# Questions tagged [cv.complex-variables]

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

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### Why are holomorphic $p$-forms parallel?

Let $X$ be a complex compact Kähler manifold, $\Omega_X$ its cotangent bundle and $D$ the Chern connection.
It is, I believe, a standard fact that parallel $k$-forms $\alpha \in \mathcal{A}^k(X)$, i.e....

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232
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### Can a doubly periodic function be locally univalent?

I am looking for a meromorphic doubly periodic function such that the function is locally univalent.
A standard meromorphic doubly periodic funtion is the Weirestrass $\wp$ function, defined as
$$\wp(...

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### Localisation principle of Szegö Kernel established by Wu and Xing

I am trying to understand the proof of the Localisation principle of Szegö established in the article "Boundary Behaviour of Szegö Kernel" by Wu and Xing. I have some doubts about a few ...

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### Expectation of complex random variable

I am researching frequency offset estimation and ended up reading a paper "Cramer-Rao Lower Bound on Frequency Offset Estimation Error in OFDM Systems With Timing Error Feedback Compensation"...

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### Extension of a Szegő Kernel to the boundary

Let $\Omega\subset\mathbb{C}^n$ be any smooth bounded pseudoconvex domain. Let $S$ denote the Szegő kernel of $\Omega$.
Recall:Szego" kernel is a kernel of Szego projection $P: L^{2}(\partial\...

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### Holomorphic Tangential Transverse property HTTP

I am having trouble showing HTTP property for a strongly pseudoconvex point that lies in a smooth boundary of a domain. Let me introduce this property:
HTTP Property: Let $\Omega\subset\mathbb{C}^n$ ...

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56
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### About jordan curves [closed]

Let $A$ be a complex set which is at most countable. Let $r>0$ be arbitrary, is there a Jordan curve $\Gamma_{r} \subset D(0,r)$ ($D(0,r)$ is the closed disk of radius $r$ centred at 0) which ...

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### Upper bound on $\sum_{i=1}^n \log |1+z_i|$ subject to $\sum_{i=1}^n z_i=0$

Given $z_1,z_2,\ldots,z_n \in \mathbb{C}$ with $\sum_{i=1}^n z_i=0$, what is an upper bound on $\sum_{i=1}^n \log |1+z_i|$?
Universally there is no upper bound as we can take $z_1=-z_2$ and choose $...

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### Can a non-Kähler complex manifold be rationally connected?

Let $X$ be a compact complex manifold. Suppose that $X$ is rationally connected in the sense that any two points lie in the image of a rational curve $\mathbb{CP}^1 \to X$. Are there any non-Kähler ...

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90
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### Limiting behaviour of Cauchy integral near boundary

Let $D \subseteq \mathbb{C} $ be bounded and simply connected, $\Gamma:= \partial D \in C^2 $, $\phi, \psi \in C^{0,\alpha}(\Gamma)$,
$$
f(z):= \frac{1}{2\pi i} \int_{\Gamma} \frac{\phi(\zeta)}{\zeta -...

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### Complex Hölder space

I already posted this question on math.stackexchange, but got no response and was suggested to post it here.
I came across a space in an ergodic theory paper, which I am calling here a (complex) ...

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96
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### Construction of holomorphic function

I was trying to construct a holomorphic function $f$ on $\mathbb{C}$ such that
$|f|^2(z)=e^{(|z|^2-\frac{1}{2})^2}$.
I will be happy if someone can give me an idea how to do that. I would like also ...

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### Behavior of $|f'(z)|/(1+|f(z)|^2)$ as $|z| \rightarrow \infty$?

Let $f(z)$ be an entire holomorphic function in $\mathbb{C}$, and consider the real-valued function
$$g_f(z)=\frac{|f'(z)|}{1+|f(z)|^2}.$$
If $f(z)$ is a polynomial, then it is easy to prove that $\...

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### Reconstructing a function from its residue and discontinuity

I am trying to work through the Zakharov and Shabat paper on inverse scattering for the nonlinear Schrodinger equation . I am stuck on section 2, specifically this problem. I need to know how to ...

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### Invertibility of the sampling matrix

Given a function $f: \mathbb{R}^2\rightarrow\mathbb{C}$ sampled as a matrix $F_{ij}$ on some ractangle $[a,b]\times[c,d]\subset\mathbb{R}^2$ with steps $\Delta x$ and $\Delta y$ as the stepsizes so ...

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### Positive integration on P^1

Let $u: \mathbb{P}^1(\mathbb{C}) \longrightarrow \mathbb{R}$ be a smooth function s.t. $u$ is invariant under complex conjugation and $\displaystyle \int_{\mathbb{P}^1(\mathbb{C})}u \; \omega_{\mathrm{...

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### Differential equation of Van Gorder type for zeta of global fields, or: Does the zeta function of a global field satisfy a differential equation?

