Questions tagged [cv.complex-variables]
Complex analysis, holomorphic functions, automorphic group actions and forms, pseudoconvexity, complex geometry, analytic spaces, analytic sheaves.
3,152
questions
3
votes
1
answer
233
views
Fixed points on Riemann surface
It is well known theorem that for a conformal mapping $\phi$ from a bounded and planar domain $\Omega$ to itself has three fixed points , then it must be identity mapping. However, I cannot find a ...
1
vote
1
answer
178
views
Has a universality theorem been proved for the Davenport-Heilbronn L function?
The question is in the title: has a universality theorem in the sense of Voronin been proved for the Davenport-Heilbronn function, or do we expect such a theorem to hold true only for L functions that ...
30
votes
4
answers
3k
views
Distribution of roots of complex polynomials
I generated random quadratic and cubic polynomials with coefficients in $\mathbb{C}$
uniformly distributed in the unit disk $|z| \le 1$. The distribution of the roots of 10000
of these polynomials are ...
4
votes
1
answer
277
views
Inequality for certain analytic functions
Suppose that $f(z)$ is analytic for $\Re(z)>0$ and it is positive on the real axis. Suppose that for some $y_0$ we have
$$
\left| {f\left( {x + iy} \right)} \right| \le f\left( x \right)
$$
for any ...
5
votes
1
answer
195
views
Ubiquity/scarcity of non-analytically continuable functions
Suppose f(z) is a power series with positive integer coefficients centered at zero and positive radius of convergence. What is the likelihood that f has a dense set of singularities on its circle of ...
3
votes
2
answers
279
views
Is the Hausdorff dimension $Dim_{H}(J(f))$ of the Julia set less than 2 for quadratic rational map?
Let $f(z)$ be a quadratic rational map with two Siegel disks which can be normalized to be $$f(z)=z\frac{z+e^{2\pi i\alpha}}{e^{2\pi i\beta}z+1}.$$ If one of the ratation numbers $\alpha$ and $\beta$ ...
1
vote
1
answer
215
views
Forster's Theorem
I am looking for a detailed account, in English or French, of the main result and its proof from this paper by O. Forster
Zur Theorie der Steinschen Algebren und Moduln, Mathematische Zeitschrift 97, ...
1
vote
0
answers
170
views
Solving a system of rational functions
Given pairwise distinct numbers $c_1, c_2, \dots c_n \in \mathbb{C} \setminus \{0\}$, does the system of equations $$\frac{6}{c_k} + \sum_{i \ne k} \frac{2}{c_k - c_i} = \sum_{i = 1}^n \frac{1}{c_k - ...
7
votes
5
answers
11k
views
Does the inverse Laplace transform of the square root exist?
Does the inverse Laplace transform, defined by the integral,
\begin{equation}
F(t) = \mathscr L_s^{-1}\left[\sqrt s\right](t) = \int_{c - i\infty}^{c + i\infty} \sqrt s ~e^{-st} ds
\end{equation}
...
3
votes
3
answers
312
views
A functional equation concerning analytic functions
Let $P$ be a polynomial; we ask about the existence of a non-constant analytic function $f : \Bbb{C}\longrightarrow \Bbb{C}$ such for all $z \in \Bbb{C}, f(z) = f(P(z))$. Clearly when $P$ is linear we ...
3
votes
1
answer
202
views
The Integral Trick and An Equality in Nakajima's Lecture
In Nekrasov et al's series papers MNS, they calculate such kinds of integral
$$\frac{E_1 E_2}{N(2\pi i)^N(E_1+E_2) }\oint d\phi_1 \wedge d\phi_2\wedge ...\wedge d\phi_N \prod_{i<N} (-\phi_i) \prod_{...
1
vote
1
answer
861
views
Two different forms of Schwarz-Christoffel-Mapping of unit disk to rectangle. Are they identical?
I found two different equations for the Schwarz-Christoffel-mapping of a unit disk to a rectangle (which are the general form of the SC-mapping, I guess). The first, e.g. from Link, page 20, is
\...
