All Questions
Tagged with cv.complex-variables analytic-functions
61 questions
0
votes
1
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124
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Holomorphic functions of certain blow up at origin
Suppose that $D=\{z\in \mathbb C\,:\, |z|\leq 1\}$ and let $f$ be holomorphic on $D\setminus\{0\}$ such that $|f(z)|\leq e^{\frac{1}{|z|}}$ for all $0<|z|\leq 1$ and assume additionally that $\lim\...
- 4,115
0
votes
0
answers
56
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Complexity of evaluation of analytic functions
Given an analytic function $f(x)$ (say as combination of elementary functions and operators), is it possible to compute $n$ first bits of the value of the function on the whole interval $[a, b]$ ...
- 67
2
votes
1
answer
99
views
A question on Bloch functions
Let $\mathcal{B}(\Delta)$ be the space of Bloch functions in the unit disk $\Delta$. For any $f\in \mathcal{B}(\Delta)$, we define the Bloch norm by
$$
\|f\|_{\mathcal{B}}=\sup_{|z|<1}|f'(z)|(1-|z|^...
- 1,706
3
votes
1
answer
231
views
Prescribe the type of an entire function which inverse zeros are summable
According to Lindelöf's theorem, given points $z_i\in \mathbb C\setminus \{0\}$ ordered by increasing modulus with possible repetitions, we can define a function
$$
f(z)=\prod_{n=1}^\infty (1-z/z_n)e^{...
- 1,352
1
vote
1
answer
190
views
Generalisation of Paley–Wiener type results for unbounded sets
Do you know an unbounded open set $A\subset \mathbb{R}^d$, $d\geq 2$ with the following property: if some integrable function $f$ on $\mathbb{R}^d$ has its Fourier transform vanishing on $A^c$ and all ...
- 1,352
3
votes
1
answer
111
views
When entire or meromorphic map of finite type restricts to a Galois covering?
Suppose that $f \colon \mathbb{C} \to \mathbb{C}$ is an entire map of finite type, i.e., with finitely many singular values. Then we can consider the restriction $f| \mathbb{C} \setminus f^{-1}(S_f) \...
- 41
3
votes
1
answer
200
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Complex analytic function $f$ on $\mathbb{C}^n$ vanish on real sphere must vanish on complex sphere
I'm considering a complex entire function $f$: $\mathbb{C}^n\to \mathbb{C}$. Suppose $f=0$ on $\{(x_1,\cdots,x_n)\in\mathbb{R}^n:\sum_{k=1}^n x_k^2=1\}$. I want to prove
$$f=0\textit{ on } M_1=\{(z_1,\...
- 421
3
votes
1
answer
136
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Growth of preimages of singular values of finite type entire map
Let $f\colon \mathbb{C} \to \mathbb{C}$ be an entire map having precisely two distinct singular values $w^1$ and $w^2$. If $w^i$ has infinitely many preimages under $f$, we write $(z_n^i)_{n \in \...
- 41
1
vote
1
answer
106
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Characterizing the unimodular functions from the closed disk $\overline{\mathbb{D}}$ to $\mathbb{C}$ with constraints
Let $\mathbb{D}$ be the open disc. It is well known that if $f:\mathbb{D}\to\mathbb{C}$ is analytic, continuous on the boundary, and is unimodular (say with a finite number of zeros) then $f$ is a ...
- 167
7
votes
1
answer
503
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Combinatorial consequences of de Branges's Theorem?
I'm usually not a proponent of the mentality “here is a tool, what results can we prove with it?” (as I prefer to start at the other end with a well-motivated problem), but this famously entertaining ...
- 191
3
votes
0
answers
75
views
Separate holomorphicity implies holomorphicity on analytic varieties
Suppose that $M$ and $N$ are two complex analytic varities and suppose that $f\colon M\times N \to \mathbb{C}$ is a map. Further assume that $f$ is such that for every $p\in M$ the map $f(p,\cdot)\...
- 513
2
votes
0
answers
110
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Real analytic periodic function whose critical points are fully denegerated
I have asked this question on MathStackExchange. My question: is there any non-constant real analytic function $f:\mathbb{R}^n\rightarrow\mathbb{R}$ such that, $$\nabla f(x_0)=0 \Rightarrow \nabla^2 f(...
