# Questions tagged [cv.complex-variables]

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

3,208
questions

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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 ...

1
vote

1
answer

115
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### 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)\...

5
votes

0
answers

163
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### 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 ...

7
votes

2
answers

359
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### 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 $\...

1
vote

0
answers

328
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### 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 ...

27
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1
answer

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### 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 ...

1
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0
answers

164
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### 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
$$\...

2
votes

1
answer

121
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### 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!

3
votes

1
answer

91
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### 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 $I_-$ ...

0
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0
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### 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 ...

3
votes

1
answer

224
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### 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^{...

14
votes

1
answer

1k
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### 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)<...

0
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0
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126
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### 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 ...

5
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0
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239
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### 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 $\...

2
votes

0
answers

61
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### 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 ...

7
votes

0
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145
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### 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 (...

10
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3
answers

758
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### 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 $\...

3
votes

0
answers

212
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### 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, ...

0
votes

1
answer

136
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### 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 ...

0
votes

0
answers

33
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### 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 $\Delta_Y \subset \Delta_X$, $\...

0
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0
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183
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### 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 ...

0
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0
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42
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### 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, ...

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 ...

4
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0
answers

68
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### 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,\...

0
votes

0
answers

50
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### 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 ...

-1
votes

1
answer

114
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### 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 ...

0
votes

1
answer

206
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### 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}<\...

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votes

1
answer

126
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### 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}...

1
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0
answers

56
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### 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 ...

1
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0
answers

127
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### 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 \...

0
votes

1
answer

181
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### 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 ...

-1
votes

1
answer

110
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### 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 $σ > σ_0$.
...

11
votes

1
answer

944
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### 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(\{...

2
votes

0
answers

237
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### 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 ...

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) \...

1
vote

0
answers

101
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### 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^{...

3
votes

1
answer

458
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### 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 ...

5
votes

1
answer

156
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### 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) ...

1
vote

1
answer

135
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### 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 ...

2
votes

1
answer

129
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### 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-...

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 ...

2
votes

1
answer

177
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### 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,\...

3
votes

1
answer

198
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### 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 $L \subset \mathbb{C}^n$, ...

16
votes

0
answers

502
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### 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 ...

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 $gz=(az+b)/(cz+d)$...

0
votes

1
answer

166
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### 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 ...

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+\...

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 ...

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(...

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 ...