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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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Is there a term for a countour integral that disregards direction?

Is there a name for integration of the form $\oint_\gamma f(z) |dz|$? In other words, the integral that only takes into account the length of the contour and the values of the function but not the ...
Anixx's user avatar
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1 vote
1 answer
115 views

Deriving a specific bound for functions in Hardy Space

Reading some article a while ago I read the following: (here $H^2$ represents the Hardy space) Let $f\in H^2$ be such that $f(0)=1$, and let $0<\lvert\lambda\rvert<1$, then $$\lVert f(\lambda z)\...
Tomas smith Smith's user avatar
5 votes
0 answers
163 views

Partial fraction expansions of meromorphic functions

Sorry if this question (inspired by the recent flurry of activity around a "new" formula for $\pi$) is too naive. Imitating what one does with Hadamard products, one can try to do the same ...
Henri Cohen's user avatar
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7 votes
2 answers
359 views

Systematic way to compute $\sum_{n=1}^\infty P(n) / Q(n)$ for polynomials $P$ and $Q$

This may be well known so feel free to downvote. When $P = 1$ and $Q(x) = x^k$, this is of course the Riemann zeta function. But what about other cases? For instance is it always possible to express $\...
John Jiang's user avatar
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1 vote
0 answers
328 views

Recognizing when a $2\pi$-periodic function is a shifted sine

Consider a real analytic function $f(x)$ with a period of $2\pi.$ Let $t\in \operatorname{Range}(f(x))$, $$I(t)=\int_{0}^{2\pi}\log |t-f(x)|dx.$$ Prove that $I(t)$ is equal to a constant if and only ...
kris001's user avatar
  • 21
27 votes
1 answer
3k views

Proof of "Possible new series for $\pi$" without use of physics

Related post: The post Possible new series for $\pi$ is about whether the identity is new, so to avoid confusion I was advised to ask this question separately. I am looking for a proof of the ...
TheSimpliFire's user avatar
1 vote
0 answers
164 views

How to "eliminate" the log pole of a logarithmic $(p,q)$-form?

Let $X$ be a compact complex manifold, and $D=\sum_{i=1}^{r} D_i$ be a simple normal crossing divisor on $X$. Let $\alpha$ be a logarithmic $(p,q)$-form, namely, on an open subset $U$, we can write $$\...
Invariance's user avatar
2 votes
1 answer
121 views

Branched covering maps between Riemann surfaces

What is an example of a branched covering map between Riemann surfaces of infinite degree? i.e. something like a branched version of the exponential map $exp: \mathbb{C} \to \mathbb{C}^*$. Thanks!
cata's user avatar
  • 357
3 votes
1 answer
91 views

Characterization of bi-Hermitian structures with equal Lee forms

Let $(M,g,I_+,I_-)$ be a compact bi-Hermitian manifold, where $g$ is a Riemannian metric and $I_+$, $I_-$ are two complex structures that are both compatible with $g$. We assume that $I_+$ and $I_-$ ...
user avatar
0 votes
0 answers
29 views

Concerning the definition of a class of functions introduced by Nilsson

In the paper 'Some growth and ramification properties of certain integrals on algebraic manifolds' by Nilsson we find the following definitions: My question is how does one prove the remark "It ...
ResearchMath's user avatar
3 votes
1 answer
224 views

Prescribe the type of an entire function which inverse zeros are summable

According to Lindelöf's theorem, given points $z_i\in \mathbb C\setminus \{0\}$ ordered by increasing modulus with possible repetitions, we can define a function $$ f(z)=\prod_{n=1}^\infty (1-z/z_n)e^{...
kaleidoscop's user avatar
  • 1,332
14 votes
1 answer
1k views

The location of the zeros of the "new" function $\Psi(s)=\sum _{n=1}^{\infty }{\frac{1}{n!^s}}$

Defining the Psi function as $$\Psi(s)=\sum _{n=1}^{\infty }{\frac{1}{n!^s}}$$ and by studying, from a numerical point of view, the location of the zeros in the complex plane up to $0<\Im(s)<...
Roberto Trocchi's user avatar
0 votes
0 answers
126 views

Function of several complex variables with prescribed zeros

I accidentally stumbled upon a problem of complex analysis in several variables, and I have a hard time understanding what I read, it might be related to the Cousin II problem but I cannot say for ...
kaleidoscop's user avatar
  • 1,332
5 votes
0 answers
239 views

Does the Poincaré lemma (Dolbeault–Grothendieck lemma) still hold on singular complex space?

