All Questions
Tagged with cv.complex-variables reference-request
271 questions
3
votes
1
answer
173
views
Can we define $\partial\bar{\partial}(\log|z_1|^2)\wedge \partial\bar{\partial}(\log|z_2|^2)$ as a current?
In complex analysis, by Poincare-Lelong theorem, we have
$$
\frac{\sqrt{-1}}{\pi}\partial\bar{\partial}(\log|z|^2)=T_{z=0}
$$
as currents, where
$$
T_{z=0}(\eta)=\int_{z=0}\eta.
$$
Now suppose we have ...
- 9,009
3
votes
1
answer
174
views
first order quasilinear partial differential equations
I am interested in understanding complex first-order quasilinear partial differential equations. In the real setting there is a huge literature dealing with such equations but in the complex setting, ...
- 35
3
votes
1
answer
308
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 ...
- 128k
3
votes
1
answer
385
views
A question about Lelong number
If $f$ is plurisubharmonic (not identically $-\infty$) on a neighbourhood of $0$ then the Lelong number of $f$ at $0$ is defined by $$\nu_{f}(0) = \liminf_{|z|\rightarrow 0}\dfrac{f(z)}{\log|z|}.$$
My ...
- 33
3
votes
1
answer
252
views
unbounded power series
I want a reference to the literature of a power series convergent in the whole CLOSED
unit disk,but unbounded there.
- 91
3
votes
1
answer
294
views
Automorphisms of bounded symmetric domains
Let $D \subset \mathbb{C}^n$ be a bounded symmetric domain. It is known that $D$ can be realized as the unit ball of some complex norm $||\cdot||$. Using the Bergman metric on $D$, one can define a ...
- 1,159
3
votes
1
answer
499
views
methods for interpolating a function, holomorphic in the upper halfplane
Let $n,k\colon\mathbb{R}\to\mathbb{R}$ be real functions such that function $N$ given by $N(x)=n(x)-ik(x)$ is a holomorphic function in the upper half-plane. Also I know some additional properties of ...
- 1,284
3
votes
1
answer
155
views
Inequality for generalized Laguerre polynomials
Please. Does anybody know a proof of this inequality
$$\Big|\frac{n!\Gamma(\alpha+1)}{\Gamma(n+\alpha+1)} L^{\alpha}_n(x)\Big|\leq e^{\frac{x}{2}}$$ where $\alpha\geq0$ and $x\geq0$ and
$L^{\alpha}_n$ ...
- 81
3
votes
2
answers
280
views
Reference request for the integral representation of the Hadamard product of two infinite series
Define $F(x) = \sum_{n\geq 1} f_{n}x^n$ and $G(x) = \sum_{n\geq 1} g_{n}x^n$. Then the Hadamard product of $F$ and $G$ is
$$H(x):=(F*G)(x) = \sum_{n\geq 1} f_{n}g_{n}x^n.$$
The author of Riesz ...
- 43
3
votes
1
answer
305
views
The spectrum and the tangent space of the algebra of holomorphic functions on a Stein manifold
Let $A$ be a Fréchet algebra over ${\mathbb C}$, and let us call the spectrum ${\tt Spec}[A]$ of $A$ the set of all characters, i.e. continuous multiplicative linear functionals $s:A\to{\mathbb C}$, ...
- 7,426
3
votes
1
answer
280
views
Composite families of formal power series over $\mathbb C$ as algebraic variety
I was led to prove that the set of composite families $(f_j)_{j \leq k}$ of germs at $0\in \mathbb C^m$ of a holomorphic function (composite = sharing a common divisor belonging to the maximal ideal) ...
- 5,428
3
votes
1
answer
2k
views
Queries about the Skolem-Mahler-Lech theorem (integer zeros of exponential polynomials)
The Skolem-Mahler-Lech Theorem says that the integer zeros of an exponential polynomial are the union of complete arithmetic progressions and a finite number of exceptional zeros. http://terrytao....
- 1,795
3
votes
0
answers
219
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, ...
