# Questions tagged [cv.complex-variables]

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

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### 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: ...
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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 http://scottaaronson....
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### 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 ...
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### When is a holomorphic submersion with isomorphic fibers locally trivial?

A justly celebrated theorem by Ehresmann states that a proper smooth submersion $\pi: X\to S$ between smooth manifolds is locally trivial in the sense that every point $s\in S$ downstairs has a ...
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Let $F(s)=\sum_{n\geq 1}\frac{a_n}{n^s}$ be a Dirichlet series with (finite) abscissa of absolute convergence $\sigma_a$. It can be shown that $\forall \sigma >\sigma_a:$ $$\lim_{T\to\infty}\frac{1}... 3answers 764 views ### Kasteleyn's formula for domino tilings generalized? It seems a marvel when a bunch of irrational numbers "conspire" to become rational, even better an integer. An elementary example is \prod_{j=1}^n4\cos^2\left(\pi j/(2n+1)\right)=1. Kasteleyn's ... 2answers 378 views ### On the integral I_s =\int_{1}^{\infty} (\pi(x)-Li(x))x^{-s-1} dx Define \pi(x) to be the prime counting function and Li(x) the logarithmic integral. Let I_s be defined as above. Is I_s known to be convergent for any real number s<1 ? 36answers 16k views ### Demystifying complex numbers At the end of this month I start teaching complex analysis to 2nd year undergraduates, mostly from engineering but some from science and maths. The main applications for them in future studies are ... 4answers 4k views ### Conformal maps in higher dimensions In dimension 2 we know by the Riemann mapping theorem that any simply connected domain ( \neq \mathbb{R}^{2}) can be mapped bijectively to the unit disk with a function that preserves angles between ... 9answers 13k views ### Why do functions in complex analysis behave so well? (as opposed to functions in real analysis) Complex analytic functions show rigid behavior while real-valued smooth functions are flexible. Why is this the case? 10answers 13k views ### Why does the Gamma-function complete the Riemann Zeta function? Defining$$\xi(s) := \pi^{-s/2}\ \Gamma\left(\frac{s}{2}\right)\ \zeta(s)$$yields \xi(s) = \xi(1 - s) (where \zeta is the Riemann Zeta function). Is there any conceptual explanation - or ... 6answers 16k views ### Why does the Riemann zeta function have non-trivial zeros? This is a very basic question of course, and exposes my serious ignorance of analytic number theory, but what I am looking for is a good intuitive explanation rather than a formal proof (though a ... 5answers 6k views ### Liouville's theorem with your bare hands Liouville's theorem from complex analysis states that a holomorphic function f(z) on the plane that is bounded in magnitude is constant. The usual proof uses the Cauchy integral formula. But this ... 1answer 8k views ### Behaviour of power series on their circle of convergence I asked myself the following question while preparing a course on power series for 2nd year students. Let F be the set of power series with convergence radius equal to 1. What subsets S of the unit ... 3answers 5k views ### Absolute value inequality for complex numbers I asked this question on stackexchange, but despite much effort on my part have been unsuccesful in finding a solution. Does the inequality$$2(|a|+|b|+|c|) \leq |a+b+c|+|a+b-c|+|a+c-b|+|b+c-a|$$... 3answers 1k views ### Universality of zeta- and L-functions Voronin´s Universality Theorem (for the Riemann zeta-Function) according to Wikipedia: Let U be a compact subset of the "critical half-strip" \{s\in\mathbb{C}:\frac{1}{2}<Re(s)<1\} with ... 2answers 3k views ### Must the set of lines through the origin on which a nonconstant entire function is bounded be finite? If an entire function is bounded for all z \in \mathbb{C}, than it's a constant by Liouville's theorem. Of course an entire function can be bounded on lines through the origin z=r \exp(i \phi), \... 2answers 1k views ### Homotopy types of schemes Let X be a scheme over \mathbb{C}. When does the topological space X\left(\mathbb{C}\right) of \mathbb{C}-points have the homotopy type of a finite CW-complex? When does the topological ... 4answers 2k views ### Distribution of roots of complex polynomials I generated random quadratic and cubic polynomials with coefficients in \mathbb{C} uniformly distributed in the unit disk |z| \le 1. The distribution of the roots of 10000 of these polynomials are ... 5answers 2k views ### Continuous + holomorphic on a dense open => holomorphic? Let D ⊂ ℂ be the closed unit disc in the complex plane, and let C be a continuously embedded path in D between the points -1 and 1. The curve C splits D into two halfs D_1 and D_2. Let ... 2answers 1k views ### Does this product have analytic continuation? The product$$ F(s)=\prod_{p}\frac1{(1-p^{-s})^p}, $$converges for \mathrm{Re}(s)>2, when p runs over all primes. Does it admit analytic continuation beyond the line \mathrm{Re}(s)=2? Any ... 2answers 1k views ### Characterize where the Dirichlet Problem for the Laplacian is always solvable Conway's 1978 textbook Functions of One Complex Variable I gives an unsatisfying characterization of the regions for which the Dirichlet Problem can always be solved, and then comments no cleaner ... 