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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Contour integral with two essential singularity

I'm solving problems on the Gamma random variables and there is this question where it wants me to calculate the Mellin transform of sum of two independent Gamma variables from their moment generating ...
K.K.McDonald's user avatar
5 votes
1 answer
318 views

Boundary zeros of a holomorphic function $f: \Omega \to \Bbb C$

My question stems from the following result about holomorphic functions on the unit disc: "A function, continuous on the closed unit disc, holomorphic inside, and vanishing on an open subset of ...
stoic-santiago's user avatar
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210 views

Partitions, weights and polynomials with roots on the unit circle

Let us consider the set $[n]=\{1,\ldots,n\}$ and all of its partitions into exactly $m$ blocks, but let us allow each block to be internally ordered. For example, taking $n=6$ and $m=2$, we will ...
Luis Ferroni's user avatar
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7 votes
1 answer
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Help finding an analytic continuation

I am looking for the analytic continuation of \begin{align*} & f_m(v,w) := \sum\limits_{k,l=0}^\infty v^k w^l {k+l+m \choose k} {k+l+m \choose l} \ , \end{align*} where $m \in \{1,2,...\}$ is ...
Tardis's user avatar
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2 votes
1 answer
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The sum of $q^{-2}$ over nonzero Gaussian integers

I'm reading about the Weierstrass zeta function. In this context, $\phi(z)=\zeta(z)-\pi\bar{z}$ is periodic over the lattice $$\mathcal{L}=\{a+bi\mid a,b\in\mathbb{Z}\}.$$ If we take $w\in\mathcal{L}\...
isz's user avatar
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114 views

Matrix product of entire functions

Suppose I have two $d \times d$ entire matrix functions $F, G$ defined on $\mathbb{C}$ with the the property that $\|FG^*\|_{L^\infty(\mathbb{C})} < \infty$. Can anything be said about $F$ and $G$, ...
Joshua Isralowitz's user avatar
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Resolvent estimates for Laplacians on directed graphs

Let us consider a directed and weighted graph $G$ with $N = |G|$ nodes. Denote the corresponding (weighted) adjacency matrix by $W \in \mathbb{R}^{N\times N}_{\geq 0}$ and let $D$ be the diagonal in-...
Qualearn's user avatar
2 votes
1 answer
183 views

Local equality of functions implies global equality?

The following question arised in my research, and I was unable to settle it after playing with it for sometime. Let $\{a^k_i\}_{i\geq 1}$ (for $k\in \{1,2,3,4\}$) be four sequences of real numbers. ...
Amr's user avatar
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95 views

Asking a reference about the $p$-Laplacian of $|\nabla u|^p$

It is well-known that for a harmonic function $u$, i.e. $$ \Delta u=0, $$ the quantity $|\nabla u|^2$ is subharmonic, i.e. $$\Delta (|\nabla u|^2) \geq 0. $$ Reason: $$\Delta (|\nabla u|^2)= 2 \nabla (...
Hheepp's user avatar
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7 votes
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Plane curve with continuously increasing Hausdorff dimension

In a recent paper, we required the following fact. Proposition 1. There exists a simple closed curve $\gamma\subset\mathbb{C}$ with the following property. If $\phi$ is a biholomorphic map, defined on ...
Lasse Rempe's user avatar
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9 votes
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Proving the Replica Trick works

The replica trick attempts to calculate the expectation of the logarithm $X=\log(Z)$ of a random variable $Z$. The wikipedia article describes the logarithm as the limit $$ \log(Z) = \lim_{n\to 0}\...
Felix B.'s user avatar
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4 votes
1 answer
177 views

Is there a complete Kahler metric on a bounded domain?

Let $\Omega\subset\mathbb{C}^n$ be a bounded domain. By the theorems of Cheng-Yau and Mok-Yau, $\Omega$ admits a complete Kahler-Einstein metric if and only if $\Omega$ is a pseudoconvex domain. My ...
yuan's user avatar
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2 votes
2 answers
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Hardy space inclusion in the right-half plane

I'm looking for an example of a function $u \in H_2$ such that $u \notin H_\infty$, where $H_p$ is the Hardy space on the right-half plane. Since this notation is perhaps not standard, here is a ...
Laurent Lessard's user avatar
2 votes
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63 views

Differentiable functions on analytic varieties

Let $\iota\colon X\to \Omega\subseteq \mathbb{C}^n$ be a complex analytic variety $X$ in an open subset $\Omega$ of $\mathbb{C}^n$. If $N$ is a smooth manifold and $h\colon M\to X$ is a continuous map,...
Thomas Kurbach's user avatar
23 votes
1 answer
1k views

