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0
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0answers
58 views

Shifting a function with non-simple root creates complex roots [on hold]

Hello friendly people, I have seen a vague statement in an article that i couldn't prove: that if a function has a root with multiplicity greater than 1, by shifting the function (adding a small ...
0
votes
0answers
82 views

Coordinate charts on converging Riemann surfaces

Let $S$ be a $2-$dim manifold and $q \in S$. Furthermore, let $j_{n}$ be a sequence of complex structures on $S$ converging in $C^{\infty}_{\text{loc}}$ to a complex structure $j$ on $S$ as ...
1
vote
0answers
57 views

Possible generalizations of Hadamard's three line lemma

Let $f$ be an analytic function on a sector $$ S=\left\{re^{i\theta}:0<r<\infty,\; 0<\theta<\gamma<\frac{\pi}{2}\right\} $$ with opening angle $\gamma$ at the origin. Suppose $f$ is ...
5
votes
0answers
53 views

How useful is knowing every torsionfree $\mathcal O(D)$ module is flat?

One of the corollaries of Weiertrass' factorization theorem plus the theorem of Mittag Leffler is that $\mathcal O(\Bbb C)$, more generally $\mathcal O(D)$ for some region $D$ is such that every ...
0
votes
0answers
20 views

Looking for a simple tool for complex function viewer [closed]

I am searching for a simple tool (online/for windows) in which i could input a complex function and some range and see its output. Does anyone knows something like that ?
0
votes
0answers
42 views

Analytic functions space on Riemann surface

I have some questions about the analytic function space on Riemann surface and distinguished varieties: Let S be a compact Riemann surface and $\Omega\subset S$ be a domain with piecewise smooth ...
5
votes
5answers
2k views

Why $\partial$ and $\bar{\partial}$ defined in that way (the Wirtinger derivatives)?

For $\mathbb{C}$-valued functions, why are $\frac{\partial}{\partial z}$ and $\frac{\partial}{\partial \bar{z}}$ defined as $$ \frac{\partial}{\partial z}= \frac{1}{2}\left( ...
6
votes
1answer
156 views

Analytic diffeomorphisms of the circle from complex domains

Let $\gamma \subset \mathbb C$ be a simple closed analytic curve and let $\Delta$ be the closure of the disk it bounds. The Riemann mapping theorem gives two biholomorphisms: $$\phi : (D^2,S^1) \to ...
2
votes
1answer
334 views

Fatou Coordinate for function with rationally indifferent fixed point, and repelling fixed point

Lets say I have f(z)=z^2+c, with c=0.35676274578 + 0.32858194507i. Then f(z) has a fixed point=0.15450849719 + 0.47552825815i, which is rationally indifferent with a period=5. The "Complex Dynamics" ...
2
votes
1answer
107 views

Complex function for mapping a circle to a superellipse

I was wondering if anyone knows an analytic complex function that would map a circle to a superellipse, or vice versa. Any ideas, comments, or functions are much appreciated! Thanks, Kayvan
28
votes
1answer
469 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) = ...
1
vote
0answers
68 views

Uniqueness of an embedding theorem for Real differential fields

I will follow a preliminary exposition for the problem in question, which will essentially follow the format on http://www4.ncsu.edu/~singer/papers/model_diff_fields.pdf [pg. 87]: Let $K$ be a real ...
7
votes
1answer
137 views

Conformal map of polygon with circle segments

I am looking for a conformal map from a "polygon" to eg the upper half plane, which consists of circle segments instead of lines. So for example, it could be a quadrilateral ABCD, but where AB is a ...
1
vote
1answer
204 views

Residues and values of Riemann Zeta function at some points

I need the following computational results for proving something. Let $1/2 + i\gamma_0$, be the first nontrivial zero of Riemann zeta function, $\zeta(s)$, i.e. $\gamma_0\sim 14.134...$. 1) what is ...
4
votes
1answer
101 views

Gauss--Lucas type theorem for tracts and higher derivatives of a polynomial

The Gauss--Lucas Theorem states that all zeros of a degree $n$ complex polynomial $p(z)$ are contained in the convex hull of the zeros of $p$. By iteration, this implies that the zeros of ...
4
votes
0answers
108 views

residue and regulator

Let $C$ be a curve defined over $\mathbb{Q}$. The regulator is a map $$ reg: K_2(C)_{\mathbb{Q}} \longrightarrow H^1(C(\mathbb{C}), \mathbb{R}). $$ Here $K_2(C)_{\mathbb{Q}}$ is the K-group tensor ...
0
votes
0answers
40 views

what is the first non-constant term in the Kronecker Limit formula?