3answers 884 views ### Is this lower bound for a norm of some complex matrices true? Let A = [a_{ij}]_{n\times n} be a Hermitian matrix, such that |a_{ij}| =1 for i \neq j, and a_{ii} = 0 for each i. I am interested in a tight lower bound of \|A\|_*:=\sum_{i=1}^n |\lambda_i(... 4answers 1k views ### Seeking a Geometric Proof of a Generalized Alternating Series' Convergence Let z \in \mathbb{C} \backslash \lbrace 1 \rbrace with |z| = 1. We consider the following infinite series, which necessarily converges:$$S(z) := \sum_{n = 1}^{\infty}\frac{z^n}{n}$$Note that S(... 1answer 911 views ### Extending an assignment property from Q to R (or C) Property of any odd number of nonnegative integers: Given x_1 \leq \ldots \leq x_{2n + 1} with each x_i \in \mathbb{Z}_{\geq 0}, suppose that for any x_i we remove, the remaining numbers can be ... 2answers 995 views ### Are there irreducible polynomials with all zeros on two concentric circles? This is somewhat similar to this recent question, but extending in a different direction. Let f(x) be an irreducible polynomial of degree n with integer coefficients. Call such f a bicycle ... 1answer 208 views ### Conformal map onto a circle, from an identification space composed of five squares I am looking to derive a conformal map for the problem illustrated in this image. I've read a bit about how to map a square onto a circle, but I'm struggling to extend the concepts for the domain at ... 2answers 470 views ### Basic questions on the Hilbert scheme/ Douady space Let X be a complex projective scheme (resp. complex analytic space). The Hilbert scheme (resp. Douady space) parameterizes closed subschemes (resp. complex analytic subspaces) of X. More precisely,... 2answers 516 views ###  2|f^{'}(0)| = \sup_{z, w \in D} |f(z)-f(w)| if and only if f is linear I know the following is a well-known result. Let D = B(0,1) \subset \mathbb{C}  a disc, f holomorphic on D. Show that$$ 2|f^{'}(0)| \le \sup_{z, w \in D} |f(z)-f(w)|$$Furthermore, there is ... 0answers 453 views ### On properties on a certain functional Consider the following function:$$F(z) = \omega(z)\sin^2\left(\frac{c\Gamma(z)}{z}\right)Here, \omega(z) is a weight we have to construct and c is a constant. The following three conditions ... 1answer 311 views ### Explicit form for hermitian structure h with respect to \omega Let (M,\omega) be a symplectic manifold. and \pi:L\to M be a complex line bundle , we denote h as hermitian structure,i.e. if s,s' are smooth sections of L and if X is a vector field on M... 1answer 307 views ### Off-diagonal holomorphic extension of real analytic functions on \mathbb{C}^n \times\mathbb{C}^n I am struggling trying to understand an statement in a paper I am reading: Let M be a complex manifold of dimension 2n. Let's consider a function \xi: M \rightarrow \mathbb{C} whose ... 1answer 467 views ### meromorphic extension of a function Let \Lambda\in \mathbf{C} be a discrete subset. We assume that \mathrm{Re}(\lambda)<0 for all the \lambda\in \Lambda. For i\in \mathbf{N}, \lambda\in \Lambda, let m_{i,\lambda}\in \... 2answers 518 views ### Reference request: Oldest complex analysis books with (unsolved) exercises? Per the title, what are some of the oldest complex analysis books out there with (unsolved) exercises? Maybe there are some hidden gems from before the 20th century out there. I am aware of the ... 1answer 513 views ### Half spaces free of roots of a given polynomial I thank Loic Teyssier and Emil Jerabek who helped me to revise the two previous version This question is motivated by the following fact in complex variable:(I learned this fact from the book of ... 0answers 220 views ### Multivariate solution to Lambert W / product-log function Consider solving the following system for x \begin{align*} a - b e^{\theta x} - cx = 0 \end{align*} According to your favorite computer algebra program, one possible (and the simplest) is \begin{... 2answers 901 views ### Question on Hartogs's Extension Theorem Does Hartogs's extension theorem hold if one replaces the word holomorphic by analytic (of course still in several variables)? For Hartogs's Extension Theorem see here: http://en.wikipedia.org/wiki/... 2answers 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 ... 3answers 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 .... 7answers 5k views ### Elementary Proof of Riemann-Roch for Compact Riemann Surfaces I am supposed to give a talk about the Riemann-Roch theorem to a seminar of first and second year graduate students. I want to do Riemann-Roch for compact Riemann surfaces, but I am open to perhaps ... 5answers 6k views ### References for complex analytic geometry? I'm looking for references on the "algebraic geometry" side of complex analytis, i.e. on complex spaces, morphisms of those spaces, coherent sheaves, flat morphisms, direct image sheaves etc.... 5answers 3k 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 ... 2answers 2k views ### Reason for studying coherent sheaves on complex manifolds. Hello everybody! I would be interested in knowing, what the reason is for investigating coherent sheaves on complex manifolds. By definition a sheaf F on a complex manifold X is coherent, when it ... 2answers 2k 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 ...
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 $\gamma$...