The complex-analysis tag has no wiki summary.

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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 ...

**57**

votes

**1**answer

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### Is there a “classical” proof of this $j$-value congruence?

Let $j: \mathbf{C} - \mathbf{R} \rightarrow \mathbf{C}$ denote the classical $j$-function from the theory of elliptic functions. That is, $j(\tau)$ is the $j$-invariant of the elliptic curve ...

**49**

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**32**answers

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### 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 ...

**43**

votes

**16**answers

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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 ...

**32**

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**7**answers

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### Is there an integration free proof (or heuristic) that once differentiable implies twice differentiable for complex functions?

The title pretty much says it all. I am revisiting complex analysis for the first time since I "learned" some as an undergraduate. I am trying to wrap my head around why it should be the case that a ...

**30**

votes

**3**answers

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### How is the Julia set of $fg$ related to the Julia set of $gf$?

Let $f$ and $g$ be complex rational functions (of degree $\geq 2$ if that helps). What can be said about the relationship between $J(fg)$ and $J(gf)$, the Julia sets of the composite functions $f ...

**28**

votes

**1**answer

471 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) = ...

**27**

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**2**answers

2k views

### Restriction of a complex polynomial to the unit circle

I am pretty sure that the following statement is true. I would appreciate any references (or a proof if you know one).
Let $f(z)$ be a polynomial in one variable with complex coefficients. Then there ...

**21**

votes

**3**answers

947 views

### Universality of zeta- and L-functions

Voronin´s Universality Theorem (for the Riemann zeta-Function) according to Wikipedia: Let $U$ be a compact subset of the "critical half-strip" $\{s\in\mathbb{C}:\frac{1}{2}<Re(s)<1\}$ with ...

**20**

votes

**1**answer

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### Provable zero-free region for any entire function that analytically is similar to zeta(s)

Is there an entire function $f:\mathbb C\rightarrow\mathbb C$ such that for some $\delta>0$:
$f(z)$ is bounded when $\Re z>1+\delta$
$f(z)$ is unbounded when $\Re z=1$
$f(z)$ grows ...

**19**

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**2**answers

539 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 ...

**18**

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**4**answers

1k views

### What is the naming reason of poles in complex analysis?

A function $f: \textbf{C} \to \textbf{C}$ has a pole of order $k$ if $f(z) = \frac{g(z)}{(z-z_0)^{k}}$ where $g(z)$ is a nonzero analytic function. Why do we call it poles?

**17**

votes

**0**answers

866 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 ...

**15**

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**12**answers

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### 2D Problems Which are Easier to Solve in 3D

It sometimes happens that 1D problems are easier to solve by somehow adding a dimension. For example, we convert linear differential equations for a real unknown to a complex unknown (to use complex ...

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**4**answers

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### Why are lacunary series so badly behaved?

Hi!
I just came across the Ostroski-Hadamard gap theorem, and while I can understand the proofs as well as the principle that the series $\sum_{n=0}^\infty z^{2^n}$ ought to have a singularity at ...

**15**

votes

**3**answers

1k views

### Perron, Fourier

Perron´s formula is in some sense just Fourier inversion, but I have never seen proven it that way in a textbook. I take this must be because the conditions for the Fourier inversion formula to hold ...

**14**

votes

**5**answers

995 views

### What is the spectrum of the ring of entire functions?

Let $\mathcal{O}(\mathbb{C})$ be the ring of entire functions, that is, those functions $f : \mathbb{C} \to \mathbb{C}$ which are holomorphic for all $z \in \mathbb{C}.$ For each $z_0 \in \mathbb{C}$.
...

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votes

**3**answers

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### Is the Euler product formula always divergent for 0<Re(s)<1?

It is known that the Euler product formula converges for Re(s)>1.
(which represents the Riemann zeta function.)
My question: Is the Euler product formula always divergent for
0 < Re(s) < 1 ?
...

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votes

**2**answers

830 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 $\widetilde D = \mathbb C \setminus D$
...

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**2**answers

885 views

### Complete metric on the space of Jordan curves?

I was interested in putting a complete metric on the space of Jordan curves. Say, just planar Jordan curves contained in $B(\bar{0}, 2) \backslash B(\bar{0}, 1)$ which separates $\bar{0}$ and ...

**13**

votes

**2**answers

408 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 $ ...

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**3**answers

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### Meromorphic 1-form and Picard's theorem

Let $D$ be the open unit disk in the complex plane and $U_1,U_2,\,\ldots\,,U_n$ be an open cover of the puntured disk $D^*= D\setminus\{0\}$. Suppose on each open set $U_j$ there is an injective ...

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**1**answer

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### Certain functional equations for the Riemann Zeta function?

Referring to this question I asked on math.SE.
I am posting a more generalized question here, for answers and further inquiry.
For the Riemann zeta function, we know of the standard functional ...

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**2**answers

846 views

### Determining rational functions by their critical points

Fix an integer $d > 1$ and $2d-2$ points $P_1, \ldots, P_{2d-2}$ in the Riemann sphere (not necessarily distinct). Thanks to the work of Eisenbud and Harris on limit linear series (Inventiones, ...

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**2**answers

556 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 ...

**13**

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**0**answers

653 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) = ...

**12**

votes

**1**answer

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### On equation f(z+1)-f(z)=f'(z)

Original Problem
If $f$ is an entire function such that
$$ f(z+1)-f(z)=f'(z) $$
for all $z$.
Is there a non-trivial solution? ($f(z)=az+b$ is trivial)
And here is something uncertainty
If we use ...

