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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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Singular Points and Residues of $ f(z)=\frac{\cot z + \cos z}{\sin 2z} $.

$ f(z)=\frac{\cot z + \cos z}{\sin 2z} $ Where $\cot z =\frac{\cos z}{\sin z}$ is the complex contangent function. a) Find all the singular points of f(z) and classify them. b) Choose one singular ...
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112 views

Proof of Casorati-Weierstrass [on hold]

A First Course in Complex Analysis by Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka Ch9.2 I have questions on the proof of Casorati-Weierstrass Theorem (Thm 9.7) - If $z_0$ is ...
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60 views

Holomorphic $f$ has a pole $\iff f(z) = \frac{g(z)}{(z-z_0)^m}$

A First Course in Complex Analysis by Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka Ch 9.2 Cor 9.6 of Prop 9.5(*) Suppose $f$ is holomorphic in $\{0<|z-z_0| < R\}$. ...
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Identifying poles of $\cot(z)$

$\cot(z)=\dfrac{\cos z}{\sin z}$ and I am supposed to find poles at $z=k\pi \quad k=0,\pm1,\pm2... $. But derivative of $\dfrac{d}{dz}\cot(z)=-{\csc}^2(x)$ and it is also singular at $z=k\pi$. So why ...
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24 views

Sparse Matrix inversion some time singular some time get a big value

I want to invert a matrix which is a "band" diagonal matrix. The structure of the matrix is The blue strip represents the elements that are non zero.All other element in white area are of zero value. ...
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2answers
54 views

Removable Singularity with Entire Function

Let $f$ be an entire function with $\sup_{z\in\mathbb{C}}|f(z)/z|<\infty$. Show that $z=0$ is a removable singularity of $g(z):=f(z)/z$. To prove the claim, I need to show that $0=lim_{z\to 0}(z-0)...
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Expression for the density function of a smooth function

I am working on tomographic methods in which the data is the "distribution" of values along a line rather than an integral. Given a measurable function $f:[0,1] \rightarrow \mathbb{R}$ one can define ...
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1answer
20 views

Source of non-linear Laplace equation

Consider the following non-linear generalization of the Laplace equation $$\Delta \phi - \frac{\sum_i (\partial_i \phi)^2}{2 \phi} = 0$$ I am looking for spherically symmetric solutions, so I assume ...
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30 views

What happens if I have an essential singularity and a pole for the same $z$?

for instance $$\dfrac{\sin(\dfrac{1}{z})}{z}$$ $z=0$ is a pole for the denominator but $z=0$ is an essential singularity for the numerator too. So how does it work ? i have two residues ? or it's ...
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Proving that the derivative always diverges faster than the original function

Let $f$ be a differentiable real function. What is the simplest/neatest way of proving that $\lim_{x \to a} f(x) = \infty$ implies that $ \lim_{x\to a} \frac{f'(x)}{f(x)} = \infty$? It seems like such ...
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Gradient Blowup for a Parabolic (Heat) Equation

Let $u(x,t)$ be a solution to the following parabolic PDE: With $\alpha \in (0,1)$, \begin{align} \partial_t u(x,t) &= \alpha (1-t)^{\alpha - 1} \partial_x u(x,t) + \frac{1}{2} \partial_{xx} u(x,...
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1answer
29 views

Measure for how singular a square matrix is in the range [0,1]

I am interested in estimating how close a square matrix is to being singular such that I can compute a value $s \in [0,1]$ where $s=1$ would mean the matrix is singular, and $s=0$ means it is as far ...
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33 views

Laurent Series for singularities and poles

Hi guys I was wondering how I can understand if the sin and the cos has essential singularities. for instance if I want to understand if 0 which singularity is i, can write the Laurent series only of ...
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55 views

Singularity Type Of $f(z^2+z)$

$f(z)$ has essential singularity at $z=0$, what type of singularity $f(z^2+z)$ has? $f(z)$ has essential singularity at $z=0$ so it can be written has $\sum_{n=0}^{-\infty} c_nz^n=c_0+\frac{c_{-1}}{z}...
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39 views

The order of a pole is an integer

I try to solve a complex analysis question about isolated singularities, the question said: Suppose that $f(z)$ has an isolated singularity at $z=z_0$, and that $\lim_{z\rightarrow z_0}{(z-z_0)^\alpha ...
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The ordinary generating function for the square-free kernel: reference request about singularities and its phase plot

Let $n\geq 1$ an integer, in this post I denote the product of distinct prime numbers dividing dividing $n$ as $$\operatorname{rad}(n)=\prod_{\substack{p\mid n\\p\text{ prime}}}p,$$ is the famous ...
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1answer
26 views

Let $0$ be an isolated singularity of $f$. Prove that if $|f(z)|\leq |z|^{-\alpha}$, this singularity is removable.

