# Questions tagged [cv.complex-variables]

Complex analysis, holomorphic functions, automorphic group actions and forms, pseudoconvexity, complex geometry, analytic spaces, analytic sheaves.

2,306 questions
Filter by
Sorted by
Tagged with
53k 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 ...
64k 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: ...
14k 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 ...
16k views

### Why does the Riemann zeta function have non-trivial zeros?

This is a very basic question of course, and exposes my serious ignorance of analytic number theory, but what I am looking for is a good intuitive explanation rather than a formal proof (though a ...
16k 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 ...
13k 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 ...
5k 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 ....
14k 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 http://scottaaronson....
5k views

4k views

### Is the Riemann zeta function surjective?

Is the Riemann zeta function surjective or does it miss one value?
3k views

### Pathology in Complex Analysis

Complex analysis is the good twin and real analysis the evil one: beautiful formulas and elegant theorems seem to blossom spontaneously in the complex domain, while toil and pathology rule the ...
2k views

1k views

### Which of the proofs of the fundamental theorem of algebra can actually produce bounds on where the roots are?

One of the old classic MO questions is a big-list of proofs of the fundamental theorem of algebra. Here is a second big-list question about this big list: Which of the FTA proofs can, even in ...
5k 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 Yau'...
3k 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 ...
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 $m,n$, ...