# Questions tagged [singularity]

This tag is for questions relating to singularity, which is a point where a mathemtical 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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### Which type of singularity does this complex function have at z = ∞?

I'm trying to classify the type of singularity at z = ∞ (the point at infinity) of the complex function: click to see the equation Up to now, I've just been able to prove that ∞ is not a pole. So I ...
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### Existence of a complex sequence with given property

How to show the existence of a complex sequence $(z_n)$ with $z_n\ne 1, \forall n$ but $\lim_{n\to\infty}z_n=1$ such that $\lim_{n\to \infty}\sin(\frac{1}{1-z_n})=100$? Can such a sequence be ...
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### Removing coordinate singularities

Consider the Riemannian metric given in this picture. Wikipedia claims that this Riemannian manifold has the topology $\mathbb{R}^2\times S^2$, but the coordinate expression given in the picture seems ...
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### Resolution of branch point singularity

Let $\pi:Y\to\mathbb{P}^2$ be a Galois cover of the projective plane which is branched along $r$ lines $L_1, L_2,...,L_r$ in $\mathbb{P}^2$. Suppose the lines $L_1, L_2,...,L_r$ all pass through the ...
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### How to compute the residue of $\frac{1}{e^{\frac{1}{z}}-1}$ around $z=0$?

Here is a problem in my complex analysis notes, i.e. compute the residue of $\frac{1}{e^{\frac{1}{z}}-1}$ around $z=0$. I think that this isolated singularity is an essential singularity. The problem ...
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### Prove that if the only singularities of a function are poles then the function must be rational.

Prove that if the only singularities of a function are poles then the function must be rational. So originally this was an iff statement, but I have solved the other direction of the proof. I'm just ...
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### Find the singularities of $f(z) =\frac{1}{(2\sin z - 1)^2}$.

Find the singularities of $f(z) =\frac{1}{(2\sin z - 1)^2}$. I am just learning about singularities and I was wondering if someone could give me feedback on my work. So I think, for this function, ...
Looking at the function $f(z) = \frac{z^2+1}{z^2(z+1)}$, I have found the singularities to be at $z=0$ and $z=-1$. My question is if they are removable. I expanded this into the Laurent series $\frac{... 1answer 30 views ### Question about proof of Riemann removable singularity theorem. Theorem: Let$f:D^*(z_0,r)=D(z_0,r)-\{z_0\}\to \Bbb C$be holomorphic and bounded. Then$\lim_{z\to z_0}f(z)$exists and the function$\hat{f}:D(z_0,r)\to \Bbb C$defined by $$\hat{f}(z) = \begin{... 0answers 60 views ### Milnor number/Mapping Degree I am reading in John Milnor's Book Singular Points of Complex Hypersurfaces and struggle to do a similar computation he did. Namely, how can I compute explicitly the Milnor number \mu(f) on the ... 4answers 66 views ### Why is the pole of \frac{x}{\sinh(x)} a simple pole and not a removable singularity? I'm doing my homework for my Complex Analysis class and I'm asked to solve a definite integral of \frac{xdx}{\sinh(x)}. This is not a problem for me, however the problem tells us directly that there ... 1answer 20 views ### Invertible X^TX - what happens when you clone rows of X? My question is inspired by https://stats.stackexchange.com/questions/70899/what-correlation-makes-a-matrix-singular-and-what-are-implications-of-singularit, in particular ttnphns's answer where they ... 1answer 80 views ### Unexpected singularities in an integral I am doing an indefinite integral,$$f=\int\frac{r^2}{(r^2 + d^2 -2rd\cos{\theta})^2}dr$$where 0\leq r< \infty, 0\leq d< \infty and 0\leq \theta \leq \pi. The integral that I am getting (... 0answers 32 views ### Residue theorem, where did I make a mistake? I have to compute$$\int_0^{2\pi}\frac{dx}{7+6\cos(x)}$$First, I was going to find a contour, which is the unit circle.$z(\theta) = e^{i\theta}$,$0 \leq \theta \leq 2\pi$. Now,$\cos(\theta) = \...
I have a function $f(z) = \frac{\cos(z)}{z^6}$. I have to find the singularities and the corresponding residues. I think there is a single pole at $z=0$, which has order 6. For the residue, I did this:...