I'm doing this exercise and I must be doing something wrong. Here it goes: Let $0$ be an isolated singularity of f. Prove that if $|f(z)|\leq |z|^{-\alpha}$, $\alpha\in(0,1)$, this singularity is ...
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1answer
32 views

Calculating Residue of $f^2$ with pole of order 2

The question: The function $f$ has a Pol of order 2 in $z_0$. Calculate the residue of $f^2$ in $z_0$ using the Laurentcoefficients of $f$. My attempt: I tried to use the fact that if $f(z) = (z- ...
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Linear transformations - singularity

My question is about linear transformations: Is there a unique linear transformation $T:R^3\to R^3$ such that the image of the plane $M=\{t(1,4,0)+s(1,1,1)+(2,2,1)|t,s$$ \in$$R \}$ is the point $(0,3,...
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1answer
34 views

infinite sequence of poles with limit point implies Casorati-Weierstraß [closed]

Let $\{z_k\}_{k\in\mathbb{N}}$ be a sequence of poles of $f(z)$. Suppose that $\lim\limits_{k\to\infty}z_k=z_0$ and that $f$ is holomorphic in a neighborhood of $z_0$ except for $z_0$ and $\{z_k\}$. ...
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Numerical integration of ODE with singular r.h.s

I have the Cauchy problem: $$ \frac{dx}{dt} = \frac{f(t, x)}{g(x) - t^3} \;,\qquad x(t_0) = x_0 \;. $$ It can be shown that when the solution reaches a vicinity of a certain point $(t_\ast, x_\ast)$ ...
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What are the possible poles of the meromorphic function $G$?

Consider the function $$F(z)=\int_{1}^{2} \frac {1} {(x-z)^2}dx,\ \mathrm {Im} (z) > 0.$$ Then there is a meromorphic function $G(z)$ on $\Bbb C$ which agrees with $F(z)$ when $\mathrm {Im} (z) >...
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Singularity of polynomial at infinity.

What can we say about the singularity of $f(z)=z^3$ at $\infty$. Has an Essential singularity at $\infty$ Has a Pole of order $3$ at $\infty$ Has a Pole of order $3$ at $0$ Is analytic at $\infty$ I ...
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26 views

What is the nature of $f$ at the given points?

Let $f(z)=\frac{e^\frac{c}{z-a}}{e^\frac{z}{a}-1}$,then $(a)z=0$ is a removable singularity. $(b)z=a$ is isolated essential singularity. $(c)z=a$ is non isolated singularity. $(d)...
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1answer
60 views

Divisor class group of the quadric cone in $\mathbb{P}^3$

I would like to compute the divisor class group of the projective quadric cone $$ Q=\mathrm{Proj}(\mathbb{C}[X_0,X_1,X_2,X_3]/(X_1X_2-X_3^2)). $$ It has as an open subset the quadric cone $U$ in $\...
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Type of singularity of $h(z)=\frac{1}{\sin(4z-\pi)}$, $z=\frac{\pi}{4}$

Let $$h(z)=\frac{1}{\sin(4z-\pi)}$$ What kind of singularity does function $h(z)$ at the point $z=\frac{\pi}{4}$? I know from theory that the point is essential singularity if and only if $\lim_{z \...
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Poles of $\sin(1/z)$ [closed]

i was studyng this function, and on wolfram alpha it says that there are no poles. But why is $z=0$ not a pole? (sorry for my bad english)
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Using Homogeneous Coordinates in Differential Equations

Recently, I asked a question on the Mathematica Stack Exchange website regarding the use of homogeneous coordinates in differential equations. The question is about extending the interval of ...
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1answer
33 views

Codimension of singular locus of quotient V/G

I guess this is really standard but I simply wasn't able to figure it out. Let $V$ be a finite-dimensional complex vector space and let $G \subset GL(V)$ be a finite group. Consider the variety $V/G$. ...
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1answer
31 views

Explanation of the statement of Great Picard Theorem

Suppose an analytic function $f$ has an essential singularity at $z=a$. Then in each neighbourhood of $a$, $f$ assumes each complex number, with one possible exception, an infinite number of times. ...
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If $f$ is an entire function which of the following is/are correct?

Consider the function $$f(z)=\frac {\sin (\frac {\pi z} {2})} {\sin (\pi z)}.$$ Then $f$ has poles at $1.$ $\ $ all integers. $2.$ $\ $ all even integers. $3.$ $\ $ all odd integers. ...
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48 views

Why does the inverse of a singular matrix plus a small-norm matrix have same columns/rows?

Question: Given two matrices A and B, where A is a singular matrix and the sum of each row of A is zero, and B is an arbitrary matrix with $\left\| {\bf{B}} \right\| \ll \left\| {\bf{A}} \right\|$, ...
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1answer
26 views

Singularities. Residues

Classify the singularities and calculate the residual of each of them of the function $$f(z) = \frac{1}{z-1} e^{\frac{1}{(z-1)^2}} + \frac{z}{sin(z)}$$
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How to prove the singularity of a variety

Consider the Veronese map $\mathbb{P}^{2}\rightarrow\mathbb{P}^{5}$ and let $X\subset\mathbb{P}^{5}$ be the cubic hypersurface defined by $\det(A)=0$. Now $X$ is the image of all reducible conics. I ...
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34 views

How to interpret a model for the angle of attack, when the translational velocities are tiny?

