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.

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 […]

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?

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 […]

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 […]

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 […]

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 $\lim_{z\to 0}z\log z=0$, […]

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 […]

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 $f$.

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