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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14 views

Can we find a holomorphic representation in these mappings

$\DeclareMathOperator\PSL{PSL}\DeclareMathOperator\PSU{PSU}$Let $U\in \mathbb C^2$ be a given open set, $A$ be the set composed by maps (not necessarily continuous) $f:U\to \PSL(2,\mathbb C)$. We call ...
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75 views

Upper bound for the complex Beta function

The question is almost the same as here. What is the upper bound for a complex Beta function $\displaystyle B(s,z)=\frac{\Gamma(s) \Gamma(z)}{\Gamma(s+z)}$ with $0<Re(s)<1$ and $0<Re(z)<1$;...
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Is this an outer function

I think I have read this theorem but I cannot find it. Let us suppose that $\Theta(z)$ is an inner function in $H^\infty(\mathbb{C}^+)$. Is it true that $1-\Theta(z)$ is an outer function? It is clear ...
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Functions bounded by polynomials [closed]

Under which minimal regularity conditions is a real or complex function bounded above and below by polynomials necessarily a polynomial? More precisely suppose $f : k \rightarrow k$ is such that $p(x) ...
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1answer
99 views

Complex manifolds making Liouville fail

Let us consider $g\colon X\to Y$ holomorphic, where $X$ is a complex manifold and $Y$ is a Stein manifold. I am searching for all the pairs $(X,Y)$ such that we can find some non constant $g$ with ...
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holomorphic functions on Stein manifolds reaching maxima in a given point on the boundary (Shilov boundary)

Let $M$ be a Stein manifold with smooth, strictly pseudoconvex boundary, and $x$ a point on its boundary. Is there a holomorphic function $f$ on $M$, smooth on the boundary, with strict maximum of $|f|...
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Regarding starlike sets in $\mathbb{C}^n$

A domain $S$ in $\mathbb{C}^n$ is called starlike about the point $x_0\in S$ if for every point of $S$, the segment of the straight line from the point to $x_0$ lies in $S$. Let $S$ in $\mathbb{C}^n$ ...
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1answer
171 views

Upper bound for maximum modulus of polynomial on unit circle in term of the distribution of its roots

Let $P(z) = \prod_{i = 1}^n (z - z_i) \in \mathbb{C}[z]$ be a monic polynomial having all roots $z_1, \dots, z_n$ on the unit circle $\mathbb{T} := \{z \in \mathbb{C} : |z| = 1\}$. What is known about ...
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1answer
99 views

Integral of $I = \int_{a}^{\infty}dx \frac{x^s}{(1+x)^{n}}$ [closed]

I have been trying to evaluate the following integral: $$I = \int_{a}^{\infty}dx \frac{x^s}{(1+x)^{n}}$$ If $a=0$, then this is the Mellin transform of $\frac{1}{(1+x)^n}$. However, suppose $a \neq 0$....
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Solving an equation containing Laplace transform

Consider the equation \begin{equation} \frac{f(p)}{f(s_{1})}\mathcal{L}(y)(s_{1})+\frac{g(p)}{g(s_{2})}\mathcal{L}% (y)(s_{2})=\mathcal{L(}y)\mathbf{(}p), \end{equation} where $\mathcal{L}$ is the ...
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413 views

An extension of the Carlson's theorem in complex analysis

For the statement of Carlson's theorem please see, https://en.wikipedia.org/wiki/Carlson%27s_theorem. There is an extension of Carlson's theorem that says that the condition that $f$ needs to vanish ...
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143 views

Approximation of the product $(\bar{z} - a)^{-1} \cdot (z - b)^{-1}$

I would like to construct an approximation of the product \begin{equation} f(z) = \frac{1}{\overline{z}-a} \frac{1}{z-b}, \end{equation} where $a, b \in \mathbb{C}$, and $|{a}/{z}|, |{b}/{z}| <1$. ...
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146 views

What is a holomorphic foliation?

For a smooth foliation $F$, there are three equivalent definitions: the leaves of $F$ are tangent to a smooth vector field; the foliation chart $\phi:U\to \mathbb R^k\times \mathbb R^{n-k}$ is ...
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85 views

Are there extensions of Euler's infinite product for sine function? [migrated]

Euler product about sine function is $\frac{\sin(x)}{x} = \prod_{n=1}^\infty \left ( 1- \left(\frac{x}{n\pi}\right)^2 \right)$ I wonder if there is known results about slight modification of above ...
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162 views

Coefficients for Expansions of $1-\zeta_p$

Let $\mathbb{Q}_p(\zeta_p)$ be the cyclotomic extension of the $p$-adic field $\mathbb{Q}_p$. Then $1 - \zeta_p$ is a uniformizer for this field. Recall that $$\sum_{i=1}^{p-1} \zeta_p^i = -1.$$ So ...
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Walsh-Lebesgue type theorem in $\Bbb R^{2m}$ for $m>1$

Is someone aware of any analogue of the Walsh-Lebesgue theorem in $\mathbb{R}^{2m}$ for $m>1$ and dealing with polyharmonic polynomials? In this post, $\phi$ is said to be polyharmonic in $\mathbb{...
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Is an entire function $\mathbb{C}^n\to\mathbb{C}$ a composition of polynomials, univariate entire functions and integrals?

