All Questions
Tagged with cv.complex-variables reference-request
271 questions
7
votes
1
answer
244
views
Volume of solution sets for polynomials in $\mathbb{C}[x]$
Denote $\pmb{a}=(a_1,\dots,a_d)\in\mathbb{R}^d$ and consider the set
$$\mathcal{E}_d=\{\pmb{a}\in\mathbb{R}^d: \text{each root $\xi$ of $x^d+a_dx^{d-1}+\cdots+a_2x+a_1=0$ lies in $\vert\xi\vert<1$}\...
- 43.1k
1
vote
0
answers
49
views
Permutation of eigenvalues induced by a loop
A friend of mine just mention me what I think is a very fun phenomena and I would be very interested to learn more about it:
Let $A,B\in \mathbb{C}^{n\times n}$ two matrices. And let $\lambda_1(z), \...
- 4,361
2
votes
0
answers
180
views
Multiple zeta values related to fractional calculus and an Appell polynomial sequence
There is an Appell sequence of polynomials $p_n(z)$ related to an infinitesimal generator for one rep of the fractional calculus that have coefficients involving the Riemann zeta function values at ...
- 10.5k
4
votes
0
answers
261
views
Reference request for some result of de Bruijn on zeros of some holomorphic function
In a video lecture on Youtube, ''Vaporizing and freezing the Riemann zeta function'', Terry Tao states that ''de Bruijn proved that if for some $t_0$ the zeros of $H_{t_0}$ are contained in the strip $...
- 57
6
votes
0
answers
288
views
Complex factorization of the angular part of the Laplacian
Some time ago some research led me to the following equality:
\begin{equation}
\frac{1}{\sin^2 \phi }\frac{\partial^2 }{\partial \theta^2} +\frac{\partial^2 }{\partial \phi^2} +\cot \phi \frac{\...
6
votes
0
answers
360
views
Flat base change in the complex analytic setting
On page 255 of Hartshorne's Algebraic Geometry, it is shown that "cohomology commutes with flat base extension":
Proposition III.9.3: Let $f : X \to Y$ be a separated morphism of finite type of ...
- 61
6
votes
2
answers
240
views
Continuity of a differential of a Banach-valued holomorphic map
Originally posted on MSE.
Let $U$ be an open set in $\mathbb{C}^{n}$ let $F$ be a Banach space (in my case even a dual Banach space), and let $\varphi:U\to F$ be a holomorphic map. I seem to be able ...
- 5,529
2
votes
0
answers
154
views
Algebra of meromorphic functions on a Riemann surface
Let $C$ be compact Riemann surface of high genus. Let $p$ be a general point on $C$ and $z$ a local coordinate around $p$.
Given a meromorphic function on $C$, regular outside $p$, we can look at its ...
- 2,384
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
6
votes
1
answer
324
views
Almost complex manifold of dimension 2... locally isomorphic to ℂ?
I know that this is supposed to be standard, but I don't know how to search for it... hence the question:
Let $J$ be an almost complex structure on $M:=\mathbb R^2$, i.e., a $C^\infty$ section of $\...
- 43.2k
2
votes
0
answers
61
views
Criteria for a limit to be a proper function
This question is obviously broad; turning this broadness into something sharp is part of the problem.
Given a sequence of functions defined on a Riemann Surface $R$, valued in $\Bbb C^2$, what ...
- 779
2
votes
0
answers
117
views
automorphic form associated with Apollonian Gasket
In /Indra's Pearls/, it's mentioned one can associate automorphic forms with limit sets. Is there an explicit description of the one associated with the Apollonian gasket (up to some appropriate ...
- 975
6
votes
1
answer
259
views
Do analytic functionals form a cosheaf?
Let $X$ be a complex-analytic manifold.
Consider the sheaf of holomorphic functions $\mathcal{O}_X$ as a sheaf with values in the category of locally convex vector spaces. For $U\subseteq X$ open, we ...
- 890
7
votes
2
answers
590
views
Dependence of a solution of a linear ODE on parameter
Is the following theorem known, or can be easily derived from known results?
Consider the differential equation
$$w''-kz^{-1}w'=(\lambda+\phi(z))w,$$
where $k>0$ is fixed, $\lambda$ is a large (...
- 91.8k
4
votes
2
answers
984
views
Reference request: Oldest complex analysis books with (unsolved) exercises?
Per the title, what are some of the oldest complex analysis books out there with (unsolved) exercises? Maybe there are some hidden gems from before the 20th century out there. I am aware of the ...