3
votes
2
answers
675
views
Convergence of Dirichlet series ("at the boundary")
I apologize if this is something standard and/or elementary, but I was unable to find anything relevant via Google.
Consider a Dirichlet series
$$
f(s) = \sum_{n=1}^\infty \frac{a_n}{n^s}
$$
and ...
3
votes
0
answers
163
views
topology generated by irreducible componets of $\Gamma$-invariant closed sets
For an analytic space $U$ equipped with an action of a group $\Gamma$,
call a subset $Z\subseteq U$ $\Gamma$-closed iff
it is a closed analytic subset and each of its irreducible components
is an ...
5
votes
1
answer
214
views
Function transformation of exponentials
I came across the following function transformation:
$$
\sum_{j=-\infty}^{\infty} e^{(-j^2\cdot t)} = \sqrt{\frac{\pi}{t}} \cdot \sum_{j=-\infty}^{\infty} e^{(-\frac{\pi^2}{t}\cdot j^2)}
$$
where $ j ...
2
votes
1
answer
333
views
Questions about expansion of $f(x)=\sum_{i=1}^{\infty} a_i x^i$
In complex field, assume $$f(x)=\sum_{i=1}^{\infty} a_i x^i$$ where $a_i \in {\bf N}$ or $a_i = 0$, and $f(x)$ converges in an area.
Question 1: are there $$f(x)=p(x)+\sum_{i=1}^{\infty}r_i(x), $$
or $...
2
votes
0
answers
112
views
How can we describe explicitly the "infinitely complex differentiable" complex-valued local martingales?
Let $\mathcal{F}_t$ be a continuous filtration on a probability space, and let $B$ be a standard $\mathbb{C}$-valued $\mathcal{F}_t$-Brownian motion. Let's call a complex-valued process $X$, possibly ...
2
votes
0
answers
98
views
Is it obvious that the defining conditions to obtain a particular singularity are well-defined on the quotient space?
Let $~f:\mathbb{C}^2 \rightarrow \mathbb{C}$ be a holomorphic function
vanishing at the origin, with
the following properties:
$$ f_{00}, ~f_{10}, ~f_{01}, ~f_{20}, ~f_{11} =0,~~f_{20} \neq 0 \...
1
vote
0
answers
544
views
Local System and Gauss-Manin connection
Fix a complex manifold $X$. Then if we have a line bundle $L=\mathcal{O}(D)$ together with Gauss-Manin connection $\nabla: L \rightarrow L \otimes \Omega^{1}_X$, we get the locally constant sheaf $F$ ...
0
votes
2
answers
718
views
Poisson inequality for subharmonic functions
This is probably a very basic matter, but I am looking for a proof of the Poisson inequality for subharmonic functions, which reads
$$\varphi(r \mathrm{e}^{\mathrm{i} \theta})\leq\frac{1}{2\pi} \...
1
vote
1
answer
125
views
Finite construction of lacunary functions using algebraic and certain analytic operations
Algebraic functions have a discrete set of singularities. Lacunary functions, e.g. $f(z)=\sum_{n=0}^\infty z^{2^n}$, have a continuum of singularities at every point of the boundary of their disk of ...
4
votes
2
answers
366
views
Comparing two Delaunay tessellations on a hyperbolic surface
Let $S$ be a closed hyperbolic surface (i.e. a compact Riemann surface of genus $\geq 2$) and let $P=\{p_1,\ldots,p_m\}$ be a non-empty finite subset of $m$ points in $S$. Let $\pi:\mathbb H\...
2
votes
0
answers
120
views
A parametrix for the $\bar\partial$ operator adapted to a holomorphic foliation
Let $X$ be a compact complex manifold with (regular) holomorphic foliation given by a holomorphic subbundle $\mathcal F$ of the tangent bundle. The foliation induces a filtration on differential forms....
2
votes
2
answers
497
views
What is the closure of space of polynomials in a dense subspace along with a marked point equal to?
