Questions tagged [cv.complex-variables]
Complex analysis, holomorphic functions, automorphic group actions and forms, pseudoconvexity, complex geometry, analytic spaces, analytic sheaves.
3,298 questions
3
votes
1
answer
473
views
Is a function which is finitely multiple-valued in each variable separately, also finitely multiple-valued in all its variables jointly?
It is well known that under suitable conditions, a function which is:
a polynomial in each variable separately is a polynomial in all its variables jointly.
a rational function in each variable ...
- 371
0
votes
1
answer
198
views
An integral arising in statistics(2)
The integral I am interested in is:
$$t(x)=\int_{-K}^{K}\frac{\exp(ixy)}{1+y^{2q}}dy$$
$K<\infty$, q natural number
For q=1 one can use contour integration.
So for K>1 we have :
$$\pi/2-\...
- 267
11
votes
1
answer
812
views
Approximation to divergent integral
Hi everyone,
I'm a physicist working on stochastic processes and I've come up against an integral that I'm not able to approximate using steepest descent (I don't have a large or small parameter), ...
- 111
6
votes
3
answers
3k
views
Zeros of the Weierstrass $\wp$-function
This question was prompted by the post here, and I asked this earlier, deleted it, and due to pressure exerted by Ilya Nikokoshev, I am asking it again. Apologies to Pavel Etingof.
Q1. Let $\Lambda$ ...
- 7,442
2
votes
2
answers
242
views
Simultaneous convergence of powers of unit complex numbers
Let $z_1,\ldots,z_n$ be complex numbers of modulus one. Does it exist an increasing sequence $k_j\in\mathbb{N}$ such that $\lim_{j\to\infty}z_i^{k_j}=1$ for all i?
- 971
13
votes
1
answer
860
views
What does the incidence algebra of the lattices in C tell us about modular forms?
I have two different and probably unrelated questions that can both be superficially described by the title, so I hope you'll forgive me if I ask them together. They both fall under the category of ...
- 118k
0
votes
1
answer
412
views
An integral arising in statistics
The integral I need:
$$t(x)=\int_{-K}^{K}\frac{\exp(ixy)}{1+y^{2q}}dy$$
$K<\infty$, q natural number
For q=1 this integral is
$$\pi/2-\int_{Arc}\frac{\exp(ixy)}{1+y^{2}}dy $$
Where Arc ...
- 267
106
votes
6
answers
19k
views
Why does the Riemann zeta function have non-trivial zeros?
This is a very basic question of course, and exposes my serious ignorance of analytic number theory, but what I am looking for is a good intuitive explanation rather than a formal proof (though a ...
- 29k
1
vote
1
answer
1k
views
How can I calculate the characteristic function of these distributions? [previously: difficult integral]
How to compute this integral in general case?
$$t(x)=\int_{-\infty}^{\infty}\frac{\exp(ixy)}{1+y^{2q}}dy$$
Mathematica can compute it when q is known. For example,for q=1 this integral is
$$\exp(-{\...
- 267
11
votes
6
answers
3k
views
Explicit Spin Structures on the Torus
Basically, I am trying to build explicit examples of Dirac operators. To this end, I'm looking at the surface E = C/(Z + λZ) - for some λ in H \ SL(2,Z) - with the Euclidean metric and ...
- 22.8k
5
votes
1
answer
1k
views
Mode of convergence of a power series
I am looking for a power series $\displaystyle f(z) = \sum_{n=0}^{+\infty} a_n z^n$ that converge uniformly on $\mathcal{D} = \Big\{ z \in \mathbb{C} \ / \ \vert z \vert \leq 1 \Big\}$ but not ...
- 53
5
votes
1
answer
1k
views
Why can't subvarieties separate?
I'm posting my answer to this question as its own question:
Let $V$ be an irreducible projective variety over $\mathbb{C}$. Let $U$ be a Zariski open set in $V$. I'll use $V(\mathbb{C})$ and $U(\...
- 10.6k
3
votes
1
answer
895
views
Bernstein inequality for multivariate polynomial
Let $P$ be a polynomial in $k$ variables with complex coefficients, and $\deg P=n$. If $k=1$ then there is Bernstein's inequality:$||P'||\le n||P||$, where $||Q||=\max_{|z|=1} |Q(z)|$.
So, are there ...
- 1,575
1
vote
1
answer
10k
views
Region and domains? [closed]
Is every region a domain? Am I correct that I understood the definition of domain to be an open, connected set? Does every region have to be a domain? For example:
$|z-1+i|\le 3$ is a region if I've ...
- 55
7
votes
1
answer
456
views
Reference for equivalent definitions of the genus
Let $X$ be a (edit: nonsingular) projective complex algebraic curve. The genus of $X$ can be defined as the dimension of the space of holomorphic $1$-forms on $X$, which in turn can be defined either ...
6
votes
3
answers
2k
views
Complex projective space as a $U(1)$ quotient
$\DeclareMathOperator\U{U}\DeclareMathOperator\SU{SU}$As is well known, one can view $\mathbb{CP}^n$ as a quotient of the unit $(2n + 1)$-sphere in $\mathbb{C}^{n+1}$ under the action of $\U(1)$, ...
- 751
-2
votes
2
answers
2k
views
Taylor series of a complex function that is not holomorphic
I want to create Taylor series of a complex function that has complex conjugate in it. Obviously I cannot do a total derivative but derivations over real and imag parts exist.
Bonus question: Can I ...
3
votes
2
answers
625
views
Continuation up to zero of a Dirichlet series with bounded coefficients
Let $a_n$ be a bounded sequence of positive real numbers. Is it the case that the Dirichlet series $\sum \frac{a_n}{n^s}$ can be meromorphically continued up to the right of zero, with at the most a ...
