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16
votes
1answer
109 views

Rearrangements of a power series at the boundary of convergence

Take some power series $f(z) = \sum a_n z^n$ with a finite non-zero radius of convergence. I can rearrange the terms of the series, say, to get a different infinite series $$f_{\sigma}(z) = ...
0
votes
1answer
68 views

A question for hyperbolic metric in the proof for Bohr's lemma

Recently I was reading an interesting proof for Bohr's lemma by the tool of hyperbolic metric, however I have a following question: Given a holomorphic map $f$ on $D$, $f(0)=0$, and $|f|<1$ on ...
-5
votes
1answer
78 views

computing integral of dz/(z+1) on unit circle [closed]

I guess it must be a simple matter in complex analysis, but I would like to compute the the following integral: $$ \oint_C\frac{dz}{z+1} $$ and $$ \int_C\frac{dz}{z+1} $$ where $C=\{z\in \mathbb{C}: ...
1
vote
0answers
44 views

Intuitive explanation for Hardy-Littlewood maximal function

I came across the Hardy-Littlewood maximal function in an analysis course. Could someone help me intuitively understood what the purpose of this map is, and why it is useful? Thank you. Regards ...
-1
votes
1answer
118 views

Holomorphic maps on $\mathbb{R}^{n}$ (for n not necessarily even)

Edit according to the comment of user36931 I remove the "motivation" from the previous version and I add an statement to the first question We consider the following two classes of smooth maps on ...
3
votes
1answer
220 views

Good book on analytic continuation?

This is a cross-post from MSE. For my Bachelor's thesis, I am investigating divergent series summation methods. One of those is analytic continuation. There are quite a few books on complex analysis ...
-2
votes
0answers
14 views

Why pdx+qdy is locally exact in U iff the integration of pdx+qdy along R is 0,where R is any rectangle with sides parallel to the axes in U [migrated]

Why is pdx+qdy locally exact in U iff the integration of pdx+qdy along R is 0, where R is any rectangle with sides parallel to the axes in U? This is a problem from Complex Analysis by Ahlfors, page ...
5
votes
1answer
179 views

Is there a holomorphic Morse-Bott lemma?

It asks for a generalization of the question in the post Normal form for a holomorphic Morse function Suppose $f$ is a holomorphic function on a complex manifold $M$ which has Bott type critical ...
1
vote
1answer
134 views

A family of examples of (Brody) hyperbolic surfaces

Let $P(x)$ and $Q(x)$ be two general polynomial of the same degree $d$ (d can be arbitrary large). Consider the surface $S : z^2 = P(x) Q(y)$ in the projective space $\mathbb{P}^3$. I want to prove ...
0
votes
2answers
184 views

Green's function - Hyperbolic Riemann surface

A Riemann surface is said to be: -Potential-theoretically hyperbolic if it has a non-constant bounded subharmonic function. -Poincaré hyperbolic if it is covered by the unid disk. Are this ...
18
votes
0answers
825 views

When does a modular form satisfy a differential equation with rational coefficients?

Given a modular form $f$ of weight $k$ for a congruence subgroup $\Gamma$, and a modular function $t$ with $t(i\infty)=0$, we can form a function $F$ such that $F(t(z))=f(z)$ (at least locally), and ...
13
votes
2answers
758 views

Who first defined _simply connected_, reference?

The following definition is due to Donald J. Newman: A connected open subset $D$ of the plane $\mathbb C$ is simply connected if and only if its complement $\widetilde D = \mathbb C \setminus D$ ...
2
votes
0answers
137 views

Separating the eigenvalues of a Hermitian matrix with a special block structure

I have a square matrix $J \in \mathbb{C}^{2n \times 2n}$ where, $J=\begin{pmatrix} A&B \\\bar{B} & A \end{pmatrix}$ $A \in \mathbb{R}^{n \times n}$ and is ${\bf diagonal}$. $B \in ...
-1
votes
1answer
124 views

Order of the zero of a meromorphic function under the action of $Gal(\mathbb{C},\mathbb{Q})$

Let's take $X$ a Riemann surface as an algebraic curve in $\mathbb{P}^n$. The group $Gal(\mathbb{C},\mathbb{Q})$ (automorphisms of $\mathbb{C}$ which act as the identity on $\mathbb{Q}$) acts on $X$ ...
9
votes
0answers
145 views

Reverse mathematics of meromorphic functions on Riemann surfaces

Various sources touch briefly on the reverse mathematics of measure theory and complex analysis. But I have found none on the uniformization theorem for Riemann surfaces or the existence of ...
1
vote
1answer
68 views

Polynomials are dense in $A_{B(0,1)}$

Let $D(0,1)$ be the disk of center 0 and radius 1 and call $A_{D(0,1)}= \{ f:\overline{D(0,1)} \rightarrow \mathbb{C} : f \text{ is continuous and } f|_{D(0,1)} \text{ is holomorphic} \}$. Can ...
0
votes
1answer
141 views

