Questions tagged [cv.complex-variables]
Complex analysis, holomorphic functions, automorphic group actions and forms, pseudoconvexity, complex geometry, analytic spaces, analytic sheaves.
2,298
questions
-4
votes
0answers
70 views
On the function $f(\sigma)=\int_{-\infty}^{\infty} | \frac{1}{(\sigma + it)\zeta(\sigma + it)}|^{2} \mathrm{d}t$
Define $$f(\sigma)=\int_{-\infty}^{\infty} \Big| \frac{1}{(\sigma + it)\zeta(\sigma + it)} \Big|^{2} \mathrm{d}t$$ where $\zeta$ denotes the Riemann zeta function and $i$ the imaginary unit. Is $f(\...
0
votes
1answer
125 views
Injectivity of analytic functions
Suppose $f : \mathbb{R} \rightarrow \mathbb{R}^n$ is a real analytic function on $(a, \infty)$. I have two questions:
Suppose $||f(x)|| \rightarrow \infty$ as $x \rightarrow \infty$. I know without ...
0
votes
0answers
30 views
Finding all possible set of functions
Let $\{ h_n(x)\}_{n=1,..,N}$ a set of $2\pi$ periodic functions such that they satisfy the reflection property
\begin{equation}
e^{h_n (x+\pi) + i\bar{h}_n (x+\pi)} = \sum_m C_{nm} e^{h_m (x) + i \...
1
vote
0answers
86 views
General class of functions satisfying growth condition on a given functional
The question is inspired by Abel Plana summation formula :
Is there a general class of functions $f$ that are positive valued on the positive real axis, and which satisfy the following
$$\int_0^\infty ...
1
vote
1answer
138 views
Existence of entire function that yields periodicity
I have the following question:
Does there exist an entire function $f(z)$ where $z=x+iy$ such that
$$g(x,y) =e^{-2\pi y^2}f(z)$$
is periodic in both $x$ and $y$ direction, i.e. $$\forall x,y: g(1,y)=g(...
7
votes
1answer
209 views
Functions $f \geq 0$ on $\mathbb{R}$ which are of the form $f = |g|^2$ for some entire function $g$
I think the answer to this question must be well known. Is it possible to characterize those functions $f \colon \mathbb{R} \to \mathbb{R}_+$ which are of the form $f(x) = |g(x)|^2, x \in \mathbb{R},$ ...
4
votes
1answer
281 views
Solving equation of matrix valued functions
Given $n\times n$ matrices with entire functions entries (holomorphic on all of the complex plane $\mathbb{C}$)
$A(z)=[a_{ij}(z)],B(z)=[b_{ij}(z)]$,
i.e.,
$a_{ij}(z),b_{ij}(z)$ are entire functions ...
-1
votes
0answers
68 views
Existence of a harmonic function on the upper half plane unbounded on all points of the x-axis
Does there exists a harmonic function on the upper half plane which diverges at all
points of the boundary x-axis? Answers in
Extension of harmonic function at infinity
seem relevant but I am not sure ...
2
votes
1answer
109 views
Analytic continuation over boundaries
In D.J Newman's paper
A simple analytic proof of the prime number theorem
there is the following theorem:
Suppose $|a_n|<1$ and form the Dirichlet series $F(s)=\sum_{n=1}^{\infty}\frac{a_n}{n^s}$ ...
2
votes
1answer
261 views
Relationship between volume and area
Let $\mu(z) dV_n$ be a measure in $\mathbb{C} ^n$.
Let $B_n(r) := \{z \mid \|z\| < r\}$ be the ball of radius $r$ in $\mathbb C^n$, and $\partial B_n(r) $ be the corresponding sphere.
In $\mathbb{C}...
0
votes
0answers
46 views
The role of a combination of Eneström-Kakeya and Gauss-Lucas theorems: reference request or soft question, asking for this combination as tool
In past days I was trying to create problems or direct applications invoking Eneström—Kakeya and Gauss-Lucas theorems for certain arithmetic functions that I know from analytic number theory. These ...