EDIT
Let $\mathbb{C}^{m*}$ be the space of non zero polynomials of degree at most
$d$ in two variables. So an element of this space is essentially
$$ f:=f_{00} + f_{10} x + f_{01} y + \ldots f_{0d}...
6
votes
0
answers
392
views
semiclassical proof of Wigner semicircle
In Terence Tao's discussion of the Gaussian Unitary Ensemble, he derives the Dyson and Airy kernels. The GUE is the probability distribution of the eigenvalues of a random Hermitian matrix.
\[ \int ...
6
votes
1
answer
338
views
interpolation with derivative of rational fraction
Studying a problem in conformal geometry, I am facing to the following interpolation problem.
Let $P$ and $Q$ two coprime polynomials. Then let $A$ and $B$ two coprime polynomials such that
$$\frac{...
14
votes
3
answers
2k
views
Do all the roots of the polynomial lie in the unit disk?
How to prove (or to disprove) that all the roots of the polynomial of degree $n$ $$\sum_{k=0}^{k=n}(2k+1)x^k$$ belong to the disk $\{z:|z|<1\}?$ Numerical calculations confirm that, but I don't see ...
29
votes
1
answer
3k
views
Zeros of polynomials with real positive coefficients
The following problem arose in collaborative work with Subhro Ghosh:
Question: To any polynomial $P_n(z)=\sum_{i=0}^n a_i z^i =a_n \prod_{i=1}^n(z-z_i)$, attach the empirical measure of zeros
$L_n=n^{...
3
votes
0
answers
143
views
What is the relationship between complex time singularities and UV fixed points?
In this paper it is described how the turbulent kinetic energy spectrum and the flatness (a measure for intermittency) are governed by the position of the (dominant) singularities of the solutions of ...
2
votes
1
answer
225
views
Interpolating delta like functions by trigonometric polynomials of bounded modulus and fast decay
Consider a grid of points $T=\{t_0,t_1,\ldots,t_m\}$ with $0\le t_i\le 1$. I would like to find a function $f(t):[0,1]\rightarrow \mathbb{C}$ of the form
\begin{equation*}
f(t)=\sum_{k=-n}^n c_k e^{2\...
2
votes
0
answers
276
views
computing a certain contour integral [closed]
I want to compute an integral along a vertical line segment. The function I'm integrating involves the zeta-function, and usually the way such integrals are done treats the line segment as one side ...
2
votes
1
answer
264
views
Approximation Runge's Theorem
Let $X$ be a Riemann Surface and $K$ a compact subset of $X$. Every holomorphic function in $K$ be uniformly approximable on $K$ by holomorphic functions on $X$ if $X-K$ have no connected component ...
2
votes
1
answer
2k
views
construct a power series with infinitely many zeros in the complex plane, bounded coefficients???
Hi all.
I want to construct a power series $F(z)=\sum_{n=0}^\infty c_nz^n$ centered at zero and with finite radius of convergence in the complex plane, and which has infinitely many zeros (in its ...
2
votes
0
answers
560
views
The functional equation of Hofstadter's Q sequence
Hofstadter's Q sequence is defined by $Q(1) = Q(2) = 1$ and
$Q(n) = Q(n-Q(n-1)) + Q(n-Q(n-2))$ for $n \geq 3$. So far hardly anything
on this sequence has been proved -- not even that $Q(n)$ is well-...
7
votes
1
answer
679
views
Zariski's main theorem in the complex analytic category
Hello,
I am looking for a reference to something like that: if $f\colon X\to Y$ is a finite (i.e., proper with finite fibers) morphism of reduced and irreducible normal (or at least smooth) complex ...
3
votes
1
answer
432
views
Which real analytic functions of two variables locally are magnitudes of complex-analytic functions [closed]
Assume we have a real-analytic function $f(x, y)>0$ in some neighborhood of 0. When does there exist a complex-analytic function $w(z)$ such that $|w(z)|=f(x,y)$ for $z=x+iy$.
One necessary ...
12
votes
4
answers
1k
views
Constructing Riemann maps using Brownian motion?
There's a relation between two-dimensional Brownian motion and conformal maps, see e.g. Thurston's answer to this question. Given two non-empty simply-connected domains $U$ and $V$ in the complex ...