- 21
2
votes
1
answer
203
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Existence of the special entire Hardy space function with infinitely many zeros in the strip
Question. Does there exist an entire function $h$ satisfying three following assertions:
$h$ belongs to the $H^2$ Hardy space in every horizontal upper half-plane;
$zh - 1$ belongs to $H^2(\mathbb{C}...
- 577
0
votes
1
answer
200
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A holomorphic function in the open unit disk satisfying certain properties
Does there exist a function which is holomorphic in $|z|<1,$ continuous in $|z|\leq1$ and such that the series $\sum |a_n|$ is divergent, where $a_n$'s coefficients in the Taylor series expansion ...
- 165
3
votes
0
answers
61
views
lower bound for zero multiplicity of function formed from determinant of functions
I have a family of single-variable analytic functions, $D(z)$, formed as follows.
Let $L$, $n$ and $T_{0}$ be positive integers, $\varphi_{1},\ldots, \varphi_{L}$ be analytic functions in ${\mathbb C}$...
- 31
4
votes
2
answers
566
views
How to compute $\sin(\frac{d}{dx})f(x)$?
Assuming $f(x)=e^{-x^2}$ for $x$ in $[-10,10]$, I have tried the following:
Fourier transform $\mathcal{F}$: $\frac{d}{dx}$ can be diagonalized as $\mathcal{F}^{-1} i\omega \mathcal{F}$. Therefore, $\...
- 350
2
votes
1
answer
262
views
On a lemma of Łojasiewicz in complex analysis of one variable
Context. The question arises from my former question on the remainder of a power series. Precisely, I was trying to understand if the boundary behavior of power series considered by Ricci in his paper ...
- 6,357
4
votes
1
answer
150
views
Quantitative analytic continuation estimate for functions small except on a small set
This question arises as a variation of this question, which was helpfully answered in the negative. It turns out that for my application, a substantially weaker conjecture suffices, which fails to be ...
- 324
3
votes
3
answers
427
views
Quantitative analytic continuation estimate for a function small on a set of positive measure
The following conjecture about analytic functions arose as a way to show the asymptotic growth for certain PDE solutions. As I am unfamiliar with any results of this type, I thought I'd ask here.
In ...
- 324
2
votes
0
answers
129
views
Existence of analytic function in disk algebra [closed]
Does there exist an analytic function $f\in A(\mathbb{D})$, where $A(\mathbb{D})$ is the disk algebra, such that $f(0)=0$ and the real part of $f(z)$ is strictly positive?
- 149
0
votes
1
answer
122
views
Existence of an eigenpair for d-bar operator in the unit disck
Let $\overline{\partial}=\frac{1}{2}(\partial_{x}+\textrm{i} \,\partial_y)$ and let $D$ be the unit disc in the complex plane. For each $\lambda \in \mathbb C$, consider the problem:
$$ \overline{\...
- 4,115
12
votes
2
answers
489
views
Behavior of $|f'(z)|/(1+|f(z)|^2)$ as $|z| \rightarrow \infty$?
Let $f(z)$ be an entire holomorphic function in $\mathbb{C}$, and consider the real-valued function
$$g_f(z)=\frac{|f'(z)|}{1+|f(z)|^2}.$$
If $f(z)$ is a polynomial, then it is easy to prove that $\...
- 1,350
0
votes
1
answer
103
views
Measure of preimage of Jordan disk under entire map
Let $f\colon\mathbb{C} \to \mathbb{C}$ be an entire map. For simplicity assume that $f$ is of finite type, i.e., it has finite set $S(f)$ of singular values. $S(f) \subset \mathbb{C}$ is a minimal (...
- 41
10
votes
1
answer
349
views
On a variant of Carlson’s theorem
My question is on whether or not there exists some monotone strictly decreasing sequence of positive numbers $c_1>c_2>\ldots$ such that given any $f$ which is a uniformly bounded holomorphic ...
