Questions tagged [singularity]
This tag is for questions relating to singularity, which is a point where a mathematical 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.
1,090 questions
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Question for Singularities, Laurent Series of $\csc\big(\frac{1}{z}\big)$. [duplicate]
I came across this question in my complex analysis class. We were given $\csc(1/z)$, and told to classify the singularity at $z=0$, alongside computing the residue at $z=0$.
I did some digging around ...
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How to retrieve indices of dependent columns or rows in a singular matrix
In a stability analysis study of a system of first order non linear ODEs, I am facing a problem to know exactly which columns (or rows) are dependent on others in order to eliminate them and consider ...
- 11
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Removable singularity of $1/f$ at $z=0$ where $f$ has essential singularities at ${a_n}$ in $\mathbb{D}$ [closed]
Let $\{a_n\}$ be a sequence of nonzero complex numbers contained in the unit disk $\mathbb{D}$ such that it converges to 0. Assume $f$ is analytic in $\mathbb{D}\setminus(\{0\}\cup\{a_n\}_{n \geq 0})$....
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How can I find a family of hypersurfaces in $\Bbb A^n$ with normal singular points for all $n\geq 3$?
Let $k$ a field with characteristic not equal to $2$ and $f \in k[x_1,\ldots,x_n]$ which is not a perfect square polynomial. Give a family of hypersurfaces $X_n \subseteq \mathbb{A}_k^n$ with normal ...
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Dimension of Jacobian Algebra (in Singularity Theory)
Motivation: In Singularity Theory, the Milnor Number is defined to be the dimension of the Jacobian algebra and is finite in the case of an isolated singularity.
We define the Jacobian algebra of a ...
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Efficiently Solving a Large System of Nonlinear Equations with Singular Jacobian for Stability Analysis
I am analyzing the stability of signaling pathways modeled as a dynamic system. Specifically, I am working with a large system of nonlinear equations (approximately 100 equations). An example of the ...
- 137
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Cusp-singularity in metric on the plane
Consider polar coordinates $r,\theta$ with $0 \leq r, 0 \leq \theta \leq 2\pi$ on the 2D plane. The flat metric is given by $$\mathrm{d}s^2 = \mathrm{d}r^2 + a(r)^2 \mathrm{d} \theta^2$$ with $$a(r) = ...
- 320
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Doubt regarding poles and essential singularity
Actually I always try to check type of singularity by checking the limit...
I know for pole at $z_0$ $\lim_ {z\to z_0}f\left(z\right)=\infty$ and for essential singularity the limit does not exist...
...
- 1,297
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Two definitions of a normal complex surface
I am looking for the definition of a "normal surface singularity". I could not find such a definition even in the book with title Normal Surface Singularities (https://link.springer.com/book/...
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Solution to the confluent Heun equation with irregular singular point at $z=1$
I want to find a solution to the following ODE
$$
w''(x)+\left(\frac{c}x +\frac{d}{(1-x)^2}\right)w'(x)-\frac{\tilde{q}}{x(1-x)^2}w(x) = 0\tag{1}
$$
on the interval $(0,1)$, satisfying the boundary ...
- 426
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Show that $\int_0^1\frac{\cos(t)}{t-z}dt+\int_2^3\frac{\sin(t)}{t-z}dt$ admits a Laurent series expansion on $1<|z|<2$ and determine this expansion
Show that the function
$$
F(z) := \int_0^1 \frac{\cos(t)}{t-z} \, dt + \int_2^3 \frac{\sin(t)}{t-z} \, dt
$$
admits a Laurent series expansion in the annulus $1 < |z| < 2$, and determine this ...
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Help solving ODE with singularities
I'm trying to solve the following ODE:
$$
\frac{-l^2 (1 + 3 b u + 2 b^2)}{(1+2bu)(1 + b u)(1 - u^2)^2} \ \ p \ \ - \ \ \left( \frac{4 b^2 u^3 + 3 b u^2 + (\frac{1}{2}-b^2)u}{(1+2bu)(1 + b u)(1 - u^2)} ...