Let
\begin{equation*}
\zeta(s):=\prod_{p\text{ prime}}\frac{1}{1-p^{-s}}
\end{equation*}
be the Riemann zeta function. Van Gorder has shown that $\zeta$ satisfies a differential equation
\begin{...

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91
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### Regarding equality of two infimums

Let $M$ be a domain in $\mathbb{C}^n$. Let $\mathbb{D}$ be the open unit disc in $\mathbb{C}$. Let $H(\mathbb{D},M)$ denote the space of all holomorphic functions from $\mathbb{D}$ to $M$. Let $m_1, ...

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89
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### Estimate for an oscillatory integral of the first kind

I am confused in finding the right bound for the following oscillatory integral
$$I = \int_\mathbb{R} (\psi(2^{-k} \xi))^2 e^{i (y \xi - 3 \eta \xi^2 t)} d\xi.$$
Where $\psi(2^{-k} \xi)$ is a smooth ...

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651
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### Explicit triples of isomorphic Riemann surfaces

Inspired by a discussion with Neil Strickland I am very interested to hear of explicit examples (one per answer, please), as follows.
A compact Riemann surface can be presented in many different ways....

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### $f' = e^{f^{-1}}$, a third time

I am of the impression the differential equation $f' = e^{f^{-1}}$ was considered on mathoverflow for the first time here: How do i solve this : $\displaystyle \ f'=e^{{f}^{-1}}$?
It was found ...

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### For estimation on the integral $g(t)=\int_{-t}^{t}\left\vert\sum_{k=1}^Ne^{ikx}\right\rvert^2dx$ for small $t>0$

For real numbers $t>0$ and $x$, let $f(x)=\sum_{k=1}^Ne^{ikx}$ and $g(t)=\int_{-t}^{t}\lvert f(x)\rvert^2dx$. Then $g(\pi)=\int_{-\pi}^{\pi}\lvert f(x)\rvert^2dx=2\pi N$.
I want to know is there ...

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### Is there a residue sum formula in quaternionic analysis?

In complex analysis, there is a formula involving the residues of complex functions that one can employ to find the value of certain infinite series.
If the function $f: \mathbb{C} \to \mathbb{C} $ ...

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### Is the proof in "On Hilbert’s 8th Problem" published on Brazilian Journal of Probability and Statistics correct? [closed]

The article can be freely accessed here. The proof is only five pages. I am quite in doubt.
A new version (2021) of that paper can be found here.

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### Are entire functions “essentially” determined by their maximum modulus function?

(Note: This has been asked on Math SE, but without an answer after almost two years and one offered bounty.)
For an entire function $f$ let $M(r,f)=\max_{|z|=r}|f(z)|$ be its maximum modulus function. ...

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163
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### Convergence of zeta Euler product with additional term

Let's consider the following Euler product ($s=\sigma+it)$:
$$ P(s)=\prod_{p \; \text{prime}} \frac{1}{1-p^{-s}} \; e^{-p^{-s}}$$
So for $\sigma>1$, it is clear the product converges and we have:
$$...

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### Karamata's Abelian/Tauberian Theorem in the complex plane

The following result is well known (a particular case of Karamata's Tauberian Theorem for Power Series in Corollary 1.7.3 of Regular variation by Bingham, Goldie & Teugels):
Fix $c, \rho>0$. If ...

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### Uniform boundedness and boundedness of Maclaurin coefficients [closed]

Suppose $f(z)$ is a bounded in some region $|z|\leq R>1 $ and analytic function and consider the Maclaurin series for $f$:
$$ f(z)= \sum_{n=0}^{\infty}\frac{f^{(n)}(0)}{n!}z^n.$$ My question is:
...

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57
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### Measure of preimage of Jordan disk under entire map

Let $f\colon\mathbb{C} \to \mathbb{C}$ be an entire map. For simplicity assume that $f$ is of finite type, i.e., it has finite set $S(f)$ of singular values. $S(f) \subset \mathbb{C}$ is a minimal (...

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140
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### Zeros of hypergeometric functions with complex variables

Let $z$ be a complex number and let $a,b,c > 0$. I would like to know the zeros of the following hypergeometric function:
$$_{2}F_{1} (a,b; c :z )=\sum_{k=0}^{+ \infty} \frac{(a)_{k}(b)_{k}}{(c)_{...

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### The monodromy in the proof of Little Picard via Klein's $J$

First of all, my question is in the context of formalising proofs in the proof assistant Isabelle/HOL, so my questions are a bit guided by what material we have in the complex analysis library there.
...

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### Complex polynomial-like functions with conjugate terms

Is there study on polynomial-like functions of the following kind?
$$f(z) = c_0 + a_1z+b_1\bar{z} + a_2z^2+b_2\bar{z}^2 + ...+ a_nz^n+b_n\bar{z}^n$$
My reason for studying it is polynomials are ...

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### Request for recommendation in probability and complex analysis

Could somebody kindly recommend to me some books which deal with the applications of the probabilistic method to problems in real and complex analysis or which consider probabilistic versions of some ...