- 4,115
8
votes
2
answers
495
views
Literature on non-Archimedean analogues of basic complex analysis results
It looks like there is some literature out there on what might be called 'non-Archimedean complex analysis' e.g. Benedetto - An Ahlfors Islands Theorem for non-archimedean meromorphic functions and ...
11
votes
1
answer
582
views
An extension of the Carlson's theorem in complex analysis
For the statement of Carlson's theorem please see,
https://en.wikipedia.org/wiki/Carlson%27s_theorem.
There is an extension of Carlson's theorem that says that the condition that $f$ needs to vanish ...
- 4,115
4
votes
1
answer
636
views
Existence of a smooth compactly supported function
Let $U$ be a bounded domain in $\mathbb R^n$. Does there exist a smooth function $f$ with compact support in $U$ such that:
$$ \| f\|_{W^{k,\infty}(U)} \leq (k!)^{2-\epsilon},$$
for some $\epsilon>...
- 4,115
6
votes
1
answer
570
views
Does it follow that $F^{(1)}(z)=F^{(2)}(z)$ for all $z \in \mathbb H$?
Let $\mathbb H= \{z \in \mathbb C\,:\, \textrm{Re}\,(z)\geq 0\}$ and for $j=1,2$ suppose that $F^{(j)}:\mathbb H\to \mathbb H$ is defined via
$$ F^{(j)}(z) = \sum_{k=1}^{\infty} \frac{a_k^{(j)}}{z+\...
- 4,115
1
vote
0
answers
112
views
A problem related to analytic function
Let, $z,w\in \mathbb{C}$. Let, $f(z)$ be an analytic function in $|z|<1$. Define, $f(z)= g(w)$ where $g(w)$ is analytic function in $\Re(w)>1/2$ and $w=\frac{1}{1+z^2}$ .
Question Prove that $$\...
user245973
2
votes
0
answers
119
views
An analytic function, asymptotically expandable in a Dirichlet series, is the sum of this series
Let there be a function $F(s)$ that is analytic in some half-plane $\sigma>\sigma_0$ (where $s=\sigma + it $). Let the function $F(s)$ have an asymptotic expansion of the form $F(s)\sim\sum\limits_{...
1
vote
2
answers
416
views
Global theory of holomorphic functions [closed]
I am trying to develop a theory explaining analytic continuation of a holomorphic function $f(z)$ defined on an open set $D \subset \mathbb{C}$. Recently, I was looking at the last chapter of Lars ...
- 3,699
1
vote
0
answers
195
views
Is a mixture of real analytic functions again analytic?
Let $$h : \mathbb{R}^2 \to \mathbb{R}^+.$$
Suppose that for each $x$, $h(x, y)$ is a real analytic function of $y$.
Let $\mu(dx)$ be a finite measure on $\mathbb{R}$, and for each $y$, suppose that
$$...
- 103
3
votes
1
answer
253
views
Looking for a sequence of analytic functions with strange behaviour
Let $K_1 \subsetneq K_2$ be two non-empty compact sets and let $D = (d_n)_{n \in \mathbb{N}}$ be a dense sequence on $K_2\smallsetminus K_1.$ Consider $f_n : \mathbb{C}\smallsetminus K_1 \rightarrow \...
- 355
1
vote
0
answers
84
views
A Riemann Hilbert problem on the unit square
Let $p_0=0$, $p_1=1$, $p_2=1+i$ and $p_3=i$ be the four vertices of a square $Q$ on the complex plane $\mathbb C$.
Let $f \in C^{\infty}_c((0,1))$ and consider the following Riemann-Hilbert problem on ...
- 4,115
-3
votes
1
answer
208
views
Conformal map from a 7-sided polyhedron to a square pyramid
I have a right-angled square pyramid, $A$, whose height and base-length are $l$. By 'right-angled', I mean that the apex of $A$ lies vertically above one of the vertices in its base. Now supposed I ...
- 167
1
vote
0
answers
80
views
Analytic continuation of function of two complex variables
Consider $f(z_1, z_2)$ a function of two complex variables, symmetric in its arguments $z_1$ and $z_2$. Consider the regions:
\begin{eqnarray}
\mathcal{R}_1= \{ (z_1, z_2) \in \mathbb{C}^2: Re(z_1)>...
