In mathematics, singularity theory studies spaces that are almost manifolds, but not quite. A string can serve as an example of a one-dimensional manifold, if one neglects its thickness. A singularity can be made by balling it up, dropping it on the floor, and flattening it. (Def: ...

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Is it a removable singularity?

In the function: $$ f(z)=2iz\frac{(1-z^{2})^{\frac{1}{2}}}{1-2z^{2}} \qquad \qquad (z \in \mathbb{Z}) \,\, , $$ There is a singularity at the point $z=\pm \sqrt{1/2}$. Is that a removable ...
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25 views

$|f(z)|\le\frac{M}{|z|^{\alpha}}$ for all $z\in U_r(0)\setminus \{0\}.$ Why is $0$ a removable singularity of $f$?

Let $0<r<1$, $f:U_r\setminus\{0\}\to\mathbb{C}$ holomorphic. Let $\alpha <1,\; M\ge 0$ such that $$|f(z)|\le\frac{M}{|z|^{\alpha}}$$for all $z\in U_r(0)\setminus \{0\}$. Prove that $0$ is a ...
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27 views

what condition of A makes transpose(A)*A nonsingular?

What contidion of A makes $$A^TA$$ nonsingular? If so, that is $$A^TA$$ is non-singular than a unique solution exists.
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26 views

Difficulty with Milnor number

I am reading through the wikipedia page of Milnor number : https://en.wikipedia.org/wiki/Milnor_number#Examples I am reading example 2 where they calculate the Milnor number of $f(x,y)=x^3+xy^2$. So ...
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1answer
40 views

$f(z)$ is an entire function , if $|f(z)|\le 100\log |z|$ for each $z$ with $|z| \ge 2$, find the value of $f(1)$

Let $f$ be an entire function on $\mathbb{C}$ such that $|f(z)|\le 100\log|z|$ for each $z$ with $|z|\ge 2$.If $f(i)=2i$ then what is the value of $f(1)$ ? How to solve this kind of question? I know ...
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31 views

Gimbal lock easier to control with quaternions?

Using quaternions doesn't resolve the issue of gimbal lock, but make is more controllable... how come? They use less memory, and are commutable, and provide an smooth rotation along nonlinear ...
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1answer
28 views

Square of a function with an essential singularity

Let $a$ be a point on the complex plane such that the function $f$ has an essential singularity. I am trying to prove that the square of that function also has an essential singularity. I suppose ...
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1answer
60 views

If $f(z)$ is meromorphic but not entire, is $\exp(f(z))$ meromorphic? Could it even be entire?

First, I can show that $f$ meromorphic is a rational function. Now, I want to consider $g=e^{f(z)}$. I have heard that there is something interesting that goes on with $g$, that there is some room ...
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16 views

$p$ a nonsingular point in $V$ and $V(f)\cap T_p(V)$ then $f|_{T_p(V)}$ has a factor of multiplicity $\geq 2$.

This is a problem from Ideals, Varieties, and Algorithms by Cox et. at. Let $V \subset k^n$ be a hypersurface with $I(V ) = \langle f \rangle$. Show that if $V$ is not a hyperplane and $p \in V$ ...
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69 views

Limit of a difference of integrals that both look almost identical,

Let $\gamma (t) = t+i(e^t-1)$ for $-1\le t \le 1$. find $$\lim_{\epsilon \to 0^+} \left[\int_{\gamma} \frac{\sin(z)}{(z-i\epsilon)^2} dz - \int_{\gamma} \frac{\sin(z)}{(z+i\epsilon)^2} dz\right]$$ ...
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3answers
678 views

Need a hint for this integral

I'm trying to evaluate the following integral $$\int_0^{\infty} \frac{1}{x^{\frac{3}{2}}+1}\,dx.$$ This is an old complex analysis exam question, so I plan to use the residue theorem. How can I ...
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0answers
35 views

Understanding singularities through Jacobian

In kinematics of mechanisms we derive the constraint equations depending on the architecture and then analyse the singularities of the mechanisms by deriving their Jacobian matrices. For example: ...
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35 views

On the dog-bone contour around [-1,1], what are the arguments of these two lines approaching the real axis from above and below?

I am using a dog-bone contour to integrate around the interval [-1,1]. (-1 and +1 are branch points of the integrand.) I am using the principal branch of log, so I am restricting its argument to ...
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1answer
20 views

Quick question on the roots and poles of a meromorphic function,

Does the degree of the polynomial in the numerator always equal the degree of the polynomial in the denominator? In other words, the number of zeros, counting multiplicity, equals the number of ...
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29 views

Local ring in singularity theory and algebraic geometry

In the book Singularity theory 1 by Arnold et al., page 13, there is a definition of a local algebra of singularity, which I tried and failed to rephrase in the language of algebraic geometry. It is ...
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2answers
93 views

How can a positive integrand integrate to 0? [duplicate]

I integrated $\dfrac{\log x}{1+x^2}$ from $0$ to infinity with residue calculus and got... $0$. This also agrees with Wolfram Alpha. How can this be? Is it due to the behavior of $\log(x)$ near ...
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2answers
192 views

Why must a meromorphic function, bounded near infinity, have the same number of poles and zeros?