Can we just use the linear term of exponential sums to sum divergent series

Suppose you want to compute the sum $\sum_{n=0}^{\infty} a_n $ You could consider the expression $f(x) = \sum_{n=0}^{\infty} e^{a_n x}$ and try to compute the coefficient of an $x^1$ term in the ...
Sidharth Ghoshal's user avatar
28 votes
2 answers
2k views

Contractibility of the space of Jordan curves

Is the space of Jordan curves in $\textbf{R}^2$ contractible? In other words, is there a canonical or continuous way to deform each Jordan curve to the unit circle $\textbf{S}^1$. If the curves are ...
Mohammad Ghomi's user avatar
5 votes
1 answer
344 views

Is there a meaningful interpretation of an $L^i$-space?

Do complex-normed spaces exist? Is there an extension of $p$-norms to $p\in\Bbb C\setminus\Bbb R$? A while ago I thought of extending $L^p$-spaces to the complex-normed setting. After some discussions,...
TheSimpliFire's user avatar
1 vote
1 answer
140 views

Bounds for the logarithmic derivative in the Selberg Class

Let $F \in \mathcal{S}$, where $\mathcal{S}$ is the set of $L-$functions in the Selberg Class. Are there established upper and lower bounds for $$\left|\frac{F^{'}(s)}{F(s)}\right|,$$ where $s = \...
Tokita Ohma's user avatar
2 votes
0 answers
212 views

Zeta function associated with a function $f$

Let the function $f(t) = \cos(at)$, where ($0 < a < 1$). Let us define $$\zeta(z, f) = \frac{1}{\Gamma(z)} \int_0^{+\infty} \frac{t^{z-1}\cos(at)}{e^t-1}\, dt. $$ Is there a general formula that ...
L.L's user avatar
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1 vote
2 answers
135 views

Location of the negative real roots of certain integer-valued polynomials

The following question on polynomials arose as a potentially helpful intermediate step on a proof of a Theorem that I want to demonstrate. Its statement is quite elementary, and I can think of a ...
Luis Ferroni's user avatar
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Vanishing components of Kähler metric

Let $(X, \omega) $ be a $n$-dimensional complex Kähler manifold such that $\omega^{n-1}=d\alpha $. Does $\partial\alpha^{n-1,n-2} =0$ (resp. $\bar\partial\alpha^{n-2,n-1} =0$) Where $\alpha^{n-1,n-2}$ ...
Samir's user avatar
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249 views

Order of growth of $\left|\frac{1}{\zeta’(\rho)}\right|$ as $\Im(\rho)\rightarrow\infty$?

Let $\zeta$ denote the Riemann zeta function, and let $\rho\in\mathbb{C}$ be a variable that takes its values among the zeros of the zeta function, so that $\zeta(\rho)=0$, and write $\rho=\sigma+it$. ...
EGME's user avatar
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1 vote
0 answers
139 views

Top cohomology of the canonical class of a compact non-Kähler manifold

Let $X$ be a complex compact manifold of complex dimension $n$. Let $K_X$ denote its canonical class. Is it true that the cohomology group $$H^n(X,K_X)$$ is one dimensional? Remark. If $X$ is Kähler ...
asv's user avatar
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14 votes
1 answer
656 views

Sums of non-surjective entire functions

Suppose Suppose $A$ and $B$ are two entire, non-surjective, functions. This means $$ A(z)=e^{f(z)}+c_1 $$ and $$ B(z)=e^{g(z)}+c_2 $$ for some entire functions $f$ and $g$, and some complex ...
Markus's user avatar
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0 votes
0 answers
137 views

A mistake on a Barnes beta like integral

Consider the Barnes beta like integral: Where Re$(a)$ ,Re$(b)$ is greater than $0$ and $c$ is not a negative integer, where $|z|<1$ and $|$arg$(-z)$|$<\pi$, $$\frac{1}{2\pi i}\int_{-i\infty}^{i\...
Dqrksun's user avatar
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3 answers
433 views

Is the hypergeometric function ${}_1F_2(1;a,a+\frac12;-x^2)$ an elementary function? How about its positivity, monotonicity, and convexity in $x$?