- 93
3
votes
1
answer
425
views
Possible condition for a many variable holomorphic map to be locally surjective
Suppose $a \in \mathbb C^n$, $U$ is a neighbourhood of $a$, and $f: U \to \mathbb C^n$ is analytic. Let $b = f(a)$ and suppose also that $f^{-1}(b) = \{a\}$. Must the image of $f$ contain a ...
- 373
2
votes
1
answer
1k
views
Essential singularity [closed]
In shaum's outline complex analysis,definition of essential point is:
An isolated singularity that is not pole or removable singularity is called essential singularity
Now in the same book there is an ...
7
votes
0
answers
461
views
On a paper of Alain Connes entitled 'Around Wilson's Theorem '
A relatively recent paper Alain Connes - Around Wilson's theorem
introduced the function
$$
S(n,x ) = \sum_{i=1}^n \sin^2\Bigl(\frac{(i-1)! x}{i}\Bigr).
$$
In the same paper, he proved that the ...
user145059
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{\...
3
votes
3
answers
1k
views
Steepest descent integration in several dimensions
The method of steepest descent provides an asymptotic approximation for integrals of the form:
$$I = \int_C \exp(M f(z))\mathrm dz$$
for large positive $M$, where $f(z)$ is analytic in the region of ...
- 884
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
11
votes
1
answer
1k
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Dual of the space of all bounded holomorphic functions
Let $\mathbb{B}$ be the open unit ball in $\mathbb{C}^n, n\geq 1$ and let $H^\infty (\mathbb{B})$ be the space of all bounded holomorphic functions on $\mathbb{B}$. It is well known that $H^\infty (\...
- 211
0
votes
0
answers
243
views
Sequence of analytic functions
Let $f_k$ be a sequence of rational functions analytic in the discs $\{ |z| < 1 + \epsilon_k\}$ (with some $\epsilon_k > 0$), which converge to an analytic function $f$ in every point $|z| < ...
- 163
8
votes
1
answer
386
views
Uniqueness theorem for conformal mapping
Let $f$ and $g$ be analytic functions in the unit disk $D$, continuous in the closed disk and locally univalent, $f'(z)\neq 0,\; g'(z)\neq 0,\; z\in D$.
Assume that each has only finitely many ...
- 91.8k
2
votes
1
answer
274
views
Generalisation of Chebyshev series to arbitrary sets
A Lipschitz continuous function $f : [-1,1] \to \mathbb{C}$ has a unique representation as a series in terms of the Chebyshev polynomials $T_k$,
$$
f(x) = \sum_{k = 0}^\infty a_k \, T_k(x)
\qquad
\...
- 243
2
votes
2
answers
206
views
Zeroes of trigonometric-like function
Consider a function $f(z)=\cos(z)\cosh(az)+\sin(z)\sinh(bz)$ for $z\in \mathbb{C}, a,b \in \mathbb{R}$. Denote $D\subseteq \mathbb{R}^2$ being the set of such pairs $(a,b)$ of parameters so that NOT ...
- 1,550
2
votes
1
answer
186
views
Modulus of Continuity for an Analytic Function on an Ellipse
Given $f\in C^{\infty} (E)$, where $E\subseteq \mathbb{C}$, define $E_{\rho} \subseteq \mathbb{C}$ as the maximal ellipse with foci at $\{-1,1\}$ where $f$ is analytic, and semi-minor + semi-major ...
- 3,574
3
votes
1
answer
2k
views
Extensions of Real Analytic to Holomorphic Functions in One & Several Variables: References?
A problem I'm working on requires the application of Cauchy's estimate for the modulus of the coefficients of a holomorphic function's power series representation, but the original functions with ...
- 139
1
vote
1
answer
181
views
Interesting property of analytic functions
Let $f:(t_0-\varepsilon, t_0+\varepsilon)\to\mathbb{C}$, be an analytic application, such that: $f(t)=0\Longleftrightarrow\ t=t_0$.
Is it true that there is an analytic function $g:(t_0-\varepsilon, ...
- 1,759