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Questions tagged [singularity-theory]

This tag is for questions relating to Singularity Theory. In singularity theory the general phenomenon of points and sets of singularities is studied, as part of the concept that manifolds (spaces without singularities) may acquire special, singular points by a number of routes.

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.

For more details follow the underline links:

https://math.berkeley.edu/~giventh/papers/arn.pdf

https://en.wikipedia.org/wiki/Singularity_theory

346 questions
13
votes
3 answers

$z_0$ non-removable singularity of $f\Rightarrow z_0$ essential singularity of $\exp(f)$

Let $z_0$ be a non-removable isolated singularity of $f$. Show that $z_0$ is then an essential singularity of $\exp(f)$. Hello, unfortunately I do not know how to proof that. To my opinion one has to consider two cases: $z_0$ is a pole of order…
user34632
3
votes
1 answer

Classify Singularities

So, I'm trying to classify the singular points of the following function: $$ f(z)=e^{\cot(\frac {1}{z})} $$ Obviously, when z is zero, the function tends to approach infinity, so that must be a singularity. To characterize the singularity, I think…
Incognito
  • 435
2
votes
0 answers

$a$ is a non isolated singularity of $f (z)$

If a point $a$ is a non-isolated singularity of a function $f(z)$ and $|f(z)|$ is bounded in some neighbourhood of $a$ then what kind of singularity of $f(z)$ occurs at $a$? Removable singularity Essential singularity Non removable Non…
khushmeet
  • 129
1
vote
1 answer

Essential singularity $z=a$

Suppose $f$ has an essential singularity at $z=a$, is it true that $f(z)$ is not equal to 3 for $z$ not equal to $a$, then $\frac{1}{f(z)-3}$ is bounded in some punctured disk $D'(a,s)$ for some $s>0$?
ilah14
  • 31
1
vote
1 answer

Isolated Singularities

Consider the following functions and determine which kind of singularities they have in $z_0$. If it is a removeable singularity, then calculate the limit; if it is a pole, then give the order of the pole and the main part. …
user34632
1
vote
1 answer

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: $\lim_{z\rightarrow z_0}f(z)(z-z_0)^{m+1}=0$ ? 2- For any $n…
Fabian
  • 671
1
vote
1 answer

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 get $${{e^{2z}} - 2{e}^{z} +1 }= 0$$ Letting…
MATH
  • 501