Writing down some easy rational functions to check this, I don't see why this must be the case. Although if the function had 3 simple zeros and 2 simple poles its rational form would be in the form ...
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0answers
13 views

How to numerically solve equation with regular singularity problem?

Good day, Here is my equation $$ (x-1)^2 (x-h)^2q_1(x,\Omega)y''(x)+(x-1)(x-h)q_2(x,\Omega)y'(x)+q_3(x,\Omega)y(x)=0 $$ ( with $q_1(x,\Omega)$[polynomial function 38 degree], $q_2(x,\Omega)$ ...
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1answer
35 views

Finding the order of pole of $f(z)=\frac{\sin z}{z-\pi}$

The problem is Kreyszig 10ed international edition : 16.2 #9. What is the order of the pole at $z=\pi$ of the function $f(z)$ below? $$f(z)=\frac{\sin z}{z-\pi}$$ I thought that it will be a simple ...
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58 views

How to get asymptotic form of the integrals with special functions?

I got difficulty when I try to plot I(x) for $m=1$ and $t=0.2$. The questions is how to get the asymptotic form of the following integral? $I(x,t)=\int_{0}^{\infty} \frac{f(y)}{2 \sqrt{\pi t}} ...
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1answer
45 views

Singularities of $\sin(z)/(1-\cos(\sqrt{z}\,))$

$\displaystyle f(z) = \frac{\sin(z)}{1-\cos(\sqrt{z}\,)}$. The assignment is to find all the singularities of $f$, determine the type of them and the residue. It is clear that the singularities are ...
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15 views

Identification of pole

I would like to know whether the following statement is true or not If $f$ is an analytic function satisfied $|f(z)|\to\infty$ when $z\to z_0$, then $f$ is a pole at $z=z_0$ I would say the ...
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3answers
100 views

Very tricky complex integral, with poles on both sides of the real line,

I am trying to evaluate$$\int_{-\infty}^{\infty} \frac {x^2 -x^4}{1-x^6}\,dx,$$ which is an old exam problem. There is a special note on this problem that reads: Note: Your answer need not be a ...
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2answers
228 views

Do the singular matrices form a topological manifold

So the definition of manifold I'm using is that of a topological manifold (a topological space with an atlas of homeomorphisms to $\mathbb{R}^n$). I have two related questions: Is the set of ...
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2answers
40 views

How can I show that this meromorphic function is a rational function of two polynomials?

Here's my updated attempt: Write$$f(z) = \sum_{n=-1}^{\infty} a_n(z-z_1)^n + ...+\sum_{n=-1}^{\infty} m_n(z-z_m)^n+\sum_{n=+1}^{-\infty} \psi_n(z)^n$$ with the last series being an expansion about ...
1
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1answer
38 views

Would a keyhole contour be advisable to use for this integration?

The integral is $$\int_0^{\infty}\frac {1}{\sqrt{x}(1+x^2)}dx$$ which is to be evaluated by contour integration. So, the integrand clearly has simple poles at $+/- i$. But what kind of pole ...
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1answer
22 views

Meromorphic Function on Extended Plane

How do I prove that every meromorphic function on the extended plane is a rational function?
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15 views

complexe analysis. Singularity points

What kind of singularity exists for sin[1/sin(1/x)]? I feel it is an essential singularity but I failed in the demontration. To develop in Laurent serie would be the solution but how??
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1answer
26 views

When computing contour integration with sines and cosines in the integrand, must we always first look at Euler's formula?

For example, in computing $$\int_{Cr}\frac {\cos(z)}{(z^2+a^2)^2}dz$$ over a semi-circular contour, must I first look at $$\int_{Cr}\frac {e^{iz}}{(z^2+a^2)^2}dz$$ compute this integral first, ...
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1answer
31 views

How can I compute the residue at this order-2 pole?

The integral is $$\int_{-\infty}^{\infty} \frac {cos(z)}{(x^2+a^2)^2}dz $$ If I use an upper semi-circular contour, then there is an order-2 pole at $z=ia$. I am trying to expand the integrand in a ...
2
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0answers
39 views

Lipschitz continuous one-to-one mapping from subset $K\subset\mathbb{R}^n$ of positive measure to $\mathbb{R}^{n-1}$

Let $f:\mathbb{R}^n\to \mathbb{R}^{n-1}$ and $K\subseteq \mathbb{R}^n$ be a set of positive Lebesgue measure. What kind of regularity do we have to impose on $f$ (e.g., $C^1$, Lipschitz) to conclude ...
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0answers
11 views

How to understand multifractal and multifractal spectrum?

A case I encountered with to illustrate multifractal is to assign different subintervals of Cantor set with different mass. In detail for each time, the interval is divided into 3 equal-length parts ...
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0answers
42 views

How to solve a singularity problem for diferential equation?

