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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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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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conocial adjoint of singular curve [on hold]

I have a question about conocial adjoint of the nonsingular curve. If curve $X$ be a singular curve, how I can compute the conocial adjoint of this curve? If $Y$ has only ordinary singularities, its ...
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44 views

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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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
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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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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
28 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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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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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
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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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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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45 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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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 ...
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1answer
27 views

Genus of the desingularization of a plane curve through meromorphic function

Could someone help me? Consider, in the complex projective plane, the curve $C$ given by the the points $[X,Y,Z]$ for which $F(X,Y,Z)=X^2Y^3+YZ^4+Z^5=0$. I have to desingularize the curve and consider ...
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51 views

How does one deal with $\frac{1}{|x|}$ at zero when multiplying two random variables?

To multiply two independent RVs, Wikipedia gives the following approach: $$f_Z(z)=\int_{-\infty}^\infty f_X(x)f_Y(z/x)\frac{1}{|x|}dx $$ https://en.wikipedia.org/wiki/Product_distribution Maybe ...
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2answers
125 views

Proof of Casorati-Weierstrass [closed]

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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Show that a holomorphic function $f$ has a pole iff we can find $g$ s.t. $f(z) = \frac{g(z)}{(z-z_0)^m}$

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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1answer
76 views

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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1answer
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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
63 views

If $f$ is entire and $f(z)/z$ is bounded, then $z = 0$ is a removable singularity of $f(z)/z$.

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}(...
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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
21 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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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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155 views

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
34 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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1answer
37 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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1answer
56 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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1answer
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
28 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
34 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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1answer
17 views

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
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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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27 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
77 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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1answer
31 views

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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1answer
45 views

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 ...