Let $X$ be a complex manifold, then we have the Poincaré lemma (or say, Dolbeault-Grothendieck lemma) (locally) on $X$, whose formulation is as follows: ( $\bar{\partial}$-Poincaré lemma) If $\...
Lelong  Wang's user avatar
2 votes
0 answers
61 views

Entire functions and Bergman spaces

Given an open set $D \subset \mathbb{C}$ with compact closure, let us consider the Bergman space $A^2(D)$ of all holomorphic functions on $D$ that are square integrable on $D$ with respect to the ...
alvarezpaiva's user avatar
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7 votes
0 answers
145 views

Example of closed non-exact torsion differential form on variety

I asked this question some time ago on MSE and received close to no interest. I feel it is appropriate for this site: I am interested in finding a particular example. I would like to find a variety (...
Thomas Kurbach's user avatar
10 votes
3 answers
758 views

The smallest nontrivial zero of the Riemann zeta function

Consider the Riemann zeta function $$\zeta(s)=\dfrac{1}{1-2^{1-s}}\sum_{n=1}^\infty\frac{(-1)^{n-1}}{n^s},\operatorname {R}(s)>0.$$ Riemann suggested that all nontrivial zeroes lie on the line $\...
わくわく's user avatar
3 votes
0 answers
212 views

Schwartz's theorem without English language reference

I'm reading the paper "Spectral Synthesis And The Pompeiu Problem" by Leon Brown, Bertram M. Schreiber and B. Alan Taylor, Annales de l’Institut Fourier 23, No. 3, 125-154 (1973), MR352492, ...
Holden Lyu's user avatar
0 votes
1 answer
136 views

Generalisation of Paley–Wiener type results for unbounded sets

Do you know an unbounded open set $A\subset \mathbb{R}^d$, $d\geq 2$ with the following property: if some integrable function $f$ on $\mathbb{R}^d$ has its Fourier transform vanishing on $A^c$ and all ...
kaleidoscop's user avatar
  • 1,332
0 votes
0 answers
33 views

Contraction of an inclusion with respect to Kobayshi hyperbolic metric

Suppose that $X = \mathbb{C}^n - \Delta_X$ and $Y = \mathbb{C}^n - \Delta_Y$, where $\Delta_X$ and $\Delta_Y$ are unions of hyperplanes in $\mathbb{C}^n$ such that $\Delta_Y \subset \Delta_X$, $\...
A B's user avatar
  • 41
0 votes
0 answers
183 views

Behavior of subtree of $\mathbb{Z}^2$ embedded in $\mathbb{C}$ under compactification of the latter to the riemann sphere

I consider a countable subtree $T$ of the integer lattice isomorphic to $\mathbb{Z}^2$ with directed edges. It shall be embedded in $\mathbb{C}$ where the edge $(u,v)$ points from $u$ to $v$ if and ...
Jens Fischer's user avatar
0 votes
0 answers
42 views

Function samplable from the past and Hardy spaces

What I am ultimately looking for is a $L^2$ function $f$ on the real line that can be sampled from the past, i.e. for each $x<0$ there are $L^2$ coefficients $c_n(x)$, $n\in \mathbb{N}$ such that, ...
kaleidoscop's user avatar
  • 1,332
1 vote
1 answer
150 views

Motivation for defining polar derivative

The polar derivative of a polynomial $p(z)$ is defined as $$ D_\alpha p(z):=np(z)+(\alpha -z)p^{\prime}(z) \qquad \alpha\in\mathbb{C}. $$ It is a polynomial of degree $n-1.$ I am new to complex ...
AgnostMystic's user avatar
4 votes
0 answers
68 views

Stability of analytic continuation

Let $(Q,\| \cdot \|)$ be a certain Banach space of entire functions, say certain functions of finite order that satisfy a growth condition of the form $|f(z)| \leq c e^{a|z|^\gamma}$ for some $c,a,\...
Mike Wiedemann's user avatar
0 votes
0 answers
50 views

Convergence of Liouville correlation functions

A key object in Liouville conformal field theory is the random Liouville measure $M$ defined heuristically as $M(d^2x) = :e^{2bX(x)}: d^2x$, where $X$ is a Gaussian free field and $:e^{2bX}:$ denotes ...
user avatar
-1 votes
1 answer
114 views

On $\Re\frac{\zeta'}{\zeta}(s) \geq -A \log(|t|+4)$ for some $A>0$

On page 438 of "Multiplicative Number Theory I" by Montgomery-Vaughan one finds the following statement and its verification : Assuming RH, there exists an absolute constant $A>0$ such ...
Alexis Σ's user avatar
0 votes
1 answer
206 views

