# Tagged Questions

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

40k 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: ...
39k 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 ...
2k 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 ? ...
11k 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....
2k 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 ...
779 views

299 views

For the chromatic polynomials of graphs we have two nice theorems which describe the behavior of their zeros: Thomassen proved that the set of real zeros of all chromatic polynomials is the union of $\... 1answer 926 views ### How to best distribute points on two concentric circles? An N-subset$\{x_1,\dots,x_N\}$of a compact set$X\subset \mathbb R^d$is called a set of Fekete points (named after Michael Fekete) if it maximizes the product $$\prod_{1\le k<j\le N}|x_k-x_j|\... 2answers 886 views ### Characterize where the Dirichlet Problem for the Laplacian is always solvable Conway's 1978 textbook Functions of One Complex Variable I gives an unsatisfying characterization of the regions for which the Dirichlet Problem can always be solved, and then comments no cleaner ... 2answers 1k views ### Does module Hom commute with tensor product in the second variable? Let A be a commutative ring, and L, M, N be A-modules. Then is it true that$$\text{Hom}_A (L, M)\otimes_A N \cong \text{Hom}_A (L, M\otimes_A N)$$as A-modules? (Note that there is a ... 4answers 1k views ### Visualizing functions with a number of independant variables I need to graph real valued functions ( for exposition and analysis) the issue is the independent variables are more so that the conventional graphing method cant be used and further i don't want to ... 1answer 2k views ### The Paley-Wiener theorem and exponential decay. Consider a function whose Fourier transform is supported on a half-ray:$$ A(t)=\int_0^\infty \omega(E) e^{-iEt}d E,$$where I can suppose$\omega(E)\geq 0$and any suitable regularity conditions on$...
I'm probably missing something elementary here, but I guess the only way to be sure is to ask here. Now, I have encountered a situation where given an nth-degree polynomial $p_n(z)$ with complex ...
This question is inspired from Etienne Ghys's talk on Knots and Dynamics from ICM 2006. The map $L \mapsto (G_4(L), G_6(L))$ gives a bijection between all lattices $L\subset \mathbb{C}$ (including ...