- 421
3
votes
0
answers
89
views
Transformation of Julia set sequence emerging from meromorphic function
I consider a sequence of meromorphic functions on the Riemann sphere $f_k:\hat{\mathbb{C}} \to \hat{\mathbb{C}}$ for $k\in\mathbb{N}$ of the form
$$f_k(z)=\sum_{j=1}^{n_k}\dfrac{1}{(z-p_j)^{c_j}}$$
...
- 91
3
votes
0
answers
184
views
"Circulant-Vandermonde" matrix: in search of a formula
An $n\times n$ circulant matrix $\mathbf{X}_n$ has the form
\begin{align}
\mathbf{X}_n= \begin{bmatrix}
x_1 & x_2 & \cdots & x_{n-1} & x_n \\
x_2 & x_3 & \cdots & x_n&...
- 43.1k
3
votes
0
answers
120
views
On tangential approach regions for general power series converging on the unit disk
Notation and premises. Here it is a list of notations more or less explicitly used in the question:
If $z\in\Bbb C$ then $z = re(t)$ where $r\in \Bbb R_{\ge 0}$, $t\in [0,1]$ and $e(t)\triangleq \exp(...
- 6,357
3
votes
0
answers
226
views
On an exact expression for the squares 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 $\{z'_1, \ldots, z'_{n-1} \}$ ...
- 429
3
votes
0
answers
135
views
Asymptotic Expansion of Seiberg-Witten Differential?
Nekrasov & Okounkov proved (https://arxiv.org/pdf/hep-th/0306238.pdf) that the Seiberg-Witten prepotential can be given by
\begin{equation}
\mathcal{F}(\mathbf{a},\Lambda) = \lim_{\hbar\rightarrow ...
- 425
3
votes
0
answers
157
views
How to decide whether a power series is algebraic? [duplicate]
I vaguely recall that there is a theorem stating, a power series is algebraic iff the coefficients of the series is automatic over every finite fields.
Could anyone give the article or the theorem ? ...
- 3,084
3
votes
0
answers
482
views
Possible automorphisms of a Jacobian
If we consider automorphisms of the Jacobian $J(C)$ of a curve $C$ which are compatible with the canonical polarization, we can describe this automorphism group in terms of $\text{Aut }C$ (see ``On ...
- 521
3
votes
0
answers
89
views
Trace of a weighted composition operator on Bergman space
I am reading a series of papers by Pollicott, Jenkinson and coauthors which make use of the following type of result:
Theorem: Let $\mathbb{D} \subset \mathbb{C}^d$ be a bounded, connected open set. ...
- 6,206
3
votes
0
answers
105
views
State of the art for univariate complex polynomials factorization with algebraic coefficients
Let $\mathbb{K}:=\overline{\mathbb{Q}}$ be the field of algebraic numbers. We choose to represent an element of $\mathbb{K}$ as its minimal monic polynomial, which is a vector in some $\mathbb{Q}^n$.
...
- 5,428
3
votes
0
answers
178
views
One-parameter groups acting on dual Banach spaces
Let $E$ be a Banach space, and $M=E^*$ (my application has $M$ a von Neumann algebra, but this is unimportant). Let $(\sigma_t)$ be a SOT cts one-parameter group on $E$: so for $t\in\mathbb R$, we ...
- 18.7k
2
votes
4
answers
1k
views
Complex differential equations
I'm looking for a gentle an concise introduction to complex-variable differential equations. Eventually, I need to look at complex PDEs, but I assume one starts with complex ODEs.
Mostly, I'm just ...
- 3,574
2
votes
2
answers
361
views
Size of $\zeta'(s)$ at its zeros
How large can the derivative of the Riemann zeta function be at its zeros?
More specifically, let $\rho$ be a zero of the zeta function with $\Im(\rho)\in (0,T]$. What can we say about $|\zeta'(\rho)|...
user536681
2
votes
3
answers
478
views
Groups of conformal isomorphisms of simply connected surfaces
By the uniformization theorem every connected and simply connected surface $M$ is conformally equivalent to one of the following three surfaces:
open disk $D$, complex plane $\mathbb{C}$, or $2$-...