1
vote
1answer
103 views
Analyze a function defined in terms of an integral
Here is a question that really has puzzled me for quite a while. I happened to see this function defined in terms of an integral
$$f(x):=\int_0^{\pi/2}\frac{2e^{x+e^x\cos y}}{1+\left(e^{e^x\cos y}\...
0
votes
0answers
65 views
Theorem 5.3 ([Okounkov-01]) in Borodin and Gorin's lecture note
In this lecture note: https://arxiv.org/pdf/1212.3351.pdf, Theorem 5.3(P28):
Suppose that the $\lambda \in \mathbb{Y}$ is distributed according to the Schur measure $\mathbb{S}_{\rho_1; \rho_2}$. ...
3
votes
1answer
90 views
Conformal mapping between two right-angled triangles
I want to derive a conformal mapping $f\!:\!A\!\to\! B$ where $A=\{ (x,y)\ |\ x\!\in\![0,1]\ \text{ and }\ 0 \leq y \leq x \}$ and $B=\{ (x,y)\ |\ x\!\in\![0,1]\ \text{ and }\ 0 \leq y \leq \frac{...
2
votes
1answer
138 views
Binomial transform of Dirichlet series
Let $\Theta(s)$ be a Dirichlet series , and let $\beta$ be its abscissa of convergence:
$$\Theta(s)=\sum_{n=1}^{\infty}\frac{\theta(n)}{n^{s}}\;\;\;\;\;\;\Re(s)>\beta$$
And let $\left\{a_{n}\right\}...
2
votes
1answer
93 views
3D similarities and quaternions?
As is well-known, in dimension 2, a linear map $f : \mathbb{R}^2 \rightarrow \mathbb{R}^2$ is a direct similarity if, once we identify $\mathbb{R}^2$ with $\mathbb{C}$, $f$ is of the form
$$\forall z \...
0
votes
0answers
69 views
Meromorphic functions on a modular curves of genus $0$ that take each value exactly once
Let $\Gamma$ be a congruence subgroup of $\operatorname{SL}_2(\mathbb Z)$, and let $\mathfrak H$ be the upper half-plane. Let $X(\Gamma)$ be the compactification of $\Gamma\backslash\mathfrak H$. Then ...
2
votes
1answer
141 views
Continuous extensions of Riemann mappings
Let $K$ be a compact set in $\mathbb C$ without interior. Suppose, additionally, that $K$ is a retract (or equivalently $K$ connected, $K$ locally connected and $\mathbb C\setminus K$ connected). ...
2
votes
0answers
72 views
Estimates for tensors using local coordinates
Suppose that we have a Kähler manifold $(M, \omega_0)$ and another $(1,1)$ form $\eta$ on $M$. Let $\varphi$ be a smooth function such that $\omega_{\varphi} = \omega_0 + i \partial \bar \partial \...
0
votes
0answers
163 views
Understanding Krantz's proof of Hefer's lemma in $\mathbb{C}^2$
Note: I initially phrased the question in a different way, and it did not receive much attention. In the hope to make it more interesting, I have included a (long) introduction to contextualize and ...
0
votes
0answers
34 views
Bounding the absolute value of a complex integral with itself
I already asked a similar question on this topic, but after a small discussion, I noted that I did must boil down the problem such that the solution space so to say to maybe have a concrete answer. I ...
0
votes
0answers
75 views
injective holomorphic mapping between unit disk and unit polydisk
In $\mathbb{C}^n,\ n\geq 2$, there is no bijection between unit disk $B^n(0,1)$ and unit polydisk $P^n(0,1)$. But if we wish to find injective holomorphic mapping from unit disk to polydisk(whose ...
-1
votes
1answer
62 views
Image of transcendental meromorphic functions
Let $f$ be a trancendental meromorphic function such that $f'(z) \ne 0$ for all $z \in \mathbb{C}$. Let $\Pi$ be the stereoprojection map from the north pole on the unit sphere. My question is the ...
1
vote
1answer
84 views
Roots for $p(w)=n+\sum_{j=1}^{m}\frac{v_{j}}{w-v_{j}}$
Let $v_{j}\in \mathbb{C}, 1\leq j\leq m$ and $w\in \mathbb{C}\setminus \{v_{j}\}_{j=1}^{m}$ and $n>0$.
