Articles of singularity theory

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.

How to get the asymptotic form of this oscilatting integral?

So the integral is like this: $$\int_1^\infty \frac{\cos xt}{(x^2-1)\left[\left(\ln\left|\frac{1-x}{1+x}\right|\right)^2+\pi^2\right]}\mathrm{d}x$$ The question is how to get the asymptotic form of this integral when $t$ is very large. The integrand is a decaying oscilating function: The denominator of the integrand is zero when $x\to 1$. Honestly, I don’t know whether this integral will converge, for it reminds […]

$\frac{1}{z}-\frac{1}{\sin z}$ at the origin – Classify singularities

I tried for a while to classifiy the singularities of $\frac{1}{z}-\frac{1}{\sin z}$ at the origin, but I am stucked. Is there someone who is able to help me at this point?

Removable singularity and laurent series

How to show $z=\pm\pi$ is a removable singularity for $\frac1{\sin z}+\frac{2z}{z^2-\pi^2}$? I tried to compute the Laurent series, specifically the coefficients for $1/z,1/z^2,…$ since if we can show those coefficients are all zeroes, we are done. Is there a nice way to compute those coefficients? I tried the contour integral formula but couldn’t get anywhere […]

Mathematical definition of the word “generic” as in “generic” singularity or “generic” map?

I’ve been trying to work out what generic means but I’m not making much progress. You can find an example of the usage of the word generic for example here: “School on Generic Singularities in Geometry” or here: “Rigidity of generic singularities of mean curvature flow” Here is what I have so far: I found […]

Proving that a function has a removable singularity at infinity

I’m having trouble with the following exercise from Ahlfors’ text (not homework) “If $f(z)$ is analytic in a neighborhood of $\infty$ and if $z^{-1} \Re f(z) \to0$ as $z \to \infty$, show that $\lim_{z \to \infty} f(z)$ exists. (In other words, the isolated singularity at $\infty$ is removable) Hint: Show first, by use of Cauchy’s […]

Type of singularity of $\log z$ at $z=0$

What type of singularity is $z=0$ for $\log z$ (any branch)? What is the Laurent series for $\log z$ centered at 0, if exist? If the Laurent series has the form $\sum_{k=-\infty}^{\infty} a_kx^k$, then certainly among $a_{-1},a_{-2},…,a_{-j},…$, at least one is nonzero (or otherwise $\log z$ would be analytic at $0$). Since $\lim_{z\to 0}z\log z=0$, […]

Could this be called Renormalization?

Quoted from   Space-Time Approach to Quantum Electrodynamics   by R. P. Feynman, Phys. Rev. 76, 769 1949 : We desire to make a modification of quantum electrodynamics analogous to the modification of classical electrodynamics described in a previous article, A. There the $\delta(s^2_{12})$ appearing in the action of interaction was replaced by $f(s^2_{12})$ where […]

$z_0$ 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 $k$ of $f$. $z_0$ is an essential singularity of $f$.