All Questions
Tagged with cv.complex-variables reference-request
271 questions
4
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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
2
answers
355
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On a variation of Hartogs' separate analyticity theorem
Let $f(z_1,z_2,\ldots,z_n)$ be a function on $\mathbf{C}^n$ such that for all $i$, the restriction
$$
[z_i\mapsto f(z_1,z_2,\ldots,z_n)]
$$
is a "rational function".
(added: to be precise ...
- 7,609
4
votes
1
answer
150
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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
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3
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687
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Finite covers of punctured Riemann surfaces
Let $X$ be a compact Riemann surface, i.e. compact smooth complex analytic (hence automatically algebraic) curve. Let $A\subset X$ be a finite subset, and $X_0:=X\backslash A$.
Let $Y_0$ be a smooth ...
- 21.8k
4
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1
answer
450
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Riemann mapping theorem with smooth boundary
This is closely related to the question here. The setup is that $U\subset\mathbb{C}$ is an open bounded simply connected domain with $C^\infty$ boundary. If $\phi:U\rightarrow\mathbb{D}$ is a ...
- 439
4
votes
2
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313
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Request for references in computational complex analysis
We know complex analysis is one of the most important branches of mathematics connecting myriad areas. It is replete with profound results and theorems and theorems. However, a good number of the ...
- 826
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1
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229
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Flat family with special fiber $\mathbb{C}\mathbb{P}^1$
Let $C=Spec \mathbb{C}[t]/(t^{n+1})$. Let $X$ be an algebraic (or complex analytic) scheme, flat over $C$ with the structure morphism $f\colon X\to C$. Assume that the special fiber is isomorphic to $\...
- 21.8k
4
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2
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177
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Ordinary generating functions with finitely many singularities at algebraic numbers are rational
I have a proof of the following fact related to ordinary generating functions, and I was curious if it was known, as it seems plausible it is classically known:
"Let $\lambda_1,\ldots, \lambda_k$ ...
- 1,429
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
4
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1
answer
206
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Super attracting fixed points have no fractional iteration
My question is really easy to state, but I'm having trouble hitting the final nail in the coffin in a proof of the result. The question concerns fractional iterations of holomorphic functions, for ...
user78249
4
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1
answer
191
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Integral of $\ln(1/|f|)$ for $f$ bandlimited
I came across the following assertion: if $f\in PW_\infty([-a,a])$, i.e. the Bernstein space of functions in $L^\infty(\mathbb{R})$ which are the Fourier transform of a distribution supported on $[-a,...
- 319
4
votes
1
answer
359
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How to classify the complex function with same natural boundary in complex plane? [closed]
There are complex functions with the same natural boundaries in the complex plane, but,they are different from each other. For example, there are lots of different lacunary power series with ...
- 3,084
4
votes
2
answers
373
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Abel–Plana formula with fractional offset
The Abel–Plana formula compares the sum $\sum_{n=0}^\infty f(n)$ to the integral $\int_0^\infty f(x)\,dx$,
\begin{equation}
\sum_{n=0}^{\infty}f\left(n\right)-\int_{0}^{\infty}f\left(x\right)dx=\frac{...
- 188k
4
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1
answer
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The cotangent sum $\sum_{k=0}^{n-1}(-1)^k\cot\Big(\frac{\pi}{4n}(2k+1)\Big)=n$
On the Wolfram Research Reference page for the cotangent function (https://functions.wolfram.com/ElementaryFunctions/Cot/23/01/), I saw the following partial sum formula
$$\sum_{k=0}^{n-1}(-1)^k\cot\...
- 195
4
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2
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700
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Basic questions on the Hilbert scheme/ Douady space
Let $X$ be a complex projective scheme (resp. complex analytic space). The Hilbert scheme (resp. Douady space) parameterizes closed subschemes (resp. complex analytic subspaces) of $X$. More precisely,...
- 21.8k
4
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1
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269
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An inequality of T. Carleman
I'm looking for the name and some references for the proof of the inequality below. I founded that is due to T. Carleman but no reference was given.
Let $f(z)$ be an analytic function on a subdomain $...
- 571
4
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2
answers
378
views
Comparing two Delaunay tessellations on a hyperbolic surface
Let $S$ be a closed hyperbolic surface (i.e. a compact Riemann surface of genus $\geq 2$) and let $P=\{p_1,\ldots,p_m\}$ be a non-empty finite subset of $m$ points in $S$. Let $\pi:\mathbb H\...