Q: Can we say anything about the m roots $w_{1},...,w_{m}$ of
$$p(w)=n+\sum_{j=1}^{m}\frac{...
3
votes
0answers
161 views
No common roots of complex polynomial and of its derivative
Our specific context
Here is our specific contour integral
$$\int_{\Gamma_{0}}F\big(\sum_{w:p_{z}(w)=0}\frac{1}{w^{a}}\frac{1}{n+\sum_{j=1}^{m}\frac{v_{j}}{w-v_{j}}} \big)\frac{dz}{z},$$
...
0
votes
0answers
51 views
Bounding the absolute value of a complex integral
I'm working on some problems involving Fourier transforms and convolution problems and there is one problem I cannot solve. In my situation we have a complex valued function $f(ix)$, with $x\in\mathbb{...
1
vote
1answer
149 views
Cauchy's Integral with quadratic exponential term
As I was studying the Cauchy's integral formula, I tried to do the integral:
\begin{equation}
I = \int\limits_{-\infty}^{\infty} \frac{1}{x - a} e^{(i A x^2 + i B x)} dx
\end{equation}
with $A>0, ...
3
votes
0answers
96 views
Analytic and algebraic definitions of intersection multiplicity of two complex algebraic curves coincide
There are two definitions of intersection multiplicity of two complex algebraic curves. One is given in An introduction to algebraic curves by Griffiths. Let $t \mapsto (t^k, y(t^k) )$ be the local ...
0
votes
0answers
41 views
Oscillatory integral independent of a parameter
Let $h: \mathbb{R} \mapsto \mathbb{C}$ be a positive definite function, continuous at the origin. (In fact, $h$ is the Fourier transform of a finite measure). Define the oscillatory integral
$$Q(t) :=...
4
votes
1answer
88 views
minimizing weighted length of closed curves
Let $\mathcal{A}$ be the family of closed smooth curves in the right half of the complex plane $\mathbb{C}$ such that any curve in the family must enlose the point $z=1$ and tangent to the $y$-axis at ...
3
votes
1answer
97 views
Reference request: The transform of a bounded random variable has a zero in the complex plane
Together with coauthors I'm working on a paper where we use the following Proposition:
If a real-valued random variable $X$ has bounded support, then except in the trivial case that $X$ has all ...
15
votes
1answer
418 views
Partial sums of $\sum_0^\infty z^n$
Let $z$ be a complex number with $|z|<1$. For every subset $A\subset\mathbb N$, the series $\sum_{m\in A}z^m$ is convergent. Denote $S(A)\in\mathbb{C}$ its sum and $\Sigma_z$ the set of all numbers ...
2
votes
0answers
80 views
Equivariant resolution of singularities with equivariant centres
From what I understand, given a complex projective variety X inside a compact complex manifold Y, according to Hironaka, there is a sequence of $r$ blowups $Y_i$ of Y along complex submanifolds (...
3
votes
0answers
168 views
Stokes's Theorem with singularities on projective line
Let $X$ be a complex manifold and $\omega\in \Omega(X\times \mathbb{P}^1)$ a form. I met the following identity:
$$\int_{\mathbb{P}^1}(\partial_z\bar{\partial}_z\omega) \log|z|^2=\int_{\mathbb{P}^1}\...
4
votes
1answer
453 views
A problem on polynomials
Let $P(z)$ be a polynomial of degree $n$ with $|P(z)|\leq 1$ on $|z|=1$ and $P_m(z)$ be a partial sum of $P(z).$ How large $P_m(z)$ can be on $|z|=1?$
2
votes
0answers
54 views
Holomorphic semigroups vs analytic semigroups
Is there any difference between the two notions in the theory of semigroups?
In the literature, we find some monographs use the farmer while others use the latter. I expect that they are always the ...
2
votes
0answers
118 views
Derivative of a polynomial $P(z)$ and the derivative of the conjugate reciprocal of $P(z)$
Let $P(z)$ be a polynomial of degree $n$ having no zeros in $|z|<1.$ Let $Q(z)=z^n\overline{P(1/\overline{z})}.$ Then it is an easy exercise to show that $\Re\left(zP'(z)/P(z)\right)= \sum_{k=1}^n\...