Let $S$ be a set of entire functions $\mathbb{C}^n\to\mathbb{C}$. To enlarge it we can take polynomial combinations of its elements, compose them with entire functions $\mathbb{C}\to\mathbb{C}$ and ...
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1answer
517 views

Existence of a smooth compactly supported function

Let $U$ be a bounded domain in $\mathbb R^n$. Does there exist a smooth function $f$ with compact support in $U$ such that: $$ \| f\|_{W^{k,\infty}(U)} \leq (k!)^{2-\epsilon},$$ for some $\epsilon>...
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Distrbution of points transformed by a family of polynomials

Consider a family of polynomials $\mathcal{F}$. Let $p$ be a single complex point or a finite set of complex points inside the unit disk. I am interested in what can we say about the distribution of $$...
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441 views

Are the fibers of a surjective holomorphic submersion $\mathbb{C}^n\to\mathbb{C}$ all homeomorphic?

Are the fibers of a surjective holomorphic submersion $\mathbb{C}^n\to\mathbb{C}$ all homeomorphic? For $n=1$ this means that a surjective entire function $\mathbb{C}\to\mathbb{C}$ without critical ...
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1answer
288 views

On convergence of entire functions

Suppose we have a sequence of entire functions $f_n$ such that $$\text{$f_n(z)\to0$ for each natural $z$}\tag{1}$$ (as $n\to\infty$). Is it possible to give general additional conditions on the ...
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63 views

Global sections appearing in Dolbeault complex with values in vector bundle

Given a holomorphic vector bundle $E$ on a compact complex Kähler manifold $X$ (I am happy to assume $X$ projective), we can compute the sheaf cohomology $H^\ast(E)$ of $E$ using the Dolbeault complex ...
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Quasi-crystaline generalization of elliptic functions

I came across some meromorphic function, call it $f(z)$, which is "quasicrystalline" in the following sense: one can write $f$ as: $$ f(z)=\frac{\sum_i a_i e^{i(q_{i,x}x+q_{i,y}y)}}{\sum_i ...
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74 views

Order of vanishing of an analytic function along its vanishing locus

Suppose $F: {\mathbb C}^n \to {\mathbb C}^k$ is a holomorphic/analytic function with vanishing locus $V_F = F^{-1}(0)$. Can one prove that there are positive real numbers $C>0$ and $\delta>0$ ...
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1answer
338 views

Ahlfors' proof of Bloch's theorem

In his pioneering paper An extension of Schwarz's lemma, Ahlfors proves the lower bound on the Bloch constant $B \geq \frac{\sqrt{3}}{4}$. The proof of this lower bound proceeds as follows: Let $W$ be ...
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112 views

How to show the set of stable polynomials equals to the set of Lorentzian polynomials in degree 2

Give a homogenous polynomial $f\in \mathbb{R}[x_1,\dots,x_n]$ of degree $2$ in $n$ variables, we can consider $f$ as a quadratic form. We call $L_n^2:=$ the set of quadratic forms with nonnegative ...
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3answers
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Putnam 2020 inequality for complex numbers in the unit circle

The following simple-looking inequality for complex numbers in the unit disk generalizes Problem B5 on the Putnam contest 2020: Theorem 1. Let $z_1, z_2, \ldots, z_n$ be $n$ complex numbers such that ...
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1answer
109 views

Existence of a global analytic solution to a linear first order PDE

Let $B=\lbrace \|z\|<1\rbrace$ be a unit ball in $\mathbb{C}^n, n\geq 2.$ Let $f_1,\cdots, f_n, f$ be holomorphic functions on $B.$ Now, consider the following first order, linear PDE: $$f_1\...
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1answer
253 views

Is there a decision procedure for analytic continuation?

Let an analytic element be a power series associated with an open disc of the complex plane over which the series is convergent. W.l.o.g. assume the series is a Taylor expansion about the center of ...
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3answers
245 views

Truncated Perron - logarithm-free error term?

Let $\{a_n\}_{n\geq 1}$, $|a_n|\leq 1$, be such that $F(s) = \sum_{n=1}^\infty a_n n^{-s}$ can be continued analytically to a neighborhood of the line $\Re s = 1$. (For instance, let $a_n = \mu(n)$.) ...
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1answer
140 views

Holomorphic map to Möbius group

$\DeclareMathOperator\PSL{PSL}$Let $U\subset\mathbb C^2$ be an open set, $f:U\to \PSL(2,\mathbb C)$ a holomorphic map. If the image of $f$ is contained in $\operatorname{PSU}(2,\mathbb C)$, I guess ...
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52 views

Question about an exact expression for the root-mean-square of the distances of the critical points to a given zero of a polynomial

Let $p(z) = \prod_{j=1}^{l+1} (z - z_j)^{M_j}$ be a complex polynomial of degree $n$, where the $z_j$ are distinct for $1, \ldots, l+1$. The first $l$ entries in the list $\{z'_1, \ldots, z'_{n-1} \}$ ...
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1answer
124 views

Frequency of large values of the Mertens function

It is known that with $M(x) = \sum_{n\le x}\mu(n)$, there are infinitely many $x$ s.t. $|M(x)|\ge x^{\frac{1}{2} - \varepsilon}$ (see Chapter 15 of Montgomery-Vaughan, for example). Is there any way ...
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1answer
168 views

When is a product of two two-parameter Mittag-Leffler functions a Mittag-Leffler function?