The Kronecker Limit formula gives the constant term in the Laurent expansion about s=1 of the Eisenstein series E(s,\tau). What is the next term? I.e., the coefficient of the first power of (s-1)? I ...
1
vote
2answers
78 views

A question on deficient values of entire functions

Recently I come cross a question about deficient values of entire functions. I find that many examples in the book about functions $f$ whose deficient values are singularities of the inverse ...
2
votes
0answers
99 views

Eigenvalue problem

I am studying torsional Alfven waves in spicules. In this concern I have encountered the following equation: $ \left(1-m^2 e^{-αz}\right)y''(z)+\left(4π i m ...
0
votes
0answers
55 views

Integrate Faddeeva function

I came across this integration in my studies. $\int_{-\infty}^{\infty}|F((w_\textbf{_} - \hat{w_\textbf{_}})\tau) |^2 . d\tau$ It uses the Faddeeva function which is $F(z) = e^{-z^2}erfc(-iz)$. I ...
1
vote
0answers
42 views

Determine the position of the contour with the value of corresponding contour integral

Let $C$ be the contour of the unit square with lower left corner at origin. We define a function $g(z)=\int_{z+C} f(w)dw$ for a given (not necessarily holomorphic) function ...
7
votes
0answers
66 views

What does this number tell me about a convex lattice polygon?

EDIT: I realized I'd tricked myself by working with a too special case of $f$, the question is now updated (boundary lattice points replaced vertices). Suppose I have a convex lattice polygon $P$, ...
12
votes
1answer
198 views

$\pm1$-polynomials with a maximal non-real root

For given $n$, consider a polynomial $\sum_{k=0}^na_kz^k$ with all coefficients $a_k\in\{\pm1\}$. I am interested in the following: How big can the modulus of a non-real root of such a ...
1
vote
1answer
162 views

The locus of rational/elliptic curves on a special surface in $\mathbb{P}^3$

Let $P$ and $Q$ be two general polynomials of the same degree $d>5$. Consider the surface $S: z^2=P(x)Q(y)$ in $\mathbb{P}^3$ (after homogenization by the variable $w$). One can show that these ...
4
votes
4answers
2k views

Does the inverse Laplace transform of the square root exist?

Does the inverse Laplace transform, defined by the integral, \begin{equation} F(t) = \mathscr L_s^{-1}\left[\sqrt s\right](t) = \int_{c - i\infty}^{c + i\infty} \sqrt s ~e^{-st} ds \end{equation} ...
223
votes
15answers
31k views

Why do roots of polynomials tend to have absolute value close to 1?

While playing around with Mathematica I noticed that most polynomials with real coefficients seem to have most complex zeroes very near the unit circle. For instance, if we plot all the roots of a ...
6
votes
1answer
264 views

Does there exist harmonic function with that property?

Can one construct a harmonic function $f$ defined in the unit disk with the condition $f(0)≥1$ such that area of $\{z∈D:f(z)>0\}$ is small enough, i. e. for every $\epsilon>0$ does there exist ...
0
votes
1answer
215 views

Is there any algorithm to decide whether a series with integral coefficiens is a algebraic function? [closed]

Given a series with integral coefficiens as following: $$F(x)=\sum_0^i a_i x^i,\text{where }a_i\in \mathbb{N}\bigcup 0 $$$$\text{and there is a computable function $\psi$ such that } \forall i ...
3
votes
2answers
440 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. ...
4
votes
1answer
361 views

Why are the angular differences of these random complex polynomial coefficients almost constant?

