**232**

votes

**15**answers

32k 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 ...

**96**

votes

**1**answer

9k views

### What are the shapes of rational functions?

I would like to understand and compute the shapes of rational functions, that is, holomorphic maps of the Riemann sphere to itself, or equivalently, ratios of two polynomials, up to Moebius ...

**82**

votes

**41**answers

28k views

### Ways to prove the fundamental theorem of algebra

This seems to be a favorite question everywhere, including Princeton quals. How many ways are there?
Please give a new way in each answer, and if possible give reference. I start by giving two:
...

**58**

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 ...

**49**

votes

**32**answers

10k views

### 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 ...

**47**

votes

**3**answers

2k views

### Does a power series converging everywhere on its circle of convergence define a continuous function?

Consider a complex power series $\sum a_n z^n \in \mathbb C[[z]]$ with radius of convergence $0\lt r\lt\infty$ and suppose that for every $w$ with $\mid w\mid =r$ the series $\sum a_n w^n $ converges ...

**46**

votes

**16**answers

10k 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 ...

**41**

votes

**4**answers

3k views

### If the Riemann Hypothesis fails, must it fail infinitely often?

That is must there either be no non-trivial zeros off the critical line or
infinitely many?
I'm sure that no one believes otherwise, but I've never seen a theorem in the
literature addressing this. ...

**39**

votes

**9**answers

8k views

### Why do functions in complex analysis behave so well? (as opposed to functions in real analysis)

Complex analytic functions show rigid behavior while real-valued smooth functions are flexible. Why is this the case?

**37**

votes

**1**answer

2k views

### Circles and rational functions

Suppose that $\gamma$ is a Jordan analytic curve on the Riemann sphere,
and there exist two rational functions $f$ and $g$ such that
$f$ maps $\gamma$ into a circle, and $g$ maps a circle into ...

**37**

votes

**4**answers

2k views

### Polynomials on the Unit Circle

I asked this question in math.stackexchange but I didn't have much luck. It might be more appropiate for this forum. Let $z_1,z_2,…,z_n$ be i.i.d random points on the unit circle ($|z_i|=1$) with ...

**36**

votes

**8**answers

5k views

### Why does the Gamma-function complete the Riemann Zeta function?

Defining $$\xi(s) := \pi^{-s/2}\ \Gamma\left(\frac{s}{2}\right)\ \zeta(s)$$ yields $\xi(s) = \xi(1 - s)$ (where $\zeta$ is the Riemann Zeta function).
Is there any conceptual explanation - or ...

**35**

votes

**1**answer

3k views

### Behaviour of power series on their circle of convergence

I asked myself the following question while preparing a course on power series for 2nd year students. Let F be the set of power series with convergence radius equal to 1. What subsets S of the unit ...

**34**

votes

**4**answers

3k views

### The maximum of a polynomial on the unit circle

Encouraged by the progress made in a recently posted MO problem, here is a "conceptually related" problem originating from a 2003 joint paper of Sergei Konyagin and myself.
Suppose we are given $n$ ...

**34**

votes

**3**answers

4k views

### On linear independence of exponentials

Problem.
Let $\{\lambda_n\}_{n\in\mathbb N}$ be a sequence of complex numbers . Let's call a family of exponential functions $\{\exp (\lambda_n s)\}_{n\in\mathbb N}$ $F$-independent (where $F$ is ...

**32**

votes

**7**answers

2k views

### 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 ...

**32**

votes

**1**answer

2k views

### $\zeta(0)$ and the cotangent function

In preparing some practice problems for my complex analysis students, I stumbled across the following. It is not hard to show, using Liouville's theorem, that
$$\pi\cot(\pi ...

**30**

votes

**3**answers

2k views

### Absolute value inequality for complex numbers

I asked this question on stackexchange, but despite much effort on my part have been unsuccesful in finding a solution.
Does the inequality
$$2(|a|+|b|+|c|) \leq |a+b+c|+|a+b-c|+|a+c-b|+|b+c-a|$$
...

**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 ...

**30**

votes

**1**answer

1k views

### Is S^2 x S^4 a complex manifold?

As observed by Calabi a long time ago, the manifold $S^2\times S^4$ admits an almost-complex structure (obtained by embedding it in $\mathbb{R}^7$ and using the octonionic product), which however is ...

**30**

votes

**2**answers

1k views

### Example of a compact Kähler manifold with non-finitely generated canonical ring?

A celebrated recent theorem of Birkar-Cascini-Hacon-McKernan and Siu says that the canonical ring $R(X)=\oplus_{m\geq 0}H^0(X,mK_X)$ of any smooth algebraic variety $X$ over $\mathbb{C}$ is a finitely ...

**30**

votes

**1**answer

615 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) = ...

**29**

votes

**2**answers

2k views

### Must the set of lines through the origin on which a nonconstant entire function is bounded be finite?

If an entire function is bounded for all $z \in \mathbb{C}$, than it's a constant by Liouville's theorem. Of course an entire function can be bounded on lines through the origin $z=r \exp(i \phi), ...

**28**

votes

**5**answers

5k views

### Interpreting the Famous Five equation

$$e^{\pi i} + 1 = 0$$
I have been searching for a convincing interpretation of this. I understand how it comes about but what is it that it is telling us?
Best that I can figure out is that it just ...

**28**

votes

**5**answers

3k views

### Liouville's theorem with your bare hands

Liouville's theorem from complex analysis states that a holomorphic function $f(z)$ on the plane that is bounded in magnitude is constant. The usual proof uses the Cauchy integral formula. But this ...

