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.

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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 ...
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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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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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Let $A$ be some matrix, If $x^TAx = 0$ for every vector $x$, then $A$ is singular. How can I prove it? Here is what I thought: first of all, for every matrice $M$, $$x^T(M + M^T)x = x^TMx + x^TM^Tx =... 1 vote 1 answer 91 views Problem in understanding Riemann's removable singularity theorem. Theorem : Suppose that f is holomorphic in the region \Omega' \subseteq \Omega which is obtained from \Omega by a removing a single point a \in \Omega. Then f has a unique holomorphic ... • 2,612 0 votes 0 answers 37 views Confused about a proof in Rational Solutions of the Fifth Painleve Equation by Kitaev et al. In the article Rational Solutions of the Fifth Painleve Equation (see here) proposition 2.3 derives the orders and residues of singularities of rational solutions to a particular differential equation ... • 113 0 votes 1 answer 56 views Is \frac{1}{\sqrt{\mid x \mid}} integrable in interval -1 to 1? I know that I can't integrate some unlimited functions while my interval of integration contains the singularity point. For example, \int_{-1}^{1}\frac{1}{x}dx is undefined, it makes sense to me ... • 17 1 vote 0 answers 69 views How pathological can a Laplace transform be? Almost every treatment of the Laplace transform that I come across talks about "the poles" of the Laplace transform function F(s), thereby seeming to implicitly assume that F(s) is ... • 6,280 0 votes 0 answers 60 views Type of singularity at z=\infty in \frac{1+e^z}{1-e^z} In my book, the poles of f(z) have been found out as z=(2n+1)πi. So, the singularity is non-isolated essential as z=\infty is a limit point of these poles. I understand this. However, if I ... 4 votes 0 answers 122 views What is a numerical method to solve IVP at irregular singular point? I have tried to search for numerical method to solve IVP with irregular singular point but didn't find any. What is the proper method I can use to solve such equations?$$y''+\frac{y'}{x^2}+y=x^3+6x+3$... • 1,266 1 vote 5 answers 156 views How to compute limit to infinity of an exponential in complex analysis? My question originates from trying to find the following limit: $$\lim_{z\to\infty} e^{-z^2}$$ Disclaimer: before getting into details, I want to point out that English isn't my first language, so I ... 2 votes 1 answer 125 views Integrating a Modified Bessel function of the second kind with a singularity Does someone know how to handle the integral $$\int_{-\infty}^{\infty} \frac{K_0\!\left(\lvert \tau \rvert \sqrt{q^2 \alpha ^2}\right)}{q^2}\cos (q x)\,\mathrm{d}q$$$\alpha$is a real number and$\...
Question: Suppose $f: \mathbb{C}\setminus P \to \mathbb{C}$ is meromorphic on the complex plane, where $P:=\{z_n: n\in\mathbb{N}\}$ is a set of infinitely many isolated poles with $\infty$ as their ...
I have modeled the interaction of two physical quantities, $S$ and $B$, by the following differential equations (the second one is a delay differential equation): S'(t) = 0.31 S(t) \Big( 1 - \frac{S(...