- 141
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Nash blowup and resolution of singularities
We work over the field of complex numbers. We recall the definition of the Nash blowup (sometimes called as canonical blowup?) from the preprint https://arxiv.org/abs/2409.19767 : Let $X \subset \...
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How to calculate the ratio of the Gamma-function and its derivative $\frac{ \Gamma'(x) }{ (\Gamma (x))^2 }$ around their poles
I stumbled across the the ratio $ \lim_{x \rightarrow 0} \frac{ \Gamma'(x) }{ (\Gamma (x))^2 }$.
$\Gamma$ is the Gamma
-function $\Gamma(x) = \int_0^\infty dt \ e^{-t} \ t^{x-1}$ and it's derivative $\...
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Determining all holomorphic functions that satisfy certain conditions
I came across a complex analysis exercise which I was not able to figure out.
Let $\Omega = \mathbb{C} \setminus \{ i - 1 , i + 1 , 2i \}$. Determine all functions $g \in H(\Omega)$ such that
$(i)$ $\...
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Is it possible to calculate $\int_1^n \frac{(n-1) n u}{(n-u) (u-1)} \, du$?
Let $n>2$ be an integer. I want to calculate - or at least set limits on - the integral
$$\int_1^n \frac{(n-1) n u}{(n-u) (u-1)} \, du$$
Clearly, the problem with the definite integral is that ...
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Singularities of $f(z) = \frac{1}{z\cos\big(\frac{\pi}{2(1-z)}\big)}$
Determine singularities and classified them into removable singularities, poles, and essential singularities of
$$f(z) = \frac{1}{z\cos\big(\frac{\pi}{2(1-z)}\big)}$$
After some calculation that if ...
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Understanding the singularity at the origin of the Riemann surface associated to $\sqrt[n]{}$
Let $U_0=\mathbb{C}\backslash[-\infty,0]$; it is a simply connected planar open Riemann surface. Le $n\geq 2$ be a fixed integer and consider the function $z\mapsto\sqrt[n]{z}$ which is holomorphic ...
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Resolution of the singularity of a lemniscate [closed]
The equation $(x^2+y^2)^2-x^2+y^2=0$ defines a lemniscate. Let $f(x,y)$ denote the LHS.
My question is:
Is there three-variable polynomials $g(x,y,z)$ , $h(x,y,z)$ that satisfies below?
The equation $...
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Is it possible to apply Frobenius's method to an ODE with coefficients over sinusoidal functions?
I have an ODE of form $y'' + \frac{p(\theta)}{\sin \theta} y' + \frac{q(\theta)}{\sin^2 \theta} y = 0$. Upon making a variable change of $u=\sin \theta$, the first order derivative here gets an extra ...
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Meromorphic function with a pole of modulus $r$ has a pole at $z=r$
I was reading Berkeley Problems in Mathematics and I'm stumped on problem 5.8.32
Problem. Let $f$ be a meromorphic function on $\mathbb{C}$ which is analytic in a neighborhood of 0. Let it's Maclaurin ...
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Question about a specific PDE with singularity
I have the following system of PDEs for a metric $h_{AB}$ and a function $f$,
$f D_t^{(2)}h_{AB} + \left(D_tf + \dfrac{1}{2}h^{AB}D_t h_{AB}\right)D_t h_{AB} -\dfrac{1}{2}f h^{CD}D_t h_{AC} D_t h_{BD} ...
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Conditions where projective hypersurface is nonsingular
Problem: Let $F(X_0,\cdots,X_n)$ be homogenous irreducible polynomial. Let $X=V_P(F)\subset \mathbb P^n$. Prove that $X$ is nonsingular if and only if $\sqrt{(F_{X_0},\cdots,F_{X_n})}=(X_0,\cdots,X_n)$...
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(Blow-up) Resolution of singularities and arithmetic genus
Suppose $X$ is a $k=\overline{k}$-variety with a node at a closed point $p$. Let $\tilde{X}$ be the blow-up of $X$ at $p$.