Is the generalized hypergeometric function ${}_1F_2\bigl(1;a,a+\frac12;-x^2\bigr)$ for $a>-1$ and $x>0$ an elementary function? How about the positivity, monotonicity, and convexity of the ...
qifeng618's user avatar
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2 votes
1 answer
199 views

An integral transform computation

In Erdelyi, Tables of Integral Transforms, p. 344 Section 7.2. they note that \begin{align} \frac{1}{2 \pi i} \int_{c-i\infty}^{c+i\infty} s^{\nu} e^{\alpha s^2} x^{-s} \, ds = 2^{-\nu/2} \pi^{-...
user506603's user avatar
4 votes
1 answer
324 views

Riemann mapping theorem with smooth boundary

This is closely related to the question here. The setup is that $U\subset\mathbb{C}$ is an open bounded simply connected domain with $C^\infty$ boundary. If $\phi:U\rightarrow\mathbb{D}$ is a ...
J_P's user avatar
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0 answers
37 views

lower bound on an ellipse

Is there anyway to estimate the lower bound for $Re(zP'(z)/P(z))=\sum_{k=1}^n\frac{z}{z-z_k}$ on an ellipse where $P(z)=\prod_{k=1}^n(z-z_k)$ with $z_k, 1\leq k\leq n$ lying entirely within the ...
user159888's user avatar
7 votes
1 answer
285 views

When $H^{p,q}_{\bar\partial}(X)$ can be seen as a subspace of $H^k(X,\mathbb C)$?

It is known that for a $\partial\bar\partial$-manifold $X$ (a compact complex manifold satisfies the $\partial\bar\partial$-lemma), the Bott-Chern cohomology $H_{BC}^{\bullet,\bullet}:=\frac{\ker \...
Tom's user avatar
  • 341
2 votes
1 answer
150 views

Bounding minimal absolute value of a point on a complex algebraic variety

Given a system of complex polynomial equations in $n$ variables, giving rise to an affine variety $V \subseteq \mathbb C^n$, is there a bound $b \in \mathbb R$ such that if $V(\mathbb C) \neq \...
Jonathan Kirby's user avatar
4 votes
0 answers
57 views

Is there a dense set of Lipschitz functions in $H^\infty(U)$, each of which maps $(1,0,\ldots,0)$ to 1, where $U$ is the unit ball in $\mathbb{C}^N$?

Let $U$ be the open unit ball in $\mathbb{C}^N$, let $A(U)$ be the algebra of functions analytic on $U$ and continuous on $\bar U$, and let $u=(1,0,\ldots,0)$. Let $\mathcal{B}=\{f\in H^\infty (U): \|...
David Walmsley's user avatar
2 votes
0 answers
163 views

Can the equation $1+z^p+z^q+z^r=z^n$ have multiple complex roots $z$?

The math overflow post asks whether the equation $1+z^p+z^q=z^n$ can have multiple complex roots where $p<q<n$ (On the irreducibility of certain trinomials and quadrinomials). Q. Let us ...
ABB's user avatar
  • 3,898
1 vote
1 answer
101 views

Dense orbits for a rational map

Given a complex rational function $f$ and $z\in \mathbb C$, let $O^+(z)=\{f^n(z):n\geq 1\}$. Let $$D(f)=\big\{z\in \mathbb C:\overline{O^+(z)}=J(f)\big\}.$$ So $D(f)$ is the set of points whose (...
D.S. Lipham's user avatar
  • 3,045
0 votes
1 answer
144 views

Residue calculation for Eulerian expansion of the cotangent

I am looking for ideas on proving the Eulerian expansion of the cotangent using residue calculation: $$\pi\cot(\pi z)=\frac{1}{z}+\sum_{n=1}^{\infty}\left(\frac{1}{z+n}+\frac{1}{z-n}\right), \ z\in\...
L.L's user avatar
  • 399
4 votes
1 answer
124 views

Sum of holomorphic squares?

Consider a variable $z \in \mathbb{R}^n$ and assume $u(z) \in \mathbb{R}^m$ and $H(z) \in \mathbb{R}^{m \times m}$. Further assume that $H(z)$ is symmetric positive definite for every $z$. Consider ...
Sébastien Loisel's user avatar
2 votes
1 answer
157 views

Is this a valid method of extending convergence of the Maclaurin series for $\frac{x}{x+1}$ from $|x|<1$ to $\Re(x)>-1$?