4
votes
1
answer
268
views
On the roots of Bernoulli polynomials
Consider the Bernoulli polynomials denoted by $B_n(z)$. Now, start plotting the set of all (combined) complex roots $\mathcal{A}_N$ of $B_n(z)$, say for $n=1,2,\dots,N$ for some enough large $N$. It ...
- 43.1k
10
votes
3
answers
2k
views
Origin of term Ahlfors-David regular
Much of the literature on analysis in metric spaces makes use of an assumption called Ahlfors regularity or Ahlfors-David regularity. Let $q>0$. A metric space $(X,d)$ is Ahlfors(-David) $q$-...
- 565
9
votes
1
answer
321
views
Cauchy path integral as a linear operator: kernel and image?
Let $\mathcal O(\Omega)$ be the algebra of functions holomorphic on the open set $\Omega\subset\mathbb C$. For $\gamma$ a simple compact curve in $\mathbb C$ consider the linear operator given by path-...
- 5,428
2
votes
0
answers
219
views
Integral with product of two infinite sums
I am looking for references and results on integrals with product of two infinite sums:
$$S_1=\int_0^{\infty} \sum_{n=0}^{\infty} f(nx) \sum_{k=0}^{\infty} \overline{ g(kx)} dx$$
Above integral is ...
- 1,199
8
votes
0
answers
277
views
Cohomology of complex manifold vs cohomology of its complex submanifold
Let $X$ be a smooth complex analytic manifold. Let $Z\subset X$ be a smooth compact analytic submanifold. Let $A$ be a holomorphic vector bundle over $X$. Assume that
$$H^i(Z, A|_Z)=0 \mbox{ for any } ...
- 21.8k
4
votes
1
answer
269
views
An inequality of T. Carleman
I'm looking for the name and some references for the proof of the inequality below. I founded that is due to T. Carleman but no reference was given.
Let $f(z)$ be an analytic function on a subdomain $...
- 571
1
vote
1
answer
518
views
Zeros of Multivariate Complex Functions [need reference]
I am looking for a good accessible reference that would summarize properties of zeros of complex analytic functions.
For my purpose, it would be interesting to see a discussion on the following ...
- 671
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
5
votes
2
answers
279
views
Vector valued disc "algebra"
I am interested in a vector-valued form of the disc "algebra" (which in this setting is not in general an algebra, hence the scare quotes). Let $E$ be a Banach space, and let $A(\mathbb D,E)$ be the ...
- 18.7k
5
votes
0
answers
268
views
Reference Request on logarithm derivative of L-functions
I'm looking for references that show almost all Dirichlet characters $\chi \mod q$ satisfy
$$|\frac{L'}{L}(1+it, \chi)|=o(\log q)$$
where $t\in \mathbb{R}$ is fixed. I have been able to adapt a method ...
- 51
1
vote
1
answer
218
views
Generating series of rational$\times \exp($rational$)$
It is known that rational functions $f\in \mathbb C(x)$, $0$ not a pole, are the sum of generating series $\sum_{n\geq 0} a_nx^n$ where $(a_n)_n$ is solution of a linear recurrence with constant ...
- 5,428
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
6
votes
0
answers
163
views
Reference request: normal form of k-differentials and flat surfaces at a puncture
Let $f(z)$ be a holomorphic function defined on a punctured neighborhood of $z=0$ with non-essential singularity of degree $d$ at $0$ (namely, $f(z)=z^dh(z)$, where $h(z)$ is a holomorphic function ...
- 1,804
5
votes
1
answer
153
views
An estimate on deviation of two smooth tangent $J$-holomorphic curves
Take $\mathbb C^2$ with coordinates $(z,w)$. Suppose that $J$ is a $C^{\infty}$ almost complex structure on $\mathbb C^2$ such that the line $w=0$ is $J$-holomorphic and $J(0,0)$ is given by $(z,w)\to ...
- 14.3k
8
votes
1
answer
594
views
Reference for flatness in complex-analytic geometry
What is a good reference for flat morphisms of complex-analytic spaces? (The book by Grauert and Remmert doesn't treat them).
Topics I'm interested in: openness of flat maps, descent for coherent ...
user95222
2
votes
0
answers
320
views
Solution to algebraic equations over $\mathbb{C}$ and $\mathbb{C}[x]$
$t^n=a$, we get one solution to the equation:
$$t=e^{\frac{1}{n}\int^a_1 \frac{1}{x}}$$ generalizing this result by replacing the exponential with an elliptic modular function and the integral with ...
- 3,084
1
vote
1
answer
655
views
Series involving factorials
Playing around with this series for natural values of $a,b$, it appears that more generally for $c\in\mathbb N$, $$\sum_{k=0}^\infty \frac{ (a+k)! \ (b+k)!}{k!\ (a+b+c+ k+1)! }=\frac{a!\ b!\ (c-1)!}{...