3
votes
2
answers
548
views
Mellin Transform
What is the inverse Mellin transform of (s-1/2)^k on the vertical line Re(s)=a where
0 < a <1 and k is a natural number?
2
votes
2
answers
175
views
What is the moduli space of germs of one-sided complex structures near the circle?
Consider a one-sided ( say, internal) neighborhood $U$ of the unit circle $S$ ( $U$ contains $S$) on the plane with a choice of smooth complex stricture $\tau$ on $U$.
By smoothness of $\tau$ on $U$ ...
3
votes
2
answers
289
views
on completeness of R_mn, the set of all rational functions of type (m,n)
It is known from finite dimensionality of $P_r$, the space of all polynomials of degree less than or equal to $r$, that $P_r$ is complete with respect to uniform norm.
Considering $R_{m,n}[a,b]=\{p/...
3
votes
2
answers
352
views
What is known about this product?
I bet the product
$$
\prod_{n=2}^\infty\frac 1 {1-n^{-s}},
$$
which is convergent for ${\rm Re}(s)>1$, has been studied before. Can it be analytically extended across the line ${\rm Re}(s)=1$? If ...
7
votes
4
answers
2k
views
Visualizing functions with a number of independent variables
I need to graph real valued functions (for exposition and analysis).
The issue is: there are more independent variables so that the conventional graphing methods can't be used, and furthermore I don't ...
8
votes
1
answer
575
views
Defining holomorphic functions in terms of Banach algebras, and similarly for C*-algebras
Let $C$ be the category of commutative Banach algebras and let $U : C \to \text{Set}$ be the usual forgetful functor. The holomorphic functional calculus guarantees that every holomorphic function $f :...
6
votes
1
answer
408
views
Is the homeomorphism class of a connected open set of C determined by its fundamental group?
Let $U,U'\subseteq\mathbf{C}$ be two connected open sets such that $\pi_1(U)\simeq\pi_1(U')$.
Q: Does this imply that $U$ is homeomorphic to $U'$?
In the case where the $\pi_1$'s are trivial then ...
2
votes
2
answers
2k
views
On the existence of a holomorphic logarithm
Hi,
The following is probably well-known, but I couldn't find anything in the literature. Any reference would be nice.
Let $\Omega$ be a domain in the complex plane, and let $f$ be holomorphic and ...
6
votes
1
answer
172
views
Is the dual of $A^1(\Omega)$ known for arbitrary domains ?
Let $\Omega$ be a domain in the complex plane, and $A^1(\Omega)$ be the space of integrable holomorphic functions on $\Omega$ equipped with the $L^1$ norm (it is called the Bergman space).
If $\Delta ...
5
votes
0
answers
246
views
proper mapping between Stein manifolds
My question is the following:
Let $b:X \rightarrow Y$ be a proper holomorphic mapping between two Stein manifolds $X$ and $Y$. For obvious reasons, $b^{-1}(y)$ are finite subsets of $X$. Is the set
$...
2
votes
2
answers
322
views
Pointwise bounds on Hardy space functions with regular boundary behaviour
Let $H^2$ denote the Hardy space on the strip $S:=\{z\in{\mathbb C}\,:\,0<\Im z <1\}$ (or the upper half plane), i.e. $H^2$ consists of all holomorphic functions $f:S\to\mathbb C$ such that for ...
2
votes
1
answer
584
views
Do there exist transcendental numbers which are not hypertranscendental?
A complex number is said to be hypertranscendental if the one is not a zero of any entire function with all rational Maclaurin coefficients. Does there exist a transcendental number which is not ...
9
votes
2
answers
2k
views
References on Taylor series expansion of Riemann xi function
I am looking for the references on Taylor series expansion of Riemann xi function at $\frac{1}{2}$.
$$ \xi (s)=\sum_0^{\infty}a_{2n}(s-\frac{1}{2})^{2n}$$
where
$$a_{2n}=4\int_1^{\infty}\frac{d[x^{3/...