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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182
votes
48answers
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: ...
15
votes
2answers
917 views

Asymptotic approximation of $x^\alpha$ by entire functions

Given a non-integral real $\alpha$, is there an entire (see http://en.wikipedia.org/wiki/Entire_function) function $h(x)$ such that $x^{-\alpha}h(x)\longrightarrow 1$ for $x\rightarrow+\infty$ (with $...
365
votes
15answers
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 ...
20
votes
1answer
3k 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 ...
20
votes
4answers
2k views

Is the Euler product formula always divergent for 0<Re(s)<1?

It is known that the Euler product formula converges for $\Re(s)>1$ (and there it represents the Riemann zeta function). My question: Is the Euler product always divergent for $0 < \Re(s) < ...
65
votes
15answers
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....
27
votes
10answers
10k 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 ...
9
votes
3answers
2k views

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 ...
13
votes
1answer
2k views

Dirichlet series expansion of an analytic function

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}...
22
votes
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 ...
3
votes
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 $s<1$ ?
73
votes
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 ...
18
votes
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 ...
63
votes
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?
66
votes
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 ...
94
votes
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 ...
40
votes
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 ...
56
votes
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 ...
44
votes
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 $$2(|a|+|b|+|c|) \leq |a+b+c|+|a+b-c|+|a+c-b|+|b+c-a|$$ ...
23
votes
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 ...
33
votes
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), \...
16
votes
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 ...
27
votes
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 ...
26
votes
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 ...
11
votes
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 ...
10
votes
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 ...
15
votes
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(...
11
votes
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(...
8
votes
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 ...
25
votes
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 ...
3
votes
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 ...
3
votes
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,...
2
votes
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 ...
5
votes
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 ...
3
votes
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 $M$...
3
votes
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 ...
1
vote
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 \...
4
votes
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 ...
2
votes
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 ...
1
vote
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{...
0
votes
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/...
124
votes
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 ...
66
votes
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 ....
27
votes
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 ...
15
votes
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....
23
votes
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 ...
14
votes
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 ...
36
votes
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 ...
44
votes
4answers
4k 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. ...
37
votes
1answer
3k 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 $\gamma$...