- 13.4k
12
votes
2
answers
1k
views
Has there been further work on Bender-Brody-Müller approach to RH?
Earlier this year (April 4, 2017), a seemingly tantalizing approach of the Riemann Hypothesis based on ideas dating back to Hilbert and Pólya by Bender, Brody and Müller was made publicly available. I ...
- 6,976
2
votes
2
answers
296
views
Planar polynomial vector field for a harmonic pair of polynomials
Has the system of ODEs
$$\frac{dx}{dt}=P(x,y)\\
\frac{dy}{dt}=Q(x,y)
$$
been studied for the special case of the polynomials $P$ and $Q$ being a harmonic pair, i.e. the real and imaginary part of ...
- 7,127
4
votes
1
answer
116
views
Can iterates of a non-polynomial function be bounded by an exponential indefinitely?
Assume $f$ is an entire non-polynomial function of arbitrarily small exponential order ('zero'th order' if you're into calling it that). Is it possible that for all $n$ we have
$$|f^{\circ n}(z)| <...
user78249
4
votes
1
answer
230
views
Poles of an integral of a meromorphic function with toric poles
Suppose I have a meromorphic function in several variables $f(x_1,\ldots,x_k,y_1,\ldots,y_m)$ and I want to integrate along the torus $T^m$ given by $|y_1|=\cdots=|y_m|=1$. It is not true in general ...
- 3,752
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
1
vote
1
answer
177
views
Conformal mapping of multiply connected domains
I am studying conformal mappings of multiply connected domains. Most of the results that I can reach concern existence and uniqueness of such mappings, whereas I can not find anything satisfactory ...
- 11
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
10
votes
2
answers
417
views
zeros on the circle of convergence
In this question some experiments were used to conjecture that the zeros of partial sums of a series converging to a function with natural boundary on the unit circle were (weakly) converging to the ...
- 96.4k
4
votes
1
answer
150
views
Linearisation of complex $S^1$ actions at fixed points
Let $(U,x)$ be an open complex $n$-manifold (say an $n$-ball) with an action of $S^1$ by holomorphic transformations that fix $x$. How to prove that there is a neighbourhood $U_1\subset U$ of $x$ ...
- 14.3k
4
votes
3
answers
667
views
Regularity for the roots of (characteristic) polynomials with given multiplicity
A classical result states that roots of a polynomial are continuous functions of its coefficients.
This is, for exemple, a direct consequence of Rouché's theorem.
Using the implicit function ...
- 2,135
1
vote
1
answer
179
views
One trig "survives" a binomial summation: why?
I've seen many trigonometric identities but here is one that I encountered for which I did not find a reference.
In case you wonder where this came from, I was investigating certain $q$-series in ...
- 43.1k
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
4
votes
0
answers
157
views
Modulus of an annulus with a cut
Let $A_r$ be a complex annulus of modulus $r>0$ obtained from a $1\times r$ rectangle in $\mathbb C$ with vertices $A=0$, $B=r$, $C=r+i$, $D=i$, by identifying isomterically $AB$ with $DC$.
Let us ...
- 14.3k
5
votes
1
answer
164
views
Reference for result on partial sums of Taylor series
I remember seeing somewhere that whenever $f$ is a holomorphic function with radius of convergence at $z$, $0<R<\infty$ the following holds
$$\limsup_{n\to \infty}(\max_{|w-z|\leq \rho R} |S_n(f,...
- 295
4
votes
0
answers
157
views
Analytic maps $\varphi: \mathbb C^n\to \mathbb C^n$ with degenerate differentials
Let $B^n\subset \mathbb C^n$ be a unit ball with center $p$ . Let $\varphi: B^n\to \mathbb C^n$ be a complex analytic map such that $d\varphi$ has rank at most $n-1$ at $p$. I would like to know if ...
- 14.3k
14
votes
1
answer
3k
views
How is the "conformal prediction" conformal?
The question is clarified by Prof.V.Vovk. See his answer below for discussion.
Recently, early works of Gammerman, Vanpnik and Vovk[4] are rediscovered by Wasserman et.al[1] and proposed it as a ...
- 8,071
1
vote
0
answers
294
views
Can an entire function have every root function?
My question is an amalgamation of two previous questions. The first question I'd like to draw attention to is here. It asks whether there can exist a non trivial semigroup defined on $\mathbb{C}$
$$\...
user78249
7
votes
1
answer
248
views
Are there such things as non-trivial entire semigroups?
I'll state the theorem I am posing up front, and then explain why I think this theorem appears to be true. I am asking if anyone can prove it, or knows references to where it is proved. Please, ...
user78249