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**3**answers

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### If a formal power series over the complex numbers satisfies a polynomial identity, does it imply that the power series has a radius of convergence?

Let $ P(z) $ be a $\textit{formal}$ power series in $z$ that a priori may not have a non zero radius of convergence. Assume that $P(0) =0$.
Let $\Phi(w,z)$ be a polynomial in two variables, that ...

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**1**answer

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 ...

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**4**answers

609 views

### non-trivial zeros of partial zeta functions

Let $N,a\in\mathbf{Z}_{\geq 1}$. Define a partial $\zeta$-function as
$$
\zeta(s;N,a):=\sum_{\substack{n\geq 1\newline n\equiv a\pmod{N}}} \frac{1}{n^s}
$$
where $Re(s)>1$. Let $\omega$ be either ...

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**3**answers

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### What holomorphic functions are limits of polynomials?

Let $\Omega$ be a connected open set in the complex plane. What is the closure of the polynomials in $\mathcal{H}(\Omega)$ the set of holomorphic functions on $\Omega$? The topology is the usual ...

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**2**answers

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### Anything special (historical?) about surface $x\cdot y\cdot z\ +\ x+y+z=0$?

QUESTION
I wanted to introduce and develop the complex logarithm from scratch. As the result I've arrived a couple of months ago at the following identity after which the road to complex logarithm is ...

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### A “holomorphic” Peano curve?

A Peano curve is a continuous map $[0,1]\to [0,1]^2$ whose image is the whole square.
I would like to know if on can obtain "holomorphic" Peano curves. Namely, is it possible to find a continuous ...

**11**

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**1**answer

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### Analysis and finitely generated groups

Dear all, this is perhaps a bit a vague question, but some references would already be very helpfull.
So let $G$ be a finitely generated group and choose some finite set of generators. This allows to ...

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**2**answers

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### Pointers for direct proof of extension of the Descartes Rule of Signs to complex polynomials?

The following describes an extension of the Descartes Rule of Signs to polynomials with complex coefficients.
First, I need to define the notion of a "sweep"... Given a complex polynomial p(z) := c0 ...

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votes

**5**answers

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### Complex Analysis applications toward Number Theory

I'm an undergrad who is taking a Complex Analysis Course mainly for its applications in number theory.
So I would like to ask some guidelines about which theorems/concepts should I focus on in order ...

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votes

**2**answers

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### Does Riemann map depend continuously on the domain?

I've always taken this for granted until recently:
In the simplest case, given Jordan curve $C \subseteq \mathbb{C}$ containing a neighborhood of $\bar{0}$ in its interior. Given parametrizations ...

**10**

votes

**1**answer

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 ...

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**2**answers

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 ...

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votes

**1**answer

894 views

### When are complex polynomial maps almost surjective?

Consider a complex polynomial map $f: \mathbb{C}^n \rightarrow \mathbb{C}^n$.
For $n = 1$, the fundamental theorem of algebra says that, for any $y \in \mathbb{C}$ there exists $x \in \mathbb{C}$ ...

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**2**answers

389 views

### Constructing Riemann maps using Brownian motion?

There's a relation between two-dimensional Brownian motion and conformal maps, see e.g. Thurston's answer to this question. Given two non-empty simply-connected domains $U$ and $V$ in the complex ...

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**2**answers

561 views

### Complexifying a real Banach space and its dual

A standard way to define the "complexification" $E_\mathbb{C}$ of a real Banach space $E$ is to define a complex linear structure on $E\times E$ by (1) $(x,y)+(u,v)=(x+u, y+v)$, (2) ...

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**1**answer

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### How manifold-like is Aut(C^n) in the holomorphic category?

This question is similar to, but not the same as this one. Take the space of automorphisms of $\mathbb{C}^n$ in the holomorphic category, with the compact-open topology. For $n=1$ this is just ...

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**1**answer

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### Properties of a matrix-valued generalization of the $\Gamma$ function

I am interested in the following function (Mellin transform of matrix exponential):
$$\int_0^{\infty} x^{s-1} e^{-A-Bx}d x$$
Where $x$ and $s$ are scalars, but $A$ and $B$ are matrices with $B\succ ...

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**0**answers

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### 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 ...

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**1**answer

878 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 ...

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votes

**3**answers

676 views

### What's the use of Malgrange preparation theorem?

The Malgrange preparation theorem,which is the $C^{\infty}$ version of the classical Weierstrass preparation theorem,says that if $f(t,x)$ is a $C^{\infty}$ function of $(t,x)\in\mathbb{R}^{n+1}$ near ...

**8**

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**1**answer

577 views

### Holomorphic line bundles on a punctured disc

Is every holomorphic line bundle on the - say - punctured unit disc $\dot{\Delta} \subseteq \mathbf{C}$ trivial? Griffiths-Harris (p. 39) prove that $H^{p,q}_{\overline{\partial}}(\Delta) = 0$ (for $q ...

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**3**answers

553 views

### j-invariant fixed point?

If we view the j-invariant of a lattice as a map from the upper-half plane to the complexes by $\tau\mapsto j([1,\tau])$, then it is surjective, holomorphic, and has quite a number of other wonderful ...

**8**

votes

**3**answers

2k views

### Harmonic level sets and boundary data

This is probably a classic problem, so a good reference book or paper to get me started on this type of question would be great:
Let $\mathbb{D} \subset \mathbb{C}$ be the unit disk with boundary ...