The complex-analysis tag has no wiki summary.

**214**

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**15**answers

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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 ...

**56**

votes

**1**answer

4k views

### 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 ...

**48**

votes

**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

9k views

### 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**

votes

**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

1k views

### 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 ...

**27**

votes

**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 ...

**27**

votes

**1**answer

438 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) = ...

**20**

votes

**1**answer

1k views

### 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 ...

**20**

votes

**3**answers

940 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 ...

**19**

votes

**2**answers

530 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**

votes

**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

861 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**

votes

**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 ...

**15**

votes

**4**answers

993 views

### 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

988 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 $z_0 \in \mathbb{C}$.
...

**13**

votes

**2**answers

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

**13**

votes

**2**answers

881 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

403 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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votes

**3**answers

1k views

### 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 ...

**13**

votes

**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 ...

**13**

votes

**2**answers

553 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**

votes

**0**answers

646 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

1k views

### 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 ...

**12**

votes

**3**answers

1k views

### 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 ?
...

**12**

votes

**3**answers

908 views

### 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 ...

**12**

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**2**answers

837 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, ...

**12**

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**1**answer

182 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 ...

**11**

votes

**4**answers

602 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 ...

**11**

votes

**3**answers

1k views

### 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

718 views

### 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 ...

**11**

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**2**answers

426 views

### 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**

votes

**1**answer

591 views

### 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 ...

**11**

votes

**2**answers

1k views

### 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 ...

**10**

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 ...

**10**

votes

**2**answers

1k views

### 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

274 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 ...

**10**

votes

**2**answers

597 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 ...

**9**

votes

**1**answer

880 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 $x \in \mathbb{C}$ ...

**9**

votes

**2**answers

386 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 ...

**9**

votes

**2**answers

550 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) ...

**9**

votes

**1**answer

484 views

### 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 ...

**9**

votes

**1**answer

406 views

### 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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votes

**0**answers

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

**1**answer

866 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 ...

**8**

votes

**3**answers

664 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**

votes

**1**answer

572 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 ...

**8**

votes

**3**answers

552 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 ...