Good day, Here is my equation $$ (x-1)^2 (x-h)^2q_1(x,w)y''(x)+(x-1)(x-h)q_2(x,w)y'(x)+q_3(x,w)y(x)=0 $$ (with $q_1(x,w), q_2(x,w), q_3(x,w)$ regular functions). The boundary conditions are $$ ...
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0answers
41 views

Is the complement of the singular locus of an algebraic variety a topological manifold?

Could someone explain to me clearly, please, why is the complement of the singular locus of an algebraic variety a topological manifold ? A lot of thanks for your help.
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38 views

About singularity

Let $X$ be a normal affine variety over $\mathbb{C}$. Q1. Let $x\in X$ be a singularity with a Cartier divisor $x\in D$. Then $\mathcal{O}_{X,x}$ is Cohen-Macaulay if and only if $\mathcal{O}_{D,x}$ ...
2
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1answer
61 views

Use contour integration to compute the Fourier transform,

The problem statement is: Use contour integration to determine the Fourier transform, $\large \hat f(ξ)=∫_{-\infty}^{\infty}f(x)e^{−iξx}dx$, of $\large f(x)=\frac{1}{2−2x−x^2}$. Some issues that I ...
4
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1answer
47 views

Taking the divergence of a field with a singularity when $\vec{r}=0$ produces a Dirac's delta.

I'm currently taking a classical electrodynamics course. I have a mathematical background and I know that the classical theorems of integral calculus (Stokes, Gauss, ...) are just particular versions ...
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0answers
38 views

Does a function with an exponential growth condition necessarily have infinitely many zeros?

This is part (2) of a question that I am working on. In part(1), I have constructed an entire function $f:=\cosh(\sqrt{z})$ that grows like $$\lim_{r \to \infty} \frac {\log M(r)}{\sqrt{r}}=1$$ ...
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1answer
40 views

Can a real symmetric matrix have 0 (Zero) as one of the eigen values?

From what i know (correct me if i am wrong): 0 as an eigen value of a real symmetric matrix implies it is Singular (Non- invertible). I am not aware of any such ...
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1answer
51 views

Singular transfer function matrix and System singularity

I have a linearized dynamic system that can be summarized as: [ΔY] = [A][ΔX] The transfer function matrix, [A], is singular for steady state. My question is ...
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20 views

Derivative of a definite improper integral

-The derivative with respect to beta, for the following definite integral is required. g = $\int_\beta^{\sqrt(\beta^2 +1}$ $erfc(\gamma z)/\sqrt(z^2 - \beta^2)$dz -I am using the leibniz formula ...
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1answer
72 views

Does every open manifold admit a function without critical point?

Assume that $M$ is a non compact smooth manifold. Is there a smooth map $f:M\to \mathbb{R}$ such that $f$ has no critical point? The motivation comes from the conversations on this post.
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44 views

How to determine the type of singularities in affine varieties

Let $X$ be the affine variety in $\mathbb C^3$ (coordinates $a, b, c$) defined as the zero set of $ac - b^2$. The variety has a "double point singularity" at the origin. This is somewhat intuitive, ...
2
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0answers
26 views

PDE Singularity and time of shock

Given the following PDE: $$∂p/∂t+p*∂p/∂x=0$$ $$p=p(x,t)$$ $$p(x,0)=e^{-x^2}$$ It is requested to find its solution, show that it is not well behaved by verifying what happens to $∂p/∂t$ and $∂p/∂x$ ...
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1answer
41 views

A Question About Poles.

I have some questions in my mind bothering me to understand poles. Let $z_0$ be a pole of order $m$ for $f(z)$. Does that mean: 1- $(m+1)$ is the smallest positive integer such that: ...
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1answer
45 views

Finding singularities of A Function

I want to find singularities of $$f(z)=\frac{{z}^{2}}{e^z + {e}^{-z} - 2}$$ I solved this problem but I am not sure about it. Is it correct? $${e^z + {e}^{-z} - 2}= 0$$ Then I divide by $$e^{-z}$$ to ...
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30 views

asymptotic matched expansion with transiently blowing inner solution

I have been trying to solve following set of equations with method of matched asymptotic expansion, $\frac{dy(t)}{dt}=k z(t) - 3 \alpha y(t) - y(t)^2 + \mu (M-z(t))^2$ $\epsilon ...
3
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0answers
53 views

General Fiber in Positive Characteristic

It is well-known that a complex polynomial, considered as a function $f:\mathbb{C}^n\to\mathbb{C}$, is a fiber bundle over a cofinite set of "atypical values" which include the singular values of $f$. ...
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1answer
40 views

Properties of resolution of singularities

Let $X$ be a complex algebraic varieties and $\pi:X' \to X$ be a resolution of singularities of $X$. Let $Y$ be a smooth (irreducible) subvariety of $X$. Is $\pi^{-1}(Y)$ smooth and irreducible? What ...
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3answers
101 views

Obtaining a Non-Singular Matrix from a Singular one by Perturbation

In a paper "http://www.math.cornell.edu/~nussbaum/papers/08-1.pdf" (page 264 Lemma 2) I encountered the following way of obtaining an invertible (non-singular) matrix from a non-invertible (singular) ...