2
votes
2
answers
296
views
Planar polynomial vector field for a harmonic pair of polynomials
Has the system of ODEs
$$\frac{dx}{dt}=P(x,y)\\
\frac{dy}{dt}=Q(x,y)
$$
been studied for the special case of the polynomials $P$ and $Q$ being a harmonic pair, i.e. the real and imaginary part of ...
- 7,127
2
votes
3
answers
515
views
Asymptotic number of zeros for Dirichlet series with functional equation
I think the usual proof for the asymptotic number of zeros of the Riemann zeta function
$$N(T) = \#\left\{\rho : \ \zeta(\rho)=0, \begin{array}{l}\scriptstyle Im(\rho)\ \in\ [0,T]\\ \scriptstyle Re(\...
- 3,403
2
votes
1
answer
518
views
When flatness of a morphism implies smoothness?
EDIT: Let $f\colon X\to C$ be a flat proper morphism of complex algebraic (or analytic) varieties. Assume the special fiber over a point $p\in C$ is smooth.
Is it true that there exists a ...
- 21.8k
2
votes
1
answer
323
views
Exact reference for Liouville theorem
It seems hard for me to find that the solution of the following equation
$$
\Delta u+e^u=0
$$
defined on a simply-connected domain $D\subset R^2$ must be of form
$$
u(z)=\log\frac{4|f'|^2}{(1+|f|^2)^2}...
- 155
2
votes
1
answer
277
views
Length-preserving Analogue of Riemann's Mapping Theorem
The Riemann mapping theorem (cf e.g. http://en.wikipedia.org/wiki/Riemann_mapping_theorem) essentially guarantees the existence of a biholomorphic mapping of a simply connected, open subset of the ...
- 13.2k
2
votes
2
answers
859
views
$ 2|f^{'}(0)| = \sup_{z, w \in D} |f(z)-f(w)|$ if and only if $f$ is linear
I know the following is a well-known result.
Let $D = B(0,1) \subset \mathbb{C} $ a disc, $f$ holomorphic on $D$. Show that $$ 2|f^{'}(0)| \le \sup_{z, w \in D} |f(z)-f(w)|$$
Furthermore, there is ...
- 125
2
votes
3
answers
632
views
How to find the almost period of an exponential polynomial
Let $u(t) = \Sigma_{k=1}^n c_k e^{i \lambda_k t} (c_k \in \mathbb C, \lambda_k \in \mathbb R) $ be an exponential polynomial of order $n$ with purely imaginary exponents. We can assume that the ...
- 1,795
2
votes
1
answer
177
views
Another combinatorial identity
Is it true that
$$\sum_{r=0}^p \sum_{i=0}^r a_{n,p,r,i}=0$$
for all natural $n$ and all natural $p\ge2n$, where
$$a_{n,p,r,i}:=\frac{(-1)^r (n+p-r-1)! (n p-i (r-i))}{i!(r-i)! (n-i)!
(p-r+i)! (n-r+i)! ...
- 128k
2
votes
2
answers
417
views
Stoilow Theorem
I want to see the precise statement and a proof for a theorem of Stoilow on "inner" functions (I do not know what this exactly means, I suppose it is an open map with other natural properties). A ...
- 85
2
votes
3
answers
511
views
Blaschke Condition for hyperbolic lattices
For $r$, $s$, small positive integers, do the complex numbers on the unit disc (without the hyperbolic metric) corresponding to the vertices of the hyperbolic tiling with Schläfli symbol $\{r,s\}$ ...
- 975
2
votes
1
answer
163
views
Jensen's Formula for Arbitrary Neighborhoods
The Jensen's formula says the following: Let $f$ be analytic on the disc $D$ of radius $R$ centered at the origin such that $f(0)\neq 0$, then
\begin{align}
\log(|f(0)|)+ \sum_{i=1}^n \log \left(\...
