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18
votes
0answers
824 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 ...
12
votes
0answers
582 views

What is the appropriate setting for Cauchy's Integral Formula?

For a $C^1$ function $f:U\to\mathbb{C}$, where $D\subset U\subseteq \mathbb{C}$ with piecewise $C^1$ boundary $\partial D$, we have the following generalized Cauchy integral formula: $$ f(\zeta) = ...
9
votes
0answers
144 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 ...
9
votes
0answers
1k views

Meaning of Cauchy integral theorem - the (co)homology viewpoint

I'm not sure what follows is not just a complicated way to deduce a blatant triviality, or if is even correct. Let's try. In the elementary theory of analytic functions of $1$ complex variable, one ...
8
votes
0answers
318 views

$C^\infty$ function $f:{\bf C}\mapsto {\bf C}$ such that $f(z)\in\overline{{\bf Q}(z)}$ for all $z\in {\bf C}$

Suppose that $f:{\bf C}\mapsto {\bf C}$ is a $C^\infty$ function such that $f(z)\in\overline{{\bf Q}(z)}$ for all $z\in {\bf C}$, ie $f(z)$ is algebraic over the field ${\bf Q}(z)$ generated by $z$ ...
8
votes
0answers
546 views

Hardy spaces: analysis <---> martingales

Let $H^p$ be the Hardy space of analytic functions on the open unit disk $D$: $f \in H^p$ if $f$ is analytic on $D$ and $\sup_{r < 1} \int_0^{2\pi} |f(re^{i\theta})|^p d\theta < \infty$. ...
7
votes
0answers
241 views

Convergence at the radius of convergence

Suppose I have (roughly speaking) a multivalued meromorphic function $f(z)$ on all of $\mathbb{C}$ that is single-valued and holomorphic on the open unit disc and has some branch points of finite ...
6
votes
0answers
608 views

Analytic continuation of the Dirichlet generating series of the multiplicative partition function

Apologies for the lengthy question, but it seems it's the only way i can convey my thoughts. Consider the Dirichlet series: ...
5
votes
0answers
141 views

the “three-point” characterization of holomorphy

I want to know the source of the following "folkloric" fact about holomorphic functions. It seems well described by the phrase: The three-point characterization of holomorphy. If F is a ...
5
votes
0answers
131 views

Perturbations of zero-dimensional algebraic varieties

Let $P(z,w):\mathbb C^2\to\mathbb C$ be a certain polynomial, and consider $p(s,t)=P(e^{is},e^{it}):\mathbb T^2\to \mathbb C$ its restriction to the real torus. Assume generic situation, so by ...
5
votes
0answers
536 views

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 ...
4
votes
0answers
246 views

Etymology of the O-notation for algebras of holomorphic functions

The notation $O(X)$ seems to be a quite standard notation for the algebra of all holomorphic functions on some connected domain in $\mathbb{C}^n$ (or a complex manifold). I would like to know where ...
4
votes
0answers
309 views

Is this an injective function ?

Hi all, I got stuck with a problem that pop up in a paper about location of zeros for some analytic functions that I am working on. The problem is the following: Fix two arbitrary positive ...
4
votes
0answers
737 views

A “Cauchy integral formula” for the Poisson kernel?

The inspiration for this question is in a certain breakdown of the analogy between holomorphic and harmonic functions. First recall the Cauchy integral formula: Let $U$ be an open subset of ...
4
votes
0answers
318 views

Adeles of Holomorphic Functions

In number theory, an adele is a kind of product of elements of the completion at each prime. For function fields, we take (a kind of) product of the completion at each point, and at non-singular ...
3
votes
0answers
131 views

elementary proof for existence of point with minimal period 2 for entire function

Fatou proved a very interesting result: for a transcendental entire function $f$, the second itarate $f^{2}$ has at least has one fixed point. (Using the technique of Picard theorem) This result ...
3
votes
0answers
102 views

What is the relationship between complex time singularities and UV fixed points?

