Questions tagged [cv.complex-variables]
Complex analysis, holomorphic functions, automorphic group actions and forms, pseudoconvexity, complex geometry, analytic spaces, analytic sheaves.
2,615
questions
0
votes
0answers
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 ...
1
vote
1answer
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 $0<Re(z)<1$;...
0
votes
0answers
48 views
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 ...
-1
votes
0answers
57 views
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) ...
2
votes
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 ...
7
votes
0answers
84 views
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|...
0
votes
0answers
83 views
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 $\mathbb{C}^n$ ...
7
votes
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 ...
1
vote
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 $a \neq 0$....
1
vote
0answers
29 views
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 ...
11
votes
1answer
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 ...
0
votes
1answer
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 $|{a}/{z}|, |{b}/{z}| <1$.
...
3
votes
0answers
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 ...
0
votes
0answers
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 ...
2
votes
0answers
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 ...
4
votes
0answers
84 views
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{...
4
votes
0answers
142 views
+50
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 ...
4
votes
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>...
0
votes
0answers
45 views
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
$$...
12
votes
1answer
441 views
Are the fibers of a surjective holomorphic submersion $\mathbb{C}^n\to\mathbb{C}$ all homeomorphic?
Are the fibers of a surjective holomorphic submersion $\mathbb{C}^n\to\mathbb{C}$ all homeomorphic?
For $n=1$ this means that a surjective entire function $\mathbb{C}\to\mathbb{C}$ without critical ...
3
votes
1answer
288 views
On convergence of entire functions
Suppose we have a sequence of entire functions $f_n$ such that $$\text{$f_n(z)\to0$ for each natural $z$}\tag{1}$$
(as $n\to\infty$).
Is it possible to give general additional conditions on the ...
1
vote
0answers
63 views
Global sections appearing in Dolbeault complex with values in vector bundle
Given a holomorphic vector bundle $E$ on a compact complex Kähler manifold $X$ (I am happy to assume $X$ projective), we can compute the sheaf cohomology $H^\ast(E)$ of $E$ using the Dolbeault complex ...
3
votes
0answers
88 views
Quasi-crystaline generalization of elliptic functions
I came across some meromorphic function, call it $f(z)$, which is "quasicrystalline" in the following sense: one can write $f$ as:
$$
f(z)=\frac{\sum_i a_i e^{i(q_{i,x}x+q_{i,y}y)}}{\sum_i ...
1
vote
0answers
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 $\delta>0$ ...
4
votes
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 ...
4
votes
0answers
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 ...
40
votes
3answers
4k views
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 ...
2
votes
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:
$$f_1\...
3
votes
1answer
253 views
Is there a decision procedure for analytic continuation?
Let an analytic element be a power series associated with an open disc of the complex plane over which the series is convergent. W.l.o.g. assume the series is a Taylor expansion about the center of ...
5
votes
3answers
245 views
Truncated Perron - logarithm-free error term?
Let $\{a_n\}_{n\geq 1}$, $|a_n|\leq 1$, be such that $F(s) = \sum_{n=1}^\infty a_n n^{-s}$ can be continued analytically to a neighborhood of the line $\Re s = 1$. (For instance, let $a_n = \mu(n)$.) ...
1
vote
1answer
140 views
Holomorphic map to Möbius group
$\DeclareMathOperator\PSL{PSL}$Let $U\subset\mathbb C^2$ be an open set, $f:U\to \PSL(2,\mathbb C)$ a holomorphic map. If the image of $f$ is contained in $\operatorname{PSU}(2,\mathbb C)$, I guess ...
2
votes
0answers
52 views
Question about an exact expression for the root-mean-square of the distances of the critical points to a given zero of a polynomial
Let $p(z) = \prod_{j=1}^{l+1} (z - z_j)^{M_j}$ be a complex polynomial of degree $n$, where the $z_j$ are distinct for $1, \ldots, l+1$. The first $l$ entries in the list $\{z'_1, \ldots, z'_{n-1} \}$ ...
4
votes
1answer
124 views
Frequency of large values of the Mertens function
It is known that with $M(x) = \sum_{n\le x}\mu(n)$, there are infinitely many $x$ s.t. $|M(x)|\ge x^{\frac{1}{2} - \varepsilon}$ (see Chapter 15 of Montgomery-Vaughan, for example). Is there any way ...
0
votes
1answer
168 views
When is a product of two two-parameter Mittag-Leffler functions a Mittag-Leffler function?
I am studying properties of the two-parameter Mittag-Leffler function.
$$ E_{\alpha,\beta}(z)=\sum_{k=0}^\infty \dfrac{z^k}{\Gamma(\alpha k+\beta)}.$$
I am particularly interested in recurrences and ...
0
votes
0answers
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,...
5
votes
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 ...
4
votes
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 ...
1
vote
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$...
3
votes
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$...
4
votes
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
votes
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 ...
1
vote
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 ...
2
votes
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
votes
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 ...
3
votes
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 ...
5
votes
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 ...
0
votes
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
votes
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 ...
1
vote
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
votes
0answers
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 ...