Sign of $\Re\frac{\xi'(s)}{\xi(s)}$ locally around a zeta zero

In the article ”On a positivity property of the real part of logarithmic derivative of the Riemann $\xi$–function” the authors check the positivity of $\Re \frac{\xi'}{\xi}(s)$ for $\frac{1}{2}<\...
Alexis Σ's user avatar
-5 votes
1 answer
126 views

Is the real part of the Eta function bounded by $2 \sum_{n=1}^{\infty}{\dfrac{(-1)^{n-1}}{n^{\alpha}}} $ [closed]

Consider the series defined by \begin{equation} f(\alpha,\beta) := \sum_{n=1}^{\infty}{\dfrac{(-1)^{n-1}}{n^{\alpha}}\cos(\beta\ln(n))} \end{equation} is it true that $$f(\alpha,\beta) \le 2\sum_{n=1}...
The potato eater's user avatar
1 vote
0 answers
56 views

Extension of meromorphic distribution

Let $W$ be a topological vector space (e.g. Frechet) with a dense subspace $V$. Let $D_s$ be a distribution on $V$ that is meromorphic in $s\in\mathbb C$ and extends continuously to $W$ with respect ...
Tian An's user avatar
  • 3,759
1 vote
0 answers
127 views

Fundamental set for families of abelian varieties

I'm considering the universal family of principally polarized g-dimensional abelian varieties with level N-structure. Let's briefly recall the usual construction as a quotient of $\mathbb{C}^g \times \...
user494203's user avatar
0 votes
1 answer
181 views

Estimates for the first coefficient of the Taylor expansion of $\zeta$ around a zero

Let's suppose that $s_0=\frac{1}{2}-\Delta+it$ with $0<\Delta<\frac{1}{2}$ is a simple zeta zero (i.e a zero not on the critical line). Then $1-\overline{s_0}$ is also a zero. If we take the ...
Alexis Σ's user avatar
-1 votes
1 answer
110 views

Riemann xi function strictly increasing along a half-plane

Matiyasevich, Saidak, Zvengrowsk proved the following result: Let $σ_0$ be greater than or equal to the real part of any zero of ξ. Then $|ξ(s)|$ is strictly increasing in the half-plane $σ > σ_0$. ...
Alexis Σ's user avatar
11 votes
1 answer
944 views

Can the topologist's sine curve be realized as a Julia set?

Does there exist a rational function $f\in\Bbb{C}(z)$ whose Julia set coincides with $$ T:=\left\{\left(x,\sin\left(\frac{1}{x}\right)\right)\,\Big|\,x\in\left(0,\frac{1}{\pi}\right]\right\}\cup\big(\{...
KhashF's user avatar
  • 3,554
2 votes
0 answers
237 views

Least number of circles required to cover a continuous function on $[a,b]$

I asked this question on MSE here. Given a continuous function $f :[a,b]\to\mathbb{R}$, what is the least number of closed circles with fixed radius $r$ required to cover the graph of $f$? It is ...
pie's user avatar
  • 473
3 votes
1 answer
109 views

When entire or meromorphic map of finite type restricts to a Galois covering?

Suppose that $f \colon \mathbb{C} \to \mathbb{C}$ is an entire map of finite type, i.e., with finitely many singular values. Then we can consider the restriction $f| \mathbb{C} \setminus f^{-1}(S_f) \...
A B's user avatar
  • 41
1 vote
0 answers
101 views

Poles/Residues of the Gamma function under action of Mobius transform $\Gamma(A(z))$

I am not sure whether this is rather an MO or MSE question but it results from my research, so I put it here. In my effort to find (or to disprove the existence of) $k,l,h\in\mathbb{N}$ such that $2^{...
Jens Fischer's user avatar
3 votes
1 answer
458 views

Bounds for Dirichlet L-functions

Let $L$ denote a Dirichlet L-function attached to the primitive character $\chi$. What are the best known bounds for $L(\sigma+it, \chi)$? PS: For $L=\zeta$ and $0\leq\sigma\leq 1$, i'm aware of a ...
Q_p's user avatar
  • 895
5 votes
1 answer
156 views

Is a $2$-form which is "almost" Kähler cohomologous to a Kähler form?

Let $X$ be a compact complex manifold. Suppose it has a closed real $2$-form $\omega$ such that $\omega$ is cohomologous to a $(1,1)$-form (but not necessarily of type $(1,1)$ itself); $\omega(v, Iv) ...
cll's user avatar
  • 2,305
1 vote
1 answer
135 views

complex optimization in the plane

I am trying to get the following condition: $\forall\,\alpha,\beta,\gamma\in\mathbb{C}\,\,\exists\,u,v\in\mathbb{T}\colon\quad|\alpha^2uv-(\beta u-\gamma v)^2|-|\alpha^2+4\beta\gamma|=|\beta u+\gamma ...
Krzysztof's user avatar
  • 361
2 votes
1 answer
129 views

Is there a scalar product which makes orthonormal the family of complex functions $ (f_n)_{ n \geq 1 } $?