- 838
4
votes
1
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116
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Can iterates of a non-polynomial function be bounded by an exponential indefinitely?
Assume $f$ is an entire non-polynomial function of arbitrarily small exponential order ('zero'th order' if you're into calling it that). Is it possible that for all $n$ we have
$$|f^{\circ n}(z)| <...
user78249
4
votes
1
answer
230
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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
4
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1
answer
1k
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definition of accretive operator
A relation T with domain and range in a Hilbert space is said to be accretive if the
transformation $ (T − \lambda)/(T + \bar \lambda\ ) $
with domain and range in the Hilbert space is contractive for ...
- 2,106
4
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1
answer
267
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variation of the obstacle in the obstacle problem
Suppose $D \subset \Bbb C$ with smooth boundary. Let $f \in C^{1,1}(D)$. Let $\varphi$ be the supremum of all members in the set
$$\lbrace g \in C^{\infty}(\overline{D})| g \ is \ subharmonic \ and \...
- 1,201
4
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1
answer
535
views
A converse of the maximum modulus Theorem
W. Rudin in Real and Complex Analysis (262) mentioned that
Theorem Suppose $M$ is a vector space of continuous complex functions
on the closed unit disc $\bar U$, with
the following properties:
(a) $...
- 43
4
votes
0
answers
168
views
Explicit bounds on gaps between zeros of $\zeta^\prime(s)$
In $\S$9.1 of "Theory of the Riemann Zeta Function", Titchmarsh uses Borel-Carathéodory and Hadamard Three Circles to show that every circle of radius 6 and center $3+iT$ contains a zeros of ...
- 11.1k
4
votes
0
answers
279
views
Order of growth of $\left|\frac{1}{\zeta’(\rho)}\right|$ as $\Im(\rho)\rightarrow\infty$?
Let $\zeta$ denote the Riemann zeta function, and let $\rho\in\mathbb{C}$ be a variable that takes its values among the zeros of the zeta function, so that $\zeta(\rho)=0$, and write $\rho=\sigma+it$. ...
- 1,018
4
votes
0
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159
views
Correct way to extend a sequence defined on the naturals into the complex plane
Preamble
Sequences $a_n$ defined on the natural numbers are clearly not uniquely interpolated by only one function. In particular, given an interpolation $f(n) = a_n$, then $f(n) + \sin(2\pi n)$ is ...
- 1,730
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173
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On the best constant for Carleson's embedding theorem
In "Interpolations by bounded analytic functions and the corona problem", Carleson introduced Carleson measures (for Hardy spaces) and proved the famous embedding theorem according to which $...
- 81
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0
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179
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As increasingly higher degree terms are added to a "random" polynomial, how fast do the roots approach the unit circle?
As increasingly higher degree terms are added to a "random" polynomial, the roots of a polynomial can be proven to approach the unit circle. For example, see the MathOverflow question Why ...
4
votes
0
answers
109
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 ...
- 219
4
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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
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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
4
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0
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157
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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
4
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0
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122
views
Complex L^1 spaces; reference request
I have been doing a fair amount of research into a complex analytic modified version of the Mellin transform. I have hit a few roadblocks, and am hoping there may already be literature on the subject. ...
user78249
4
votes
0
answers
287
views
Is there a coordinate free proof of the Morrey--Kohn--Hormander identity?
The Morrey--Kohn--Hormander identity is the key to proving vanishing/existence results on bounded pseudoconvex domains in $\mathbb{C}^n$, or more generally, Stein domains. See, for instance, the ...
- 18.7k
4
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0
answers
715
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some questions about properties of harmonic measure
The original post
The following argument appears in a paper of Nazarov (Lemma 1.2) "Local estimates for exponential polynomials and their applications to inequalities of the uncertainty principle ...
- 1,795
3
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3
answers
406
views
When can Hodge filtrations (decompositions?) be described explicitly in terms of periods?
It seems that there is no chance to explain the Hodge theory (to students) in an hour or so. Yet do there exist any cases when the Hodge filtration (or the Hodge decomposition) of the cohomology of a ...
- 16.9k
3
votes
3
answers
477
views
Undecidability and holomorphic functions (Reference request)
The goal of this question is to recall a certain mathematical fact -not in my field- that I was once briefly told and that I have fogotten, and also to collect similar results.
The fact, I think, ...
- 23.3k
3
votes
2
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1k
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A question around Liouville's theorem
Liouville's theorem of complex analysis states that a bounded entire function is constant. I am trying to understand if a sort of converse holds in the following sense: consider a closed set $S \...