In Exercise 28.3.E, Vakil asks to show that there is an exact sequence $0\to \...
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Confusion regarding a second order pole on the contour
Let us define a function
$$f(x) = \frac{1}{(x^2 + A^2) (x^2 - B^2)^2}$$
where, $A,B > 0$. We need to calculate
$$G = \int_{-\infty}^{\infty} f(x) dx $$
I am confused regarding whether $G$ diverges ...
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Question about Regular singular points
Given this definition of regular singular points, I am unsure why only checking the limit is finite is sufficient. I know that if the limit exists, the function can be made to exist but this doesn't ...
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Could this function be a traveling solution to the 1D wave equation?
I am trying to understand when solutions of the wave equation could have issues, so I made a continuous function which some extreme behaviors: has compact-support given by a smooth bump function term, ...
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What is $\ker p_*$ of a resolution of rational singularity?
Let $X$ be a rational singularity over a field $k$ of characteristic zero. Which means that there is a smooth variety (resolution of singularities of $X$) $Y$ with proper birational map $p \colon Y \...
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zeros by using partial fraction expansion and complex analysis
I want to know zeros of a complex function. The definition of Laplace transform is
$$\mathcal{L}\{\Gamma(\nu,at\}(p)~=~\Gamma(\nu)p^{-1}\left(1-\left(1+\frac{p}{a}\right)^{-\nu}\right)~~~~~~\text{Re}~...
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Can someone help explain the singularities in this surface?
I'm investigating properties for the following surface:
$$
r = 1 \ + \ A \cos \theta
$$
For $|A| > 1$, a dimple forms at $\theta = 0$. The Ricci scalar is:
$$
R(\theta) = \frac{2 A^2 \left(2 + A^2 +...
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Criteria for essential singularities
Pretty straight-forward question.
Wikipedia states the following criteria for a point in the complex plane to be an essential singularity for a complex function:
"If neither $\lim_{z \to a}f(z)$ ...
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Classify singularities of a complex function at infinity with f(1/x)
i am having some trouble classifying the singularities of the function $$f(z) = \frac{e^{z^2}}{z^3}$$ I understand it has a pole in $z=0$, but I do not understand why, according to my textbook, it has ...
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Find all holomorphic, bijective maps $\mathbb{C} \setminus \{ 0 \} \rightarrow \mathbb{C}$
I am trying to characterize all holomorphic maps $\mathbb{C} \setminus \{ 0 \} \rightarrow \mathbb{C} $ which are bijective.
I think you can say from Picard’s Great Theorem that the singularity at the ...
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Proof function $f$ has removable singularity at $a$ if $\lim_{z \to a} (z - a) f(z)^2 = 0$
While studying for my complex analysis course, I came across the following question:
Let $\Omega$ be an open subset of $\mathbb{C}$ and let $a \in \Omega$. Assume that
$f : \Omega \setminus \{a\} \to ...
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Integrating $ \int^p_1 \frac{x 10^x}{\sqrt{x^2-1}}$ by parts
I need to implement an Excel solution to the following equation for an engineering project I'm working on:
$$ \int^p_1 \frac{x 10^x}{\sqrt{x^2-1}}$$
where p is a number between 1 and 10.
After using ...
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A problom about singularity in stein Complex Analysis chapter3
Suppose that $f$ is meromorphic in the extended plane.Then $f\left( \frac{1}{z} \right)$ has either a pole or a removable singularity at $0$.
According to this,$$\lim\limits_{|z|\rightarrow \infty}f(...
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Argument Principle for poles at infinity
From Mathematical Methods in the Physical Sciences by M. Boas (3rd ed., p. 695): Show that $f(z)=z^3+4z+1$ has exactly one zero in the first quadrant.
The solution uses the Argument Principle to show ...
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Laurent expansion at $z = -1$
I want to get to the Laurent expansion of $g(z) = \frac{z^2 + 4z + 4}{z+1}$ at $z = -1$.