In this paper it is described how the turbulent kinetic energy spectrum and the flatness (a measure for intermittency) are governed by the position of the (dominant) singularities of the solutions of ...
3
votes
0answers
187 views

The functional equation of Hofstadter's Q sequence

Hofstadter's Q sequence is defined by $Q(1) = Q(2) = 1$ and $Q(n) = Q(n-Q(n-1)) + Q(n-Q(n-2))$ for $n \geq 3$. So far hardly anything on this sequence has been proved -- not even that $Q(n)$ is ...
3
votes
0answers
356 views

Real part of a function on a circle in the complex plane

Fix $0 < y <1$. Assume that $n$ is a large positive integer. Let $H(z):= \sum_{i=1}^n \ln(z^i+i-1)$. Is it true that $\Re(H(e^{\xi+i\theta}))$ has asymptotically (for large $n$) a peak at ...
3
votes
0answers
312 views

confusion about rational maps and Fatou components

Dear fellows, I have come to another conclusion which must be wrong. Let $f$ be a rational map and let $U$ be a connected but not simply connected open subset of the Fatou set such that $f(U)$ is ...
3
votes
0answers
440 views

analytic continuation of an integral involving the mittag-leffler function

greetings. we have the following integral : $$I(s)=s\int_{0}^{\infty} \frac{dx}{2x}\left(E_{s/2}((\pi x)^{s/2})-1\right)\omega(x)-\left(E_{s/2}(( x)^{s/2})-1\right)e^{-x}$$ where : $E_{\alpha}(z)$ ...
3
votes
0answers
261 views

Laplace Transform: Are there theorems similar to the Bernstein Theorem?

Bernstein's Theorem states, that if a function is completely monotonic, then it is the Laplace transform of an $L^1$-function. (E.g. Widder, "The Laplace Transform", Chapter IV, Theorem 19b) Are ...
3
votes
0answers
169 views

Monotonicity of a certain parametric integral

I would like to ask for some help (hints, ideas) in solving the following problem: Given integer $n>0$ and real $\alpha>0,\beta>1$ we want to show, that if we define for any ...
3
votes
0answers
277 views

Why is Mellin-inverse of Gamma periodic?

Specific Case The periodicity is obvious from computation: $$\cal{M}^{-1}\{\Gamma\}(x) := \frac{1}{2\pi i}\int_{c}\Gamma(s)x^{-s}d s=e^{-x}$$ However, is there a way to see directly from the integral ...
3
votes
0answers
400 views

When checking if a harmonic function is continuous on its boundary, is a dense subset enough?

Let $U$ be an open connected subset of $\mathbb{C}$ and let $u:U\rightarrow \mathbb{R}$ be harmonic and bounded on $U$. Let $f:\partial_\infty U \rightarrow \mathbb{R}$ be a continuous function, ...
2
votes
0answers
130 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 ...
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 ...
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 ...
2
votes
0answers
66 views

Introducing new poles

Suppose we have a contour integral of an entire function along a line in the plane (or perhaps a line segment). I am looking for examples of such integrals that are computed by first altering the ...
2
votes
0answers
178 views

Analytical continuation of electrostatic potentials

I'm having some trouble figuring out the properties with respect to analytical continuation of functions defined using an integral kernel. More particularly, I am working with the electrostatic ...
2
votes
0answers
142 views

Univalent functions with non-negative coefficients

Is anything non-trivial known about univalent functions with non-negative coefficients? Let $U$ be the unit disc, and $f$ a univalent (=injective) holomorphic function, $f(0)=0$, $f'(0)>0$. ...
2
votes
0answers
230 views

How well do continuously differentiable functions behave from R^2 to R^2 ?