This is based on მამუკა ჯიბლაძე's (not-)answer here. I guess it is better to make up a new thread for it. Let me repeat the setup here: We consider polynomials whose complex roots are randomly ...
0
votes
1answer
81 views

Location of the zeros set of holomorphic function [closed]

Recently I proved the following result. "If a holomorphic function $f$ maps the unit disc $\Delta$ into the unit disk $\Delta $ with $0<|f(0)|$ then $f$ doesn't vanish in the disk $D(0,|f(0)|)$. " ...
4
votes
2answers
484 views

Power series with funny behavior at the boundary

Consider a power series $$ \sum_{n=0}^{\infty}a_nz^n $$ where $a_n$ and $z$ are complex numbers. There is radius $R$ of convergence. Let us assume that is a positive real number. It is well known that ...
1
vote
0answers
36 views

singularity of the solution to an integral equation

I consider a function $x\mapsto f(x)$ which is the positive solution to the integral equation $$\int_{f(x)}^\infty \frac{du}{u^\gamma-x}=1, \quad x\ge 0,$$ where $\gamma\in (1, 2]$ is some ...
10
votes
1answer
277 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 ...
2
votes
0answers
120 views

Questions about transformation or integral transformation

I have asked several mathematicians about the following questions,but all of them think they are good questions,but can not give a complete answer.Now I have to come here to ask mathematicians all ...
10
votes
2answers
608 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 ...
8
votes
1answer
218 views

Different notions of convergence of complex subvarieties

Let $X$ be a smooth complex algebraic variety (or, better, complex analytic manifold). Let $\{C_i\}$ be a sequence of compact algebraic subvarieties (resp. analytic reduced subspaces) which converges ...
1
vote
0answers
159 views
0
votes
0answers
82 views

Boundedness and Convergence of a Complex sequence

Consider a dynamical systems over complex numbers $$ z_{n+1}=\frac{\alpha}{z_{n}}+ \frac{\beta}{z_{n-1}},\qquad n=0,1,\ldots $$ where the parameters $\alpha, ~\beta$ are complex numbers, and the ...
1
vote
3answers
779 views

Is Euler's formula a theorem or a definition? [closed]

The first time I tried to understand Euler's formula was about 2 years ago. I didn't need it, I just randomly ran across it, when trying to understand a Fourier transformation. The problem was, that I ...
1
vote
0answers
111 views

Asymptotic expansion of an integral, related to Maass forms

I am trying to compute the asymptotic expansion of the integral $I(t) = \int_{C} e^{\sqrt{1+u}(\frac{1}{t}+\frac{t^2}{\sqrt{u}})}\frac{u^\eta}{\sqrt{1+u}}du$ as $t$ is real and $t\rightarrow +\infty$, ...
-2
votes
1answer
92 views

Is there some lattice not rigid

I heard that in complex hyperbolic space setting for example CH2, there is some deformation of lattice nontrivial. What confused me is it seems contradicting Mostow Rigidity. Could someone explain ...
3
votes
1answer
154 views

Can someone tell me properties of Douady space?

I want to know the parallel properties of Douady space with respect to Hilbert scheme. For example I want to know what is the irreducible component of Douady space, what if I consider a family of ...
0
votes
1answer
169 views

is grassmannian rational connected or not [closed]

I wan to know if Grassmannians are rational connected? Any reference describe how to tell if a variety is rational connected or not?
1
vote
2answers
154 views

Measures, orthogonal to holomorphic functions

Let $G$ be a domain in $\mathbb{C}^{d}$ and let $H\left(G\right)$ be the space of all holomorphic functions on $G$. My question is how to characterize all such (Radon) measures $\mu$ on $G$, that ...
0
votes
0answers
85 views

A question related to the Grauert semi-continuity theorem

Let $f\colon X\to Y$ be a proper holomorphic map of complex analytic manifolds. Assume $f$ to be submersive for simplicity, but probably it is not important. Let $\mathcal{F}$ be a coherent sheaf on ...
2
votes
1answer
117 views

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
votes
1answer
193 views

Computing Reciprocal Gamma

Reciprocal Gamma $1/\Gamma(z)$ is an entire function and so it has a convergent Taylor series expansion which was given in its wikipedia article. ...
3
votes
0answers
135 views

Bounding an integral transform ouside a circle (or inside a strip)

Let $g$ be a symmetric unimodal probability distribution and $H$ be the right half plane. We call $$f(z) = \int_{-\infty}^\infty \frac{1}{z-i t}g(t)dt$$ the dispersion function of $g$. Now, one can ...
6
votes
3answers
169 views

Is there a effective computational criterion to all periodic points of a rational function are repelling.

I came up with a question to know the fatou component of of some types of rational function. In some sense, I may need to give a computational criterion to existence of attracting periodic basin for a ...