Let $ (f_n)_{ n \geq 1 } $ be a family of complex functions defined as follow, $ \forall n \geq 1 $, $$ f_n (z) = \dfrac{1}{n^{z}} $$ I would like to ask you if it is possible to construct a ( non-...
Angel65's user avatar
  • 595
2 votes
0 answers
95 views

Construction of an analytic function whose Fourier transformation has compact support [closed]

Is there a non-constant real analytic function $f$ on $\mathbb{R^2}$ satisfying the following properties? $f$ vanishes on $x$-axis and $y$-axis; the Fourier transformation $\hat{f}$ of $f$ has a ...
adobereader's user avatar
2 votes
1 answer
177 views

Complex analytic function $f$ on $\mathbb{C}^n$ vanish on real sphere must vanish on complex sphere

I'm considering a complex entire function $f$: $\mathbb{C}^n\to \mathbb{C}$. Suppose $f=0$ on $\{(x_1,\cdots,x_n)\in\mathbb{R}^n:\sum_{k=1}^n x_k^2=1\}$. I want to prove $$f=0\textit{ on } M_1=\{(z_1,\...
Holden Lyu's user avatar
3 votes
1 answer
198 views

Subset of a complex manifold whose intersection with every holomorphic curve is analytic

The Osgood's lemma in complex analysis says that if $f \colon \mathbb{C}^n \to \mathbb{C}$ a continuous function such that $f|_{L}$ is holomorphic for every complex line $L \subset \mathbb{C}^n$, ...
 V. Rogov's user avatar
  • 1,160
16 votes
0 answers
502 views

Gabriel's theorem for complex analytic spaces

Let $X,Y$ be noetherian schemes over $\mathbb{C}$. Then, it is known that $$ \text{Coh}(X) \simeq \text{Coh}(Y) \Rightarrow X \simeq Y, $$ by P. Gabriel(1962). Are there some results in the case of ...
Walterfield's user avatar
4 votes
0 answers
218 views

Holomorphic non vanishing modular form

Let $\mathcal{O}(\mathcal{H})^\times$ be the multiplicativee group of holomorphic functions on the Poincaré half-plane $\mathcal{H}$ that do not vanish there. Let $j(g,z)=(cz+d)$ and $gz=(az+b)/(cz+d)$...
Emmanuel Royer's user avatar
0 votes
1 answer
166 views

If there are infinite poles inside an infinite integral contour, and the sum of residues of these poles converges. Can the residue theorem hold?

The residue theorem is commonly used to calculate situations where the number of isolated singularities is limited. However, I am very curious whether the residue theorem can be extended to cases with ...
adios518's user avatar
3 votes
1 answer
626 views

Bounds for some exponential type sum

Let $m \in \mathbb{N}$. Given that $$\sum_{\frac{T}{\log^{2} T} < n < T} n^{iT} \ll T^{1/2}\log^{2} T,$$ is it true that $$\sum_{\frac{T}{\log^{2} T} < n < T, n \neq m} \frac{n^{iT}}{m^{1+\...
Reviewer 2.'s user avatar
0 votes
0 answers
55 views

Validation of complex mapping area calculation

I want to know whether the following approach and computations are correct for calculating the area of image of a polynomial under a polynomial map. Here are my thoughts : I want to to estimate the ...
AgnostMystic's user avatar
7 votes
2 answers
382 views

Perturbation of zeros of an entire function of exponential type

Suppose that $(z_n) \subset \mathbb C$ is a sequence (repetitions allowed) such that $$ F(z) = \prod_n \left ( 1-\frac{z}{z_n} \right ) $$ defines an entire function of exponential type, that is, $|F(...
J. Swail's user avatar
  • 437
2 votes
0 answers
138 views

Does the complex interpolation space $(L^1(\mathbb{R}),W^{2,1}(\mathbb{R}))_{\frac{1}{2}}$ continuously embed into $L^\infty(\mathbb{R})$?

The complex interpolation space between $(L^p(\mathbb{R}),W^{2,p}(\mathbb{R}))_\theta$ with interpolation parameter $\theta=\frac{1}{2}$ is known to be $W^{1,p}(\mathbb{R})$ for $1<p<\infty$. As ...
vmist's user avatar
  • 939

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