is analytic on $D(0,1)\setminus\{0\}$ with an essential singularity at $0,$ show f is not one-to-one [duplicate]

I am working on following Question; f is analytic on $D(0,1)\setminus\{0\}$ with an essential singularity at 0, show f is not one-to-one. I am a bit stuck. All I have been able to figure out is the ...
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1answer
23 views

Understanding a mistake regarding removable and essential singularity.

It is a well known fact that $0$ is an essential singularity of the function $e^{1/z}$ therefore it is not essential nor a pole. On the other hand we know the Rieamann's continuation theorem which ...
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0answers
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Integral evaluation via the Residue Theorem (example)

$$\int_{\gamma}\frac{e^z-1}{\sin^2z}\,dz, \quad \,\gamma(t)=4e^{it},\,t\in[0,2\pi].$$ Let $$f(z)=\frac{e^z-1}{\sin^2z},\quad z\neq n\pi,\,n\in\mathbb{Z}$$ The curve contains only the singular ...
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1answer
49 views

del Pezzo surface singularities

Following the Calabi Yau Bestiary for Physicsts, I have the following (basic) doubts. We have a Hirzebruch surface $S_{\epsilon}$ which is a member of the following configuration $$ \left[\begin{...
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2answers
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pole and essential singularity in the same point

In this case in $0%$ i have an essential singularity by $\sin(\frac{1}{s})$ but i have a ""pole"" too ( the denominator of the fractions $\frac{(...)\sin(...)}{**S**}$) , so in this case is this a ...