On complex exponential sum estimation

Let $c>0$ a real number, let $N$ a large natural number and let $e\left(x\right):=e^{2\pi ix}$. Is it true that $\forall k\in\left[1,\,2N\right]$, $k$ natural number, that ...
2
votes
2answers
428 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 ...
8
votes
1answer
794 views

If $f$ is $C^{\infty}$ and $f^2$ is analytic, then $f$ is analytic

Assume that $f:\mathbb{C}^n\rightarrow \mathbb{C}$ is a $C^{\infty}$ function such that $f^2$ is (complex) analytic. Then one can show that $f$ is analytic. (Note: Liviu Nicolaescu and Alexandre ...
4
votes
4answers
260 views

Extending a function from $\mathbb{Q}$ to the upper half plane $\mathbb{H}\cup\mathbb{Q}\cup\{i\infty\}$

Define the extended upper half plane $$\overline{\mathbb{H}}:=\{z\in\mathbb{C}: \mathrm{Im}(z)>0\} \cup \mathbb{Q} \cup \{i\infty\}.$$ To what extent can an arbitrary function on the rationals ...
19
votes
2answers
449 views

Which smooth compactly supported functions are convolutions?

If $f,g$ are smooth functions with support in the interval $[-r,r]$ for some $r>0$, then their convolution $f*g$ is smooth with support in $[-2r,2r]$. My question is about the converse: Given ...
5
votes
1answer
221 views

Asymptotic value of a multivariate integral

The following question is a simple case of a type of problem that occurs in combinatorial enumeration problems. Define $$F(x_1,\ldots,x_n) = \frac{1}{(2\pi)^{n/2}}\exp\biggl( -\frac12\sum_{j=1}^n ...
0
votes
1answer
57 views

Cardinality of Picard exceptional values of holomorphic function without an isolated singularity

Picard's theorem says that if $f\colon D\to \mathbb{C}$ is an entire nonconstant function or holomorphic with an essential isolated singularity, then $\mathbb{C}\setminus f({D})$ is either empty or a ...
4
votes
1answer
187 views

Analytic continuation of a multiple contour integral

Let $W(t_1,\dotsc,t_n)$ a holomorphic function on some connected open set $U$ of $\mathbb C^n$. Let $\mathbf t^{(0)}$ a point of $U$. Assume that there exists a cycle $\gamma$ in $\mathbb C^m$ and a ...
0
votes
1answer
81 views

Looking for methods/results for explicitly bounding iterations of rational functions

In Theorem 2.6.4 of Beardon's book, "Iteration of Rational Functions", he states the values for the first two coefficients of an iterated power series. That is, suppose that $$ ...
2
votes
0answers
208 views

Convergence of certain L-series

Suppose $|a_{n}| \leq 1$ completely multiplicative function assuming real values. Suppose further that, $ L(s)=\sum_{n} \frac{a_{n}}{n^s} $ may be continued analytically to the left of $s=1$ a bit ...
47
votes
32answers
9k views

Demystifying complex numbers

At the end of this month I start teaching complex analysis to 2nd year undergraduates, mostly from engineering but some from science and maths. The main applications for them in future studies are ...
7
votes
3answers
291 views

Summation of a series

I would like to sum the series $$ \sum_{n=0}^\infty \frac{1}{(1+a^2 (n+1/2)^2) ^{3/2}} . $$ It arose when trying to perform a calculation on superconductivity. In particular I am interested in its ...
0
votes
1answer
131 views

The dual space of the Dirac measures on an Abelian group

Let $G$ be a Hausdorff locally-compact Abelian group and $L^2(G)$ the Hilbert space of two-integrable complex functions on the group. Question. What would be natural vector space $\mathcal{R}$ ...
3
votes
1answer
139 views

A homogeneous but slightly asymmetric inequality

I need to prove the following inequality: for any $Z=(z_1,\dots,z_l)\in\mathbb{C}^l$ for any $p\geq 2$ and $l\geq 2$ \begin{equation} \left|\left|\sum_{j=1}^l ...
2
votes
0answers
51 views

Question concerning Mellin transforms

I've recently come across a result I've been trying to generalize. Say that $\phi(\sigma \pm iy) < Ce^{\frac{\pi}{2}|y|}$ in the strip $a < \sigma < b$ then then the following integral is ...
0
votes
1answer
193 views

A question on the area of the unit disc w.r.t. a complete conformal metric

Question: Let $\Delta$ be the unit disc in $\mathbb{C}$ and $\rho(z)|dz|^2$ be a complete conformal metric on $\Delta$ where $\rho(z)$ is continuous on $\Delta$. Let $a$ be the infimum of $p ...
0
votes
1answer
580 views

When may Function (meromorphic) be expanded as power series with coefficients of integers

Let F be meromorphic function,with what properties may it expanded as power series with coefficients of integers in such a form: $$F=\sum_0^{\infty}a_i x^i,a_i\in \mathcal{N} \bigcup 0,\exists M ...
12
votes
2answers
361 views

Spectra of elements of a Banach algebra and the role played by the Hahn-Banach Theorem.

