# Tagged Questions

Complex analysis, holomorphic functions, automorphic group actions and forms, pseudoconvexity, complex geometry, analytic spaces, analytic sheaves.

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### Contour integral $\int_{\gamma}e^{-t\mu}(-\mu)^{-m}\log\sqrt{-\mu}d\mu$

Motivation: In my research I try to find the asymptotics of the heat trace $\text{tr}e^{-t\Delta}$ as $t\to0$, where $\Delta$ is Laplace operator on a manifold with singularities. First I find the ...
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### Complex proof of $B(a,b)=\Gamma(a)\Gamma(b)/\Gamma(a+b)$

It is a question in spirit of this one. Is there a way to prove Euler's formula $$\int_0^1 x^{a-1}(1-x)^{b-1}dx=\frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)}$$ using contour integration (and maybe ...
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### How to take partial derivative of spherical interpolation of quaternions?

Using the standard definition of quaternionic spherical linear interpolation (slerp): $$Q(q_0,q_1,t) := q_0(q_0^{-1}q_1)^t,$$ how can I take each partial derivative? Actually, I'm confident how to ...
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### For which integer values of $k$ can we find one solution to the equation $\sum\limits_{n=1}^{n=k} \frac{1}{n^s}=0$ by iteration? [closed]

I am trying to find solutions to the well known equation: $$\sum\limits_{n=1}^{n=k} \frac{1}{n^s}=0$$ Now with this program below I have found that for certain values of the integer $k$ one can find ...
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### A question on evaluation of complex integrals

Is any general relationship between the integral $\int_{0}^{1}f(u, \sigma + it)u^{-1 + d}du$ and $f(0, \sigma + it)$ known? I have proved one such result where the main term of the given integral ...
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### Periodicity in iterated powers of sin, cos, exp

Given a complex number $z$, consider the sequence \begin{align*} a_0 & = 1\\ a_1 & = (cos(1))^z\\ a_n & = (cos(a_{n-1}))^z \end{align*} This question is about trying to understand ...
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### Complex structure on a punctured torus giving a complex structure on the torus?

Can anyone provide an idea of the proof or a reference of the fact that a complex structure on the once punctured torus extends to one on the torus? In other words, the Teichmuller space of the ...
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### What is the Fano index for Hermitian symmetric spaces of compact type?

As we know Hermitian Symmetric spaces of compact type are all Fano picard number one, we can talk about his Fano index. Suppose $X$ is one of those Hermitian symmetric spaces, $L$ is the generator of ...
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### Is the set of entire functions Borel in the space of analytic functions?

$\def\bbR{\mathbb R}\def\bbC{\mathbb C}\def\scrT{\mathscr T}\def\ssp{\kern.4mm}$More specifically, I ask whether $S$ be a Borel set in the topological space $(\Omega,\scrT)$ in the following ...
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### Stone-Weierstrass theorem for holomorphic functions?

The Stone-Weierstrass theorem has an analog for the algebras of smooth functions, called Naсhbin's theorem: An involutive subalgebra $A$ in the algebra ${\mathcal C}^\infty(M)$ of smooth ...
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### Asymptotic Robin inequality and RH [closed]

There exists several equivalent formulations of RH. Among them, there is a criterion of Robin that describes a bound on the growth rate of the sum-of-divisors function $\sigma$. Apparently yesterday (...
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### Fourier transform of the critical line of zeta?

This was asked on MSE and got a lot of upvotes but no answers, so I'm posting it here. Is there a known expression for the (distributional) Fourier transform of the Riemann zeta function, taken along ...
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### Assuming admissible functions $\rho$ are continuous in definition of conformal modulus

It's stated in Väisälä's 'Lectures on n-dimensional quasiconformal mappings' (p. 20) that, in the geometric definition of a quasiconformal mapping, that the modulus of a family of curves associated to ...
Assume that $\gamma$ is a starlike Jordan curve in the complex plane w.r.t. 0. Let $\alpha\in (0,\pi/2]$. For each $z\in\gamma$, $z\neq x$, we let $\alpha(z,x)$ denote the acute angle which the ...