I know $g(z) = \frac{(z+2)^2}{z+1}$.
I tried computing the Taylor-Series for $g(-1)$ and getting
$g(-1)+ g'(-1) ...
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Is the geometric genus of a Gorenstein variety a birational invariant?
If $X$ is a projective scheme with at worst Gorenstein singularities, then the dualising sheaf ${\omega}_X$ is a line bundle and it makes sense to talk about the geometric genus $p_g(X) := \mathrm h^0(...
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Find $\displaystyle\int_\Gamma \frac{\tan z}{z^2+\pi^2} \, dz$ where $\Gamma$ is the circle $\left|z-\frac{\pi}{2}\right|=1$
Find $\displaystyle\int_\Gamma \frac{\tan z}{z^2+\pi^2} \, dz$ where $\Gamma$ is the circle $\left|z-\frac{\pi}{2}\right|=1$.
This problem showed up on a qualifying exam where the usual "find the ...
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The space of solutions for an ODE on an interval containing a regular singular point
Consider the differential operator $L(y)=x^ny^{(n)}+a_1(x)x^{n-1}y^{(n-1)}+\cdots+a_n(x)x^0y^{(0)}$, where each of $a_i(x)$ has a power series expansion at $x=0$ converging for all $|x|<r_0$ for ...
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Does maximal contact hypersurface (m.c.h.) exist for any curve over a field of arbitrary characteristic, and if so, for what definition of m.c.h.?
Does a maximal contact hypersurface always exist for any curve over a field of arbitrary characteristic, and if so, for what definition of maximal contact hypersurface?
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Sign of a complex integral
If one consider the complex value function
$$
f(z)=\frac{1}{(z-1)^{3/2}(z-2)^{3/2}}
$$
with branch cut chosen to be between $z=1$ and $z=2$.
Consider
$$
\oint f(z) dz,
$$
where the contour is taken to ...
- 1,181
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Complex integral with fractional singularities
If one consider the complex value function
$$
f(z)=\frac{1}{\sqrt{z-1}\sqrt{z-2}}
$$
with branch cut chosen to be between $z=1$ and $z=2$. Could someone please explain why
$$
2\int_1^2 f(x)dx=\oint f(...
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Using the residue theorem to compute two integrals [closed]
Classify the singular points for the function under the integral and using the residue theorem, compute:
(a) $$ \int_{|z-i|=2} \frac{z^2}{z^4 + 8z^2 + 16} \, dz, $$
(b) $$ \int_{|z|=2} \sin\left(\frac{...
user1340098
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Residue of a removable singularity at inifinity
Exercise:
Find all the singularities of $$\frac{z^3e^{\frac{1}{z^2}}}{(z^2+4)^2},$$ classify them, and find each residue.
I found that $+2i, \ -2i$ are poles of order two. I was able to calculate ...
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1
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Smoothing projective nodal curve, is the general fiber smooth?
Proposition 29.9 of Hartshorne's Deformation theory states the following:
A reduced curve Y in $\mathbb{P}^n$ with locally smoothable singularities and $H^1(Y,O_Y(1)) = 0$ is smoothable. In particular,...
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Determine whether a matrix is non singular
The model of a spacecraft is the following:
\begin{equation*}
\dot{\sigma} = \mathbf{G}(\sigma)\omega
\end{equation*}
\begin{equation*}\mathbf{G}(\sigma) = \frac{1}{2}\bigg(\frac{1-||\sigma||^2}{2}\...
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1
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Integral with singularities
Let $n > 1$ and $1 \le k \le (n-2)$ be integers, and set
$$f(x) := (-1)^n k \left(\frac{k u \sin (k \pi u)}{n-k u}+\frac{(k+1) u \cos \left((k+1) \pi \sqrt{u}\right)}{(k+1)u-n}+\frac{n}{\pi }\...
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What is the definition of "multiple component of germ"?
Recently I read a paper and I am confused with a word "multiple components", but I don't find its definition in this paper. I guess it is about the singularity. Here is a picture.
You can ...
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