This problem was posed on Math StackExchange some time ago, but it did not garner any solutions there. I think that it is interesting enough to be posed here on Math Overflow, so here it goes. Let $ ...
3
votes
1answer
237 views

Series of the inverse quadratic trinomial

Maybe it's a very simple question, but I have a problem with the following series $$\sum\limits_{n=1}^{\infty}\frac{1}{n^2+pn+q},$$ where $p, q \in \mathbb{R}$. I know about five ways how to calculate ...
1
vote
1answer
187 views

Some questions about inner functions

To avoid trival cases, we assume that $f$ is neither a constant nor a finite Blaschke product. Two celebrated theorems of Frostman say that $f_a(z)$ is actually a Blaschke product for every ...
0
votes
1answer
89 views

Use of Jensen's inequality on a Riemann surface

Let $f:\mathbb{C}\to \mathbb{C}$ be entire and consider the composite function $g(z):=f(\sqrt{z^2 - 1})$ on $\mathbb{C}\setminus \big ((-\infty , -1]\cup [1,\infty )\big )$ on the branch of the square ...
1
vote
0answers
39 views

Proving convergence of certain Mellin transforms

I am wondering about the following modified mellin transforms and if they are absolutely converging or not. Let $\phi$ be some holomorphic function such that for all $y \in \mathbb{R}$ we have ...
0
votes
1answer
156 views

How to characterize the value which assumed by an inner function infinitely often but is an asymptotic value?

The Frostman Shift of an inner function at the value which assumed infinitely and is not an asymptomatic value is an infinite Blaschke Product. But how to charaterize it when it is an asymptotic ...
0
votes
2answers
64 views

Subharmonic function on a twice punctured complex plane

is the twice punctured complex plane parabolic or hyperbolic? In this sense: does $\mathbb{C}-\{0,1\}$ admit a nonconstant, negative, subharmonic function? Thanks,
0
votes
1answer
94 views

Bloch type function

I would like to know whether there exists an analytic function $f$ on the unit disk such that $$\sup_{|z|<1}|f(z)|(1-|z|^2)<\infty$$ and for every $|a|=1$, $$\limsup_{z\to ...
0
votes
1answer
92 views

Relation between the eigenvalue density and the resolvent?

Disclaimer: This is a cross-post from Math Underflow. Given that there is little activity on the subject (random-matrice) on the aformentioned site, and given that many interesting discussion on this ...
6
votes
3answers
1k views

Is a non-compact Riemann surface an open subset of a compact one ?

Let $X$ be a non-compact holomorphic manifold of dimension $1$. Is there a compact Riemann surface $\bar{X}$ suc that $X$ is biholomorphic to an open subset of $\bar{X}$ ? Edit: To rule out the case ...
1
vote
0answers
65 views

How to find number of points at infinity of a Riemann surface

Let $X \subset \mathbb C^2$ be a Riemann surface with boundary $\partial X \subset \mathbb C^2$ and without compact components. Let $\bar X = X \cup \{p_1,\ldots,p_N\} \subseteq \mathbb CP^2$ be its ...
2
votes
1answer
85 views

Convolution-type operator for series

Suppose $f(z)=\lambda(a_1z+a_2z^2+\cdots)$ is holomorphic in $\{|z|<1\}$ with $\lambda>0$. For each $d\geq 1$ , I am trying to define an operation, $\star_d$ , so that $f(z)\star_d ...
0
votes
1answer
108 views

Residues and Mittag-Leffler sequence

Let $X$ be a compact Riemann surface, $\omega$ a meromorphic differential on $X$ and $f$ a meromorphic function on $X$ with poles only over the points $P_1,\dots,P_d$. The product $\;f\cdot\omega\;$ ...
2
votes
2answers
234 views

Surjective entire functions without critical points

It is easy to construct surjective locally univalent holomorphic functions $f: {\mathbb D}\to {\mathbb C}$, where ${\mathbb D}$ is the open unit disk. I am pretty sure that the answer to the ...
11
votes
2answers
521 views

Belyi functions on non-compact surfaces; or: Building Riemann surfaces from equilateral triangles

Some background on (compact) Belyi surfaces $\newcommand{\Ch}{\hat{\mathbb{C}}}$ A compact Riemann surface $X$ is called a Belyi surface if there exists a branched covering map $f:X\to \Ch$ such that ...
1
vote
0answers
106 views

Extensibility of real analytic function of several variables to complex domain

My question relates to the extensibility of a real analytic function of several variables to a specific complex domain. In order to formulate the question, let me define the following complex ...
1
vote
2answers
159 views

Analytic continuation for PI(1+z^(4^n))

How to do analytic continuation for following function? $$f(z) = \prod_{n=0}^{+\infty} {(1+z^{4^n})}$$ Evidently it satisfies $f(z)f(z^2)=\dfrac{1}{1-z}$...