- 671
2
votes
1
answer
230
views
Entire composite square roots of functions of finite order
A composite square root of a function $g$ is a function $f$ such that $f(f(z)) = g(z)$. Not surprisingly, for arbitrary $g$ a function like this is hard to find. Specifically I am looking at functions ...
user78249
2
votes
2
answers
411
views
Asymptotics of the derivatives of analytic functions
Are there sources that treat questions like the following ones?
Suppose that $f\colon\mathbb{C}\to\mathbb{C}$ is an entire function such that $f(x)$ is real for all real $x$ and $f(x)\sim1/x$ as $x\...
- 128k
2
votes
2
answers
424
views
Meromorphic Functions as Distributions
For the function $\frac{1}{x}$ on the real line, one can use a modified principal value integral to consider it as a distribution p.f.$(\frac{1}{x}),$ and one can do a similar construction to make $\...
2
votes
1
answer
583
views
Brieskorn's proof of a theorem by Milnor about the Milnor number
I am looking for a reference or short explanation of a proof by E. Brieskorn.
In his famous work "Singularities of complex hypersurfaces" Milnor proves that the (nowadays called) Milnor Number (in ...
- 1,124
2
votes
1
answer
304
views
Is it possible to define pseudodifferential operator $p(x,T)$ using Cauchy integral formula?
I was wondering how I can define a pseudodifferential operator using Cauchy integral formula.
Consider a differential operator $p(T)$ ($p$ is a polynomial for instance). $p(T)$ can be defined as:
$$\...
- 350
2
votes
1
answer
210
views
Defining a map into $S^1$ as an "angle" in a non simply connected domain
Suppose that ambient space is $\mathbb R^2$, and $\Omega \subset \mathbb{R}^2 $ is a smooth domain, non simply connected domain. To fix ideas,we can assume $$\Omega = \{(x_1,x_2) : 1< x_1^2+x_2^2 &...
- 2,494
2
votes
1
answer
373
views
Does the "Ohsawa-Takegoshi theorem without bounds" have a name?
There are many theorems which now could be called "The Ohsawa-Takegoshi" theorem. Of these, the most basic is roughly the following:
Let $\Omega \subset \subset \mathbb{C}^n$ be a psuedoconvex ...
- 12.1k
2
votes
1
answer
120
views
Roots for $p(w)=n+\sum_{j=1}^{m}\frac{v_{j}}{w-v_{j}}$
Let $v_{j}\in \mathbb{C}, 1\leq j\leq m$ and $w\in \mathbb{C}\setminus \{v_{j}\}_{j=1}^{m}$ and $n>0$.
Q: Can we say anything about the m roots $w_{1},...,w_{m}$ of
$$p(w)=n+\sum_{j=1}^{m}\frac{...
- 5,474
2
votes
1
answer
186
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 $...
- 739
2
votes
1
answer
149
views
Higher dimensional analogue of Ahlfors covering surface theory
It is well known that Ahlfors covering surface theory in one dimensional is very powerful in dealing with many problems. I wonder whether there exists some generalization of this theory into higher ...
- 21
2
votes
1
answer
2k
views
The normal derivative of the Green's function
I was wondering if anything was known about the following:
Let $\mathbb{D}^2=\lbrace x^2+y^2< 1 \rbrace \subset \mathbb{R}^2$ be the open unit disk.
Consider now the Green's functions $G(z; p)$ ...
- 2,299
2
votes
0
answers
37
views
Theta series of well-rounded lattices
I've started looking into well-rounded Euclidean lattices and I was interested in learning whether their theta series have any interesting properties, but haven't found much in terms of bibliography ...
- 223
2
votes
0
answers
90
views
Computing a complex integral with many poles
For an integer $k\geq 1$, let $f:\mathbb{C}^k\to\mathbb{C}$ be such that $f$ is analytic in the region $\text{Re}(u_i) > -1$ (say) for each $1\leq i \leq k$, and decays rapidly on vertical lines (i....
- 1,990