The behaviour of complex smooth vs 1-dimensional real smooth functions is discussed in a previous question. In "Complex Analysis as Catalyst" by Steven G. Krantz, the Cauchy integral formula is ...
2
votes
0answers
311 views

Characterizing essential singularities

In the paper Picture of an essential singularity, an analogy is made between the multipolar moments of infinitesimal charge distributions and the lines of constant modulus/argument around an essential ...
2
votes
0answers
247 views

Is the complex harmonic extension of a $\mathcal{C^r}$ map from $S^1$ to $\mathbb{C}$ is smooth upto the boundary ?

Suppose we have a map $ h : S^1\to \mathbb{C} $ that we know is a $\mathcal{C^r} $ map ( in the sense of a map between 1-manifold ( or in the sense of a $2\pi$ periodic map from $\mathbb{R}\to ...
2
votes
0answers
224 views

Question in complex analysis arising from large $N$ gauge theory

This is a question in complex analysis that comes up in the treatment of large $N$ gauge theory with gauge group $SU(N)$ in the 't Hooft limit where $N$ is taken to infinity with $\lambda=g^2 N$ fixed ...
2
votes
0answers
582 views

Pole data of meromorphic matrix function

Let $T(z)$ be a meromorphic square matrix function, that is - a matrix whose entries are complex meromorphic function of one variable. Recall that such a $T$ is said to have a right pole of order $r$ ...
2
votes
0answers
214 views

Finer properties of a sequence of harmonic functions

This was a question that arose for me when I was thinking about how one proves strong unique continuation for elliptic equations. I never could come up with a satisfactory answer. Background: When ...
2
votes
0answers
284 views

What is this effect in Fourier/additive synthesis called?

Hi, I have re-synthesized a cyclic function additively, and I added a fixed offset to the frequency of each partial. So if the function was $\sum a_{n} sin(2 \pi x * n)$ and its frequencies were ...
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
vote
0answers
38 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 ...
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 ...
1
vote
0answers
105 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
0answers
69 views

analytical behaviour of holomorphic functions on boundary

Is there any survey about criterions for a holomorphic function to be analytical on the boundary of a pseudo convex domain on which it is defined ?
1
vote
0answers
150 views

complex contour integral calculation after Möbius transformation

Good day to everyone. In my scientific research I've got stuck with a contour integration problem. I would like to evaluate the following integral: $$I=\int_0^{\infty } \frac{e^{\frac{\alpha -\mathrm ...
1
vote
0answers
71 views

Pointwise bounds on Hardy space functions with regular boundary behaviour

Let $H^2$ denote the Hardy space on the strip $S:=\{z\in{\mathbb C}\,:\,0<\Im z <1\}$ (or the upper half plane), i.e. $H^2$ consists of all holomorphic functions $f:S\to\mathbb C$ such that for ...
1
vote
0answers
180 views

An integral with Gamma functions (Part 2)

I was wondering if there is a generalization of the integral discussed here to a case like, \begin{equation}\int \frac{d^dq}{q^{\nu_1}\vert \vec{q} \pm \vec{k}_1\vert ^{\nu_2}\vert \vec{q} \pm ...
1
vote
0answers
150 views

Convergent series, asymptotics and truncation

In regard to the characteristics of certain "explicit formulae" arising in number theory, I am pondering the connection between the rate of convergence of series and the asymptotic order of the ...
1
vote
0answers
109 views

an infinite series expansion in terms of the polylogarithm function

we have the complex valued function : $$f(z)=\sum_{n=0}^{\infty}a_{n}Li_{-n}(z)$$ we wish to recover the coefficients $a_{n}$ . the only thing i though would work is to try and come up with a function ...
1
vote
0answers
202 views

Mellin inverse of the Hadamard product rep. of the Riemann zeta function?

The floor function is given - by Perron's formula - as a Mellin inverse of the zeta function. namely : $$\left \lfloor x \right \rfloor=\frac{1}{2\pi ...
1
vote
0answers
98 views

Sum of univalent functions

Let $f,g$ be a pair of univalent functions on a proper subdomain $\Omega$ of $\mathbb C$. Does their sum $f+g$ necessarily omit any complex value? Similarly, can all holomorphic function be written as ...