The complex-analysis tag has no wiki summary.

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### Question on the partial differential equations in complex space

As is known most of the theory now developed for partial differential equations is in the real space, especially function space like Sobolev space, BMO space, $L^p$ space, etc. However is there some ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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) = ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 $P$, ...

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### $\pm1$-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 ...

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### 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 ...

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### 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}
...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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. ...

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### 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 ...

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### 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 $D(0,|f(0)|)$. "
...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 $t\rightarrow +\infty$, ...

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### 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 ...

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### 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 ...

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### 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?

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### 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 ...

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### 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 ...

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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 ...

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### 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.
...

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### 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 ...

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### 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 ...

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### Roots of modified polynomials

Consider the following two polynomials:
$$
g=x^3 - x^2 - (c + 2)x + c
$$
and
$$
h=x^3 - x^2 - cx + c
$$
The roots of $h$ are $1$ and $\pm \sqrt{c}$. I am interested in obtaining the roots of $g$, ...

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### Infinite product's question

Given a pair of strictly increasing functions $f,g:\mathbb{N}\to \mathbb{N}$
define:
$P_N(f,g)\doteq \left(z\in \mathbb{C}\mapsto \prod_{i=1}^{f(N)}\left(1+\frac{z}{v_i(N)}\right)\in ...

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### System of quadratic complex equations

I want to solve this system of N non-linear equations without using a numerical method:
$x_{k}^{2}= \alpha_{k }+ \sum\limits_{m=1}^{N} (\beta_{km} x_{m} + \psi_{km} x_{m}^{*})$
With
$\left| ...

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### Injective element of a commutative Banach algebra

A revision:
According to the comment of Nate Eldredge, in order to avoid the triviality, we revise the property $P$.
Assume that $A$ is a commutative unital Banach algebra. Its maximal ideal ...

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### Grunsky-Motzkin-Schoenberg formula

I found this formula in Brian McCartin's interesting book "Mysteries of the equilateral triangle" http://www.kettering.edu/news/mysteries-equilateral-triangle and it looks as follows:
Suppose that ...

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### How far do conjugated Mobius transforms move points?

I start with an automorphism $f$ of the complex unit disc $S^1 = \{ z \in \mathbb{C} : |z| \leq 1\}$. I assume that such a map is given by a Mobius transform, namely
$$
f(z) = \frac{z - ...

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### Original article about a theorem of Cartan on iterations of analytic functions

I'd like to know in which paper of H. Cartan I could find the following theorem :
Let $\Omega$ be a connected, open and bounded subset of $\mathbb{C}$. Let $a \in \Omega$ and $f \in ...

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### Homology of a region of the plane

This is related to this MO question, I don't know if it's really "research-level". As in that question, let $U$ be a domain of the complex plane $\mathbb{C}$, i.e. an open connected subset. Let
$$ ...

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### f(f(x))=exp(x)-1 and other functions “just in the middle” between linear and exponential.

The question is about the function f(x) so that f(f(x))=exp (x)-1.
The question is open ended and it was discussed quite recently in the comment thread in Aaronson's blog here ...