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.
1,052
questions
0
votes
1
answer
47
views
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 ...
1
vote
0
answers
13
views
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?
0
votes
0
answers
66
views
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 ...
- 1,139
1
vote
1
answer
106
views
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
$$
2\int_1^2 f(x)dx=\oint f(...
- 1,139
-1
votes
0
answers
52
views
$\int_0^{+\infty}\frac{\cos (ax)-\cos(bx)}{x^2}\mathrm dx$ using complex analysis [duplicate]
$$
\mbox{I defined the complex function}\quad
\operatorname{f}\left(z\right) =
{{\rm e}^{{\rm i}z} \over z^{2}}
$$
and in the end i just take the real part of my answer. I want to use Residues Theorem....
- 17
1
vote
1
answer
102
views
Using the residue theorem to compute two integrals [closed]
Classify the singular points for the function under the integral and using the residue theorem, compute:
(a) $$ \int_{|z-i|=2} \frac{z^2}{z^4 + 8z^2 + 16} \, dz, $$
(b) $$ \int_{|z|=2} \sin\left(\frac{...
- 45
0
votes
0
answers
26
views
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 ...
- 117
0
votes
1
answer
64
views
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,...
- 53
0
votes
0
answers
31
views
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}\...
- 75
1
vote
1
answer
57
views
Integral with singularities
Let $n > 1$ and $1 \le k \le (n-2)$ be integers, and set
$$f(x) := (-1)^n k \left(\frac{k u \sin (k \pi u)}{n-k u}+\frac{(k+1) u \cos \left((k+1) \pi \sqrt{u}\right)}{(k+1)u-n}+\frac{n}{\pi }\...
0
votes
0
answers
42
views
What is the definition of "multiple component of germ"?
Recently I read a paper and I am confused with a word "multiple components", but I don't find its definition in this paper. I guess it is about the singularity. Here is a picture.
You can ...
- 59
0
votes
0
answers
24
views
restricting a function changes its singular points and analyticity?
Let define $f(z) = \frac{1}{z-2}$ for $z\in\mathbb{C}\setminus\{2\}$. Then it is clear that, $f(z)$ has singular point at $z=2$ (Namely pole of order 1 at $z=2$).
However, if I update the definition ...
1
vote
0
answers
81
views
Singularities of $\frac{1}{1-z^n}$
I'm looking to classify the singularities of $g(z) = \frac{1}{1-z^n}$ and compute the residue at each pole.
Now, $g(z)$ has singularities at the roots of unity $w^k$, where $w=\exp\{\frac{2\pi \textbf ...
- 467
1
vote
0
answers
27
views
Singularity in Poisson's Equation
Consider an instance of Poisson's equation in spherical coordinates for the radial dimension:
$$
\nabla \cdot \nabla \phi(r) = \frac{1}{r^2} \frac{d}{dr}\left(r^2 \frac{d\phi}{dr} \right) = -\sin(r).
$...
- 135
0
votes
0
answers
11
views
What is the relationship between regular singular points and the Cauchy Euler equation?
In my PDE course, my professor shows me how we can take a generic homogenous polynomial coefficient 2nd order differential equation:
$$P(x)y'' + Q(x)y' + R(x)y = 0$$
And transform it into the ...
- 21