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### Vanishing of a sum of roots of unity

In my answer to this question, there appears the following sub-question about a sum of roots of unity. Denoting $z=\exp\frac{i\pi}N$ (so that $z^N=-1$), can the quantity
$$\sum_{k=0}^{N-1}z^{2k^2+k}$$
...

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### Finite set of numbers whose powers sum up to irrational number

It is well-known that $e/\sqrt{2}$ is irrational.
Indeed, if it was rational, i.e. $p/q$ then $e^2/2 =p^2/q^2.$ Thus, $q^2e^2=2p^2,$ which would imply that $e$ is a root of $q^2x^2=2p^2.$
Now my ...

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### On a variant of Carlson’s theorem

My question is on whether or not there exists some monotone strictly decreasing sequence of positive numbers $c_1>c_2>\ldots$ such that given any $f$ which is a uniformly bounded holomorphic ...

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### Can a holomorphic vector field have an attractor homoclinic loop?

It is well known that a holomorphic vector field $z'=f(z), z\in \mathbb{C}$ does not have any limit cycle.See the last paragraph of this post
Orbits space of real-analytic planar foliations
One can ...

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### Reference request for value distribution theory of bicomplex meromorphic functions

While there is abundant literature available on value distribution of meromorphic functions, I am interested to know whether the value distribution theory for bicomplex meromorphic functions has been ...

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### Maps that preserve winding numbers

This question is a cross post from the Math StackExchange since it got no attention at all there: https://math.stackexchange.com/questions/4414601/maps-that-preserve-winding-numbers
I am looking for a ...

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### Is $\frac{1}{L(1+it)}$ unbounded?

Let $\chi$ be a Dirichlet character and $L(s, \chi)$ be the corresponding L-function. Is $$\frac{1}{L(1+it, \chi)}$$ unbounded for $t \in \mathbb{R}$? I'm aware that this is true if $L=\zeta$, but I'm ...

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### Integral of $\ln(1/|f|)$ for $f$ bandlimited

I came across the following assertion: if $f\in PW_\infty([-a,a])$, i.e. the Bernstein space of functions in $L^\infty(\mathbb{R})$ which are the Fourier transform of a distribution supported on $[-a,...

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### Less strict holomorphy

Using the Grünwald-Letnikov definition of fractional derivative for complex-valued functions, can there be complex function $f(z)$ over all plane, that is not holomorphic but there is $r \in (0,1)$ s....

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### Component wise convergence of a sequence of complex harmonic functions

It is well known that a complex harmonic function $f$ on a simply connected domain $D$ has a canonical decomposition of the form $$f=g+\bar{h},$$ where $g$ and $h$ are analytic functions on $D.$ In ...

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### Appearances of the basic hypergeometric series ${}_0\phi_1(;z;q;q^l z)$

On wikipedia one can find the general definition of a unilateral basic hypergeometric series ${}_r\phi_s$. The special case ${}_0\phi_1(;z;q;q^l z)$ has the expansion
$$
{}_0\phi_1(;z;q;q^l z) = \sum_{...

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### Estimate the imaginary part of Stieljes transformation Im $m(E+i\eta)$ given an estimation of Im $m(E)$ (the continuous extension from $\mathbb{C}_+$)

This is a step of proof from the paper "Anisotropic local laws for random matrices". In the proof of Lemma A.4, the authors say:
we have ... $\operatorname{Im} m(E) \asymp \sqrt{\kappa(E)} \...

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### Constructing proper holomorphic self-mappings of the unit disk with a given set of branch points and corresponding ramification degrees

I was trying to solve the following problem:
Let $f: D \longrightarrow D$ be proper holomorphic (so that means it is a Blaschke product with finitely many factors). Suppose $\{ a_1, ..., a_n \} \...

3
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1
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151
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### A question about average deviation of given $n$ complex numbers

This question just came to my mind and I have no idea as to how to approach it. Let $z_1,z_2,\dots,z_n$ be $n$ be any complex numbers in the unit disc $|z| \leq 1.$ Consider a complex function on the ...

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### Gysin homomorphism of an inclusion to Kähler tubular neighborhood

Let $Z\subset U$ be a Kähler tubular neighborhood of a compact manifold $Z$ of codimension $r$.
Consider de Rham complexes of smooth differential forms $\Lambda^{*,*}(Z),\Lambda^{*,*}(U)$, let $\...

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### Trigonometric sum and residues

I am interested in the sum
$$
\sum_{n=1}^k 2\biggl[\sin\biggl(\frac{n\pi}{2k+2}\biggr)\biggr]^{-2g}
$$
where $k$, $g$ are integers. It is not too hard to show that this can also be expressed as
$$
-1-...

35
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5
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### Looking for some interesting complex integration contours

I am currently working on some tools to make contour integration in a proof assistant less painful and I'm looking for interesting examples of contours in the complex plane used in the literature. I ...