2
votes
0answers
75 views
Theta-function in the lower half-plane
Standard theta function
$$\vartheta(q)=\sum_{n=-\infty}^\infty q^{n^2} \qquad\qquad(1)$$
has a natural boundary of analyticity at $|q|=1$. This means that it can not be used to regularize expressions ...
0
votes
0answers
27 views
Fixing coefficients using analytic structure
I am trying to understand an exercise in a set of lecture notes on random matrices - Eynard - Random matrices - given in the paragraph following (4.6.60) (pp. 70–71).
Specifically, I am trying to fix ...
0
votes
1answer
84 views
Logarithms of $L$-functions of irreducible characters of Galois group
We know that the $L$ functions of Dirichlet characters $\chi$ of $(\mathbb Z / m\mathbb Z)^\times$ satisfy the property that $\log L(s, \chi)$ is holomorphic for $\Re(s) \geq 1$ if $\chi$ is a ...
2
votes
1answer
86 views
Existence of analytic continuation of Dirichlet series corresponding to the indicator sequence of a complement of a special multiplicative set
Let $K/ \mathbb Q $ be a finite Galois extension and let $X$ be a proper non-empty subset of the Galois group $G=Gal(K/ \mathbb Q)$ that is closed under conjugation. Consider a set of integer primes $...
3
votes
1answer
78 views
A question on preimage of a locally injective meromorphic function
Let $f: \mathbb{C} \rightarrow \mathbb{C}$ be a meromorphic function such that $f'(z) \ne 0$ for all $z \in \mathbb{C}$. Let $\Gamma \subset \mathbb{C}$ be a curve which has no self intersections. If ...
3
votes
1answer
193 views
To show holomorphicity of a certain infinite series of functions
I have the following sequence of holomorphic functions on $(f_n(s))_{n \geq 1}$ on the closed region $R:= \{ s \in \mathbb C : \Re(s) \geq 1\}$ by
$$f_n(s) :=
\begin{cases}
\log(1+p^{-s})-p^{-s}\...
2
votes
0answers
211 views
On the logarithmic derivative of an analytic function
I have reached upto this situation while exploring some properties of analytic functions which are bounded on the unit circle.
Suppose $f(z)$ be such an analytic function in the closed unit disc ...
4
votes
1answer
209 views
Fun examples relating to Hopf surfaces
A Hopf surface is a compact complex surface whose universal cover is complex analytically isomorphic to $\mathbb{C}^2 \setminus \{ 0 \}$. I would like to know whether anyone has any of the following ...
4
votes
1answer
116 views
Distribution boundary value of analytic function and wave front sets
Assume $f(z)$ is analytic in the tube domain $\mathbb R^n\oplus iC$, where $C\subset \mathbb R^n$ is a convex cone. Under the assumption $|f(x+iy)|\leq 1/|y|^k$, we know by a Theorem of Martineau (see ...
10
votes
1answer
314 views
Sendov's conjecture
It has been more than fifty years for famous Sendov's conjecture which states that if $p(z)$ is a polynomial of degree $n$ having all its zeros in the unit disc $|z|\leq 1$ then each of the n ...
1
vote
0answers
58 views
zero extension of positive currents are always positive
In Demailly's Complex Analytic and Differential Geometry page 139:
He said the trivial (zero) extension of the positive current $T$ (on $X\setminus E$), which denoted by $\tilde T$ is always positive ...
6
votes
1answer
113 views
The state of art of the singular Levi problem — and hyperkähler quotients
One of the versions of the classical Levi problem asks the following:
Let $X$ be a complex manifold. Is it true that $X$ is Stein iff
$X$ admits a smooth exhaustion strictly plurisubharmonic ...
4
votes
0answers
104 views
Resources for divergent / asymptotic series
This series is divergent; therefore, we may be able to do something with it. -- Oliver Heaviside
Other than the usual references given in Wikipedia and Mathworld, which resources have you found ...