I am studying properties of the two-parameter Mittag-Leffler function. $$ E_{\alpha,\beta}(z)=\sum_{k=0}^\infty \dfrac{z^k}{\Gamma(\alpha k+\beta)}.$$ I am particularly interested in recurrences and ...
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54 views

Arranging the $k$ solutions of $r(z)=te^{i\theta}$ into $k$ continuous functions of $(t,\theta)$

I have originally opened this question on MSE, but I migrated here, since I realized this environment is more suitable. Let $r$ be a rational function, that is, quotient of two coprime polynomials $p,...
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0answers
207 views

Are the two-valued homogeneous harmonic functions classified?

Question. Is there a classification of homogeneous two-valued harmonic functions on $\mathbf{R}^n$, valid in dimensions $n \geq 3$? For reference, multi-valued functions are familiar objects in ...
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1answer
160 views

Reverse residue theorem without using Serre's duality

In V. Talovikova's text "Riemann-Roch Theorem", a key part of the proof of Riemann-Roch theorem is the following proposition (4.6 in the text): Let $\{a_1, \dots,a_n\}$ be a set of points in ...
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1answer
53 views

Existence of continuous family of uniformising parameters

I asked this question on MSE a while ago but didn't receive any useful answers. Suppose I have a $1$-parameter family continuous maps $f_t: \mathbb{S}^2\rightarrow \mathbb{C}P^1$ from a topological $2$...
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1answer
229 views

Boundary behavior of an analytic function

Let $f$ be a function holomorphic in a simply-connected domain $D$; for simplicity, assume that the boundary $\partial D$ of $D$ is piece-wise analytic with positive inner angles. Let $0\in \partial D$...
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1answer
238 views

Lagrange inversion formula in positive characterisic

Does there exist an analog of Lagrange inversion formula in positive characteristic? Obviously, the formula is still valid for coefficient with index not divisible by the characteristic, but for the ...
4
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2answers
226 views

$\log |f|$ is subharmonic

It is known that the logarithm of the modulus of an analytic function $f: D \subset \mathbb C \rightarrow \mathbb C$ ($D$ is a domain) is subharmonic. I have two questions: (1) Are there some weaker ...
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2answers
96 views

Convergence of a sequence of entire functions on an open dense subset

Let $f_n\colon \mathbb{C} \to \mathbb{C}$ be a sequence of entire functions, such that $f_n$ converges to the zero function on an open dense subset $U$ of $\mathbb{C}$ pointwise (or equivalently ...
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0answers
172 views

Understanding a more intricate Schwarz reflection principle--A question about Tetration

everyone. This is going to be a long question as it requires a good amount of back story in theory. This question is mostly along the lines: "I think this should happen, and I think my proof is ...
6
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2answers
743 views

How to treat Puiseux series as functions?

I have been reading about Puiseux series in the context of the Newton–Puiseux algorithm for resolution of singularities of algebraic curves in $\mathbb{C}^2$. Given a curve $f(x,y)=0$ with $f$ a ...
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1answer
183 views

Smooth functions with vanishing normal derivatives

Let $B_1 := \{z ∈ C : |z| \le 1\}$, and let $C_0(B_1,\mathbb C)$ be the space of continuous complex-valued functions on $B_1$ equipped with the uniform convergence topology. How to show that, the set ...
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1answer
139 views

Efficients method for finding a zero of a multilinear complex polynomial in an specified region

Let P be a given multilinear polynomial in $\mathbb{C}[z_1,\dots,z_n]$ and $D\subset \mathbb{C}$ be a given disc in the complex plane. Does there exist an efficient method for checking that $P$ has a ...
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0answers
107 views

Ratio condition on polynomials

Consider the set $$\mathscr{P}_n := \{ p \in \mathbb{R}[x] \mid p(A)\ge 0,~\forall A\in \mathsf{M}_n(\mathbb{R}),~A\ge 0 \}.$$ Recently, I and an undergraduate student showed that $p \in \mathscr{P}_2$...
3
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1answer
168 views

Short proof of the error bound in PNT assuming a zero-free strip?

I am looking for a short proof of the fact that $\zeta(z)\neq 0$ for $\Re z>a$ implies the prime number theorem with an error bound $O(x^{a+\varepsilon})$ for any $\varepsilon>0$, which would be ...
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0answers
32 views

Uniformization of triangulation on a sphere up to Moebius transformations

This is not the most precise question but rather a hope that someone has seen something like this. I am given a triangulation of the 2-sphere $S^2$ which I only know up to Moebius transformations. I ...
10
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269 views

the (non-existent) group of conformal transformations

In physics intros to 2d conformal field theory, people often talk about the "group of conformal transformations". Of course, that's not a group but rather a pseudo-group... that's not what ...

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