I originally asked this question on Math StackExchange a few months ago and no answers or even comments have yet been posted, so I'm asking this question again here on Math OverFlow. This Math ...
Steven Clark's user avatar
  • 1,061
5 votes
1 answer
106 views

Jordan curve boundaries of Fatou components

Let $f:\mathbb C\to \mathbb C$ be a rational map and let $J(f)$ and $F(f)$ denote the Julia and Fatou sets of $f$, respectively. Let $\mathcal S$ be the set of all boundaries of Fatou components. ...
D.S. Lipham's user avatar
  • 3,045
3 votes
1 answer
186 views

Can the equation $1+z+z^q=z^n$ have multiple complex roots $z$?

It is proved here that the equation $1+z+z^2=z^n$ have no multiple complex roots. Q. Let us consider the equation $1+z+z^q=z^n$ where $q$ and $n$ are natural numbers with $1<q<n$. Any ...
ABB's user avatar
  • 3,898
12 votes
3 answers
750 views

Can the equation $1+z+z^2=z^n$ for natural $n$ have multiple complex roots $z$?

The question is stated in the title of this post. It is easy to see that, if $z$ is a multiple root of $p_n(z):=1+z+z^2-z^n$, then $(n-2)z^2+(n-1)z+n=0$, so that we can successively express $z^2,\dots,...
Iosif Pinelis's user avatar
3 votes
0 answers
289 views

Demailly regularisation on singular complex spaces

Let $X$ be a compact (Hausdorff reduced) complex space. It is asserted (and used in an essential way) in a famous paper by Demailly and Păun ("Numerical characterization of the Kähler cone of a ...
Mingchen Xia's user avatar
2 votes
0 answers
82 views

Finding a branch cut or a branch point [closed]

Is there a way to find a branch cut or a branch point, through which a curve over a complex function goes, or in general in some region of complex function, say $\ln(f(z))$, using numerical methods or ...
roignoirewg's user avatar
1 vote
0 answers
109 views

Existence of meromorphic one-form with a fixed order pole

Let $X$ be a compact Riemann surface of genus $g$. We identify it with $4g$ polygon $\{a_i,b_i, a_i^\prime, b_i^\prime\}_{i=1}^g$. For a meromorphic 1 form $\omega$, we define $A_i(\omega)= \int_{...
zapkm's user avatar
  • 541
1 vote
0 answers
120 views

Asymptotic location of zeros of of a sequence of analytic functions

Assume we have a sequence of functions $f_n$ analytic in a bounded domain $\Omega \subset \{ |z|\ge 1 \}$ of the complex plane, such that the sequence $$ g_n(z) = f_n(z) - z^n $$ converges to an ...
Andrei MF's user avatar
  • 692
6 votes
1 answer
287 views

Log-convexity of determinant

Let $f(z):=\langle g(z),g(z)\rangle,$ where $z \mapsto g(z)$ is holomorphic and $\langle \bullet,\bullet\rangle$ is an inner-product on some function space, such as $L^2$, such that $\langle g(z),g(z)\...
António Borges Santos's user avatar
0 votes
0 answers
50 views

Integrability of logarithm of a function in Hardy class

Assume that f is in $H^1$ (The Hardy space of holomorphic functions in the unit disk). I need an easy reference to the following fact $ \log |f|$ is in $h^1$, i.e. it belongs to harmonic Hardy space.
user67184's user avatar
0 votes
0 answers
139 views

Is there a Cauchy integral formula for complex manifolds?

Is there a Cauchy integral formula for holomorphic functions on complex manifolds?
Mathgrad's user avatar
  • 293
10 votes
0 answers
186 views

Local cohomology and residues of rational functions at 0 and $\infty$

Let $a_1,\dots,a_s$ and $b_1,\dots,b_t$ be positive integers, where $s,t>0$. Choose $c\in\mathbb{Z}$. Let $M_c$ be the real vector space spanned by all monomials $x^\alpha y^\beta=x_1^{\alpha_1}\...
Richard Stanley's user avatar
0 votes
0 answers
90 views

Sum power series not continuous unit circle

This is (probably) not a research question and I already asked it on StackExchange but I got no answer over there. Let us consider the sequence $(a_n)_{\geq 1} = \left(\frac{\cos(2\sqrt{2n}+\frac{\pi}{...
Libli's user avatar
  • 7,210
4 votes
1 answer
179 views

Finding a real-analytic diffeomorphism

Let $U_1\subset \mathbb R^3$ be a simply connected bounded open set with a smooth boundary and let $U_2$ be a neighborhood of $U_1$. Does there exist a real-analytic diffeomorphism $\psi: U_2 \to W_2$ ...
Ali's user avatar
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