**28**

votes

**1**answer

6k views

### What, exactly, has Louis de Branges proved about the Riemann Hypothesis?

Hi,
I know this is a dangerous topic which could attract many cranks and nutters, but:
According to Wikipedia [and probably his own website, but I have a hard time seeing exactly what he's claiming] ...

**28**

votes

**1**answer

1k views

### Zeros of polynomials with real positive coefficients

The following problem arose in collaborative work with Subhro Ghosh:
Question: To any polynomial $P_n(z)=\sum_{i=0}^n a_i z^i =a_n \prod_{i=1}^n(z-z_i)$, attach the empirical measure of zeros
...

**27**

votes

**5**answers

3k views

### Why does so much recent work involve K3 surfaces?

I've been noticing that a whole lot of papers published to the Arxiv recently involve K3 surfaces. Can anyone give me (someone who, at this point, knows little more about K3 surfaces than their ...

**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

1k views

### Computing Self-Intersections with Complex Analysis

It is possible to find the winding number of a path $C \subset \mathbb{C}$ using complex analysis:
$$n = \oint_C\frac{dz}{z}.$$
You can also count the number of roots of $f(z) = 0$ inside a close ...

**26**

votes

**2**answers

898 views

### Can the holomorphic image of $(\mathbb{C}^*)^n$ be open but not dense

Let $M$ be a compact complex connected [but not necessarily kähler] $n$-manifold, and suppose we have a holomorphic map $$(\mathbb{C}^*)^n \to M$$ such that the image is open. Is the image ...

**26**

votes

**3**answers

879 views

### Rational functions with a common iterate

Let $f$ and $g$ be two rational functions. To avoid trivialities, we suppose that their degrees are
at least $2$. We say that they have a common iterate if $f^m=g^n$ for some positive integers $m,n$,
...

**26**

votes

**0**answers

3k views

### A paper to the question, if the six dimensional sphere is a complex manifold

Hi,
for a few days a paper was published on arxiv.org with the title "The six dimensional sphere is a complex manifold": http://arxiv.org/PS_cache/math/pdf/0505/0505634v3.pdf
Because I am not able to ...

**24**

votes

**3**answers

3k views

### Which almost complex manifolds admit a complex structure?

I was reading Yau's list of problems in geometry, and one of them is to prove that any almost complex manifold of complex dimension $n \geq 3$ admits a complex structure. It's been some time since ...

**23**

votes

**3**answers

2k views

### What is special about polylogarithms that leads to so many interesting identities and applications?

I have heard that Polylogarithms are very interesting things. The wikipedia page shows a lot of interesting identities. These functions are indeed supposed to have caught the attention of Ramanujan. ...

**23**

votes

**2**answers

977 views

### A 14th and 26th-power Dedekind eta function identity?

Given the Dedekind eta function $\eta(\tau)$. Let p be a prime and define $m = (p-1)/2$.
Let p be a prime of form $p = 12v+5$. Then for $n = 2,4,8,14$:
$$\sum_{k=0}^{p-1} \Big(e^{\pi i m k/12} ...

**23**

votes

**2**answers

492 views

### Why is there no connection between fast-growing functions and complex analysis

I found myself wondering the other day whether the fast-growing functions from natural to naturals that are studied by people like proof theorists are the restriction to the naturals of analytic ...

**21**

votes

**4**answers

1k views

### Distribution of roots of complex polynomials

I generated random quadratic and cubic polynomials with coefficients in $\mathbb{C}$
uniformly distributed in the unit disk $|z| \le 1$. The distribution of the roots of 10000
of these polynomials are ...

**21**

votes

**4**answers

2k views

### Genealogy of the Lagrange inversion theorem

A wonderful piece of classic mathematics, well-known especially to combinatorialists and to complex analysis people, and that, in my opinion, deserves more popularity even in elementary mathematics, ...

**21**

votes

**3**answers

964 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 ...

**21**

votes

**1**answer

576 views

### Comparison map between de Rham cohomology of analytic and formal neighborhoods of singularities

Suppose that $X$ is a complex algebraic (or complex analytic) variety, and $x \in X$ is a singular point. I am interested in two types of local differential forms at $x$: analytic and formal.
First, ...

**20**

votes

**2**answers

569 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 ...

**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

**0**answers

373 views

### Decidability of equality of elementary expressions

In the following definition the term expression is to be understood as a finite tree built from formal symbols without any predefined meaning assigned to them.
Define the set $\mathcal{E}$ of ...

**20**

votes

**0**answers

940 views

### Is analytic capacity inner regular?

For a compact set $K$ in the complex plane, define the analytic capacity of $K$ by
$$\gamma(K) := \sup |f'(\infty)|$$
where the supremum is taken over all functions $f$ holomorphic and bounded by $1$ ...

**19**

votes

**1**answer

4k views

### Analogue of the Chebyshev polynomials over C?

I've been driven up a wall by the following question: let p be a complex polynomial of degree d. Suppose that |p(z)|≤1 for all z such that |z|=1 and |z-1|≥δ (for some small δ>0). Then what's the ...

**19**

votes

**5**answers

946 views

### Topology on the set of analytic functions

Let $H(D)$ be the set of all analytic functions in a region $D$ in $C$ or in $C^n$.
Everyone who worked with this set knows that there is only one reasonable topology
on it: the uniform convergence on ...

**18**

votes

**2**answers

2k views

### Riemann mapping theorem for homeomorphisms

How do you prove to any two simply-connected domains in the plane are homeomorphic without using the Riemann mapping theorem? An elementary proof would be nice.

**18**

votes

**10**answers

5k views

### Uniformization theorem for Riemann surfaces

How does one prove that every simply connected Riemann surface is conformally equivalent to the open unit disk, the complex plane, or the Riemann sphere, and these are not conformally equivalent to ...

**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?