- 1,407
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2
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243
views
Is every planar bounded $C^2$ domain finitely connected?
Let $\Omega \subset \mathbb R^2$ be a bounded $C^2$ domain. Is $\Omega$ then finitely connected? As I learned recently a domain in $\mathbb R^2$ is finitely connected iff “[its] complement has ...
- 313
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2
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625
views
Intuition behind $\zeta(2) = \frac{\pi^2}{6}$ [closed]
In literature, there are many proofs of the well-known result $$\zeta(2) = \frac{\pi^2}{6}.$$
However, as far as I know, they do not offer an intuitive explanation of why this result should be true. ...
user60665
3
votes
2
answers
129
views
Vector bundles over a Stein space are projective
It is a "well known" fact that
locally free sheaves over a Stein space $X$ are projective as $\mathcal{O}_X$-modules
(see e.g. just after Lemma 1.6 in O'Brian-Toledo-Tong's "The trace ...
- 1,109
3
votes
1
answer
178
views
Analytic continuation to the Mittag-Leffler star using Mittag-Leffler summation
This is a reference request for a theorem I thought I had read in a book by Steven Krantz, but I can no longer find it.
Searching for Mittag-Leffler star, I can find references to the following result....
- 1,084
3
votes
1
answer
249
views
Is Toeplitz operator on the Bergman space bounded iff its symbol is bounded?
It is fairly well known that if $T_\varphi$ is a Toeplitz operator on the Hardy-Hilbert space, then $\lVert T_\varphi \rVert = \lVert \varphi \rVert _{\infty}$.
Now, if $\varphi \in L^\infty (\mathbb ...
- 151
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2
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312
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Complex Hermite polynomial orthogonality on weighted space
Consider the "probabilist's" Hermite polynomials given by $$H_n(x)=(-1)^ne^{\frac{x^2}{2}}\partial_x^ne^{-\frac{x^2}{2}}.$$
These polynomials trivially extend to functions of $w\in\mathbb{C}$...
- 854
3
votes
1
answer
326
views
Holomorphic homotopy conjecture
Let $X$ be a smooth projective variety over the complex numbers, and let $\text{Coh}(X)$ be the category of coherent sheaves on $X$. Consider the dg-category $\text{Perf}(X)$ of perfect complexes on $...
- 31
3
votes
1
answer
458
views
Limit of an infinite series with quadratic arguments
I have encountered a limiting process on some infinite series. So, I would like to ask:
QUESTION. Assume $n$ is an even positive integer. Is this true?
$$\lim_{r\rightarrow1^{-}}\sum_{j=1}^{\infty}\...
- 43.1k
3
votes
1
answer
326
views
Polynomial and rational approximation of continuous functions in $\mathbb{C}$
I am wondering what the state of the art is for polynomial and rational approximations to continuous/holomorphic functions in $\mathbb{C}$. The particular domains of interest are the closed unit ball $...
- 465
3
votes
1
answer
214
views
Holomorphic vector fields with a non-degenerate isolated zero
Let $v$ be a holomorphic vector field defined in a neighbourhood of $0$ on $\mathbb C^n$ with an isolated zero at $0$. Let $\sum_{i,j}{a_{ij}}z_i\frac{\partial}{\partial z_j}$ be the linear term of $...
- 14.3k
3
votes
1
answer
219
views
Reference request: The transform of a bounded random variable has a zero in the complex plane
Together with coauthors I'm working on a paper where we use the following Proposition:
If a real-valued random variable $X$ has bounded support, then except in the trivial case that $X$ has all ...
- 5,492
3
votes
1
answer
171
views
A question on preimage of a locally injective meromorphic function
Let $f: \mathbb{C} \rightarrow \mathbb{C}$ be a meromorphic function such that $f'(z) \ne 0$ for all $z \in \mathbb{C}$. Let $\Gamma \subset \mathbb{C}$ be a curve which has no self intersections. If ...
- 1,350
3
votes
1
answer
361
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bounded analytic function as a power series
Suppose $$f(x)=\sum_{k=0}^\infty a_k\frac{(i x)^k}{k!}$$ where $$a_k=k!\int_0^1 p_k(y_{k-1})\int_0^{y_{k-1}}p_{k-1}(y_{k-2})\cdots \int_0^{y_1}p_1(y_0) dy_0\cdots dy_{k-2}\;dy_{k-1}$$ for functions $...
- 587