I'm playing with a model for the angle of attack of a flying machine, "M". If M's angle of attack is modeled by, say, $$ \text{AoA} = \arctan \big( \frac{\dot{y}(t)}{\dot{x}(t)} \big)$$ and the ...
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Singular Solutions to ODE

Give is the following differential equation: $$f' = e^f\sin(t) $$ which has the following general solution: $$f(t) = -\ln(\cos(t) + c),$$where $c$ is a constant. Now my book goes on to say that the ...
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1answer
17 views

Order of pole of function

Question Let $f(z)$ be meromorphic function given by $f(z)=\frac{z}{(1-e^z)\sin{z}}$ then which of the following are correct? (1) $z=0$ is pole of order $2$ (2) for every $k$ in $\mathbb{Z}$, $z=...
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1answer
50 views

double integral with singularity (not exactly singularity)

I'd like to solve $\iint_R \tan^{-1} \frac{y}{x}dA$, where $R=\{ (x,y) : x\ge0, y\ge 0, x^2+y^2 \le 4\}.$ I'm wondering if the following calculation is true: $$ \begin{split} \iint_R \tan^{-1} \frac{...
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45 views

How to compute residues

I studied in Principles of algebraic geometry by Phillip Griffiths and Joseph Harris,there is a generalization of residue for isolated hypersurface singularity. For isolated hypersurface $f$ and ...
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57 views

Type of singularity of $a$, for which $a $ is a limit point of a sequence of zeros of the analytic function $f(z)$

Suppose $f(z)$ is an analytic function in a region $D$, and $f(z)$ is not identically equal to zero there. Let $z_n$ be a sequence of $f(z)$ in $D$. If $\lim_{n\to \infty}z_n=a$, then $a$ must be ...
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1answer
96 views

Classifying singularities of $\frac {z^{1/2}-1}{\sin{\pi z}}$

I am trying to classify the singularities of $$\frac {z^{1/2}-1}{\sin{\pi z}}$$ where $-\pi<\arg z<\pi$. I am confused by this because of the branch cut of $\sqrt z$ but here is my (bad) ...
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Extracting singular points from an equation of this form $y'' - y' + (a-x^2) y = 0$

I put this equation in \begin{equation} y'' - y' - (a-x^2)y=0 \end{equation} into Wolframapha, and it gave a linear combination of two solutions; the first was a Hermite polynomial solution and the ...
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2answers
38 views

Calculating Residues using L'Hopital?

I am given that the complex function $$f(z)=\frac{(e^{z-1}-1)(\cos(z)-1)}{z^3(z-1)^2}$$ has 2 simple poles, one at $z=0$ and another at $z=1$, and asked to calculate the Residues of the function at ...
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19 views

Remove singular points

I'm developing a tool in Matlab and since it doesn't now how manage singularities, I need to correct these two equations: $A(\theta,\phi)=\frac{1+cos\theta}{2}\frac{J_0(kasin\theta)}{2.405^2-(kasin\...
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1answer
63 views

Essential singularity at infinity of exponential function.

Show that if $f(z)$ is a non-constant entire function, then $e^{f(z)}$ has an essential singularity at $z=\infty$. This is my approach: By Liouville's theorem I know that if $f$ is a non-constant ...
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1answer
29 views

The product $(fg ^ 2)(p)$ is different of zero

Could anyone help me understand what this passing a demonstration is highlighted in yellow? Let $f, g$ be the analytic functions defined in the complex plane. Assume that $p$ is zero of order $2m$ ...
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1answer
22 views

Complex integration using Cauchy Goursat (I think)

$$\oint_\Gamma \left(\frac{1}{z^2 - 2} + \frac{z}{z-5}\right) dz, \quad \text{where } \Gamma : |z+i| = \frac{1}{25}.$$ My approach Locate singularities, namely $\sqrt{2}, -\sqrt{2}$ and $5$. As none ...
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1answer
48 views

Find all roots of $z^4 + 1 = 0$ [duplicate]

$z^4 + 1 =0$, looking at a relatively similar question I concluded that the next step is to factorise into $(z^2 + i) (z^2 - i) = 0$. However, I'm not sure what the next step should be? Do i continue ...
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1answer
28 views

Finding the singularities of a complex function.

I need to find and classify the singularities of the function: $ f(z) = \frac{z^2+1} {z^4-2}$. I'm aware that I'm going to have to first find the Laurent series corresponding to this function. I ...
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0answers
34 views

complex integral , with singularity ${1\over \pi}$

How to calculate $$\int {sin z \over ( z + 1 ) sin {1\over z}}$$ $ on$ $$| z | = {1\over 2}$$ My attempt was to Binding $\frac{1}{\pi}$ and $0$ by two distinct circle each contains only one ...