- 7,442
12
votes
1
answer
5k
views
Conformal maps of doubly connected regions to annuli.
In another question here on MO, Anweshi asks if any doubly connected region in the complex plane can be conformally mapped to some annulus. The answer to this is yes. But the fact is that two annuli ...
- 2,050
8
votes
3
answers
2k
views
Riemann mapping for doubly connected regions
Remove the closure of simply connected region from the interior of a simply connected region. Is it true that the resulting domain can be mapped conformally to some annulus?
- 7,442
14
votes
5
answers
2k
views
What is $\sum (x+\mathbb{Z})^{-2}$?
This is a simple question, but its been bugging me. Define the function $\gamma$ on $\mathbb{R}\backslash \mathbb{Z}$ by
$$\gamma(x):=\sum_{i\in \mathbb{Z}}\frac{1}{(x+i)^2}$$
The sum converges ...
- 13k
5
votes
1
answer
513
views
Field of Definition of a Meromorphic Function
Question
Let X be a smooth, projective curve over the algebraic closure of ℚ. Let f:X->ℙ1 be a meromorphic function. Assume that the zeros and the poles are defined over some number field,...
- 1,522
23
votes
5
answers
11k
views
Example of continuous function that is analytic on the interior but cannot be analytically continued?
I am looking for an example of a function $f$ that is 1) continuous on the closed unit disk, 2) analytic in the interior and 3) cannot be extended analytically to any larger set. A concrete example ...
- 757
26
votes
3
answers
2k
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 ...
- 7,127
212
votes
52
answers
82k
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:
...
31
votes
11
answers
13k
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 ...
8
votes
1
answer
2k
views
Level set of a harmonic function
Let $u$ be a nonconstant real-valued harmonic function defined in the open unit disk $D$. Suppose that $\Gamma\subset D$ is a smooth connected curve such that $u=0$ on $\Gamma$. Is there a universal ...
20
votes
4
answers
6k
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 ...
- 757
23
votes
1
answer
5k
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 ...
- 9,823
4
votes
3
answers
2k
views
Most important domains, extension theorems, and functions in several complex variables
For a new learner of several complex variables, the many domains (eg holomorphically convex, pseduconvex, Stein) and the many extension theorems (eg Riemann) and the many functions (plurisubharmonic) ...
user2529
74
votes
10
answers
18k
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 ...
- 12.3k
36
votes
6
answers
2k
views
When are some products of gamma functions algebraic numbers?
I want to know when certain expressions of the form
$ {\Gamma(r_1/m) \Gamma(r_2/m) \ldots \Gamma(r_j/m) \over \Gamma(s_1/m) \Gamma(s_2/m) \ldots \Gamma(s_j/m)} $
are algebraic numbers. These ...
- 14k
2
votes
2
answers
2k
views
Inversion of Laurent series
For a power series $f(z) = \sum_{i=0}^{\infty} a_i z^i$ with $a_1$ nonzero, Lagrange's inversion formula gives an explicit way to compute the Taylor coefficients of the inverse function.
Is there any ...
- 21k
6
votes
1
answer
780
views
What is the origin of this positive matrix characterization of bounded analytic functions on the unit disk?
Background: Let $S$ denote the so-called Schur class of complex analytic functions from the open unit disk $D$ in $\mathbb{C}$ to the closed unit disk $\overline{D}$. Given distinct points $z_1,\...
- 7,329
3
votes
1
answer
263
views
Asymptotically multiplicative functions and matrices
Hi,
Let $\mathbb{N}_{cop}^2$ denote the set of all pairs of coprime natural numbers. A function $f:\mathbb{C}\rightarrow\mathbb{C}$ is called asymptotically multiplicative, iff $\epsilon_{m,n}:=f(mn)...
- 7,127
9
votes
1
answer
695
views
Asymptotics of Power Series With Branch Singularities
I am wondering if there are analytic tools to find asymptotic formulae for the coefficients of a complex power series of a function with branch singularities. For example, it is possible to show ...
- 740
8
votes
1
answer
638
views
Composite residues with determinant denominators
I am looking for a good reference on composite residues of multi-variable contour integrals (something better and more explicit than Griffiths and Harris or Tsikh). This means I want to evaluate $\...
- 81
8
votes
2
answers
1k
views
Is the maximum domain to which a Dirichlet series can be continued always a halfplane?
Let $f(s)=\sum_n a_n n^{-s}$ be a Dirichlet series whose coefficients satisfy $\lvert a_n\rvert\leq n^{C}$. Then $f(s)$ converges absolutely in some halfplanes, and is conditionally convergent in (...
- 7,826
5
votes
0
answers
533
views
Two meromorphic functions with overlapping sets of poles
Assume that we have two meromorphic functions $f(z,w)$ and $g(z,w)$, where $z$ and $w$ are complex (we are interested only in behavior on compact sets). Fix $z$ and assume that the sets of poles of $f(...
- 228
3
votes
6
answers
1k
views
Dolbeault cohomology
Hello
I am trying to get a good book that explains the Dolbeault Cohomology, does anyone know of a good one?
- 143
22
votes
6
answers
2k
views
Elementary solutions to f(z+1)-f(z)=g(z) in entire functions
Let g(z) be an entire function of a complex variable z. Does there exist an entire function f(z) such that f(z+1)-f(z)=g(z)? As I learned several years back, the answer to this is apparently 'yes', ...
- 7,826
74
votes
15
answers
18k
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://...
- 24.7k
13
votes
2
answers
862
views
Motivation for BMO
At the moment, I don't have access to the early 1960's paper of John and Nirenberg that (from what I understand) introduced the space BMO (bounded mean oscillation). Why were John and Nirenberg ...
- 261
72
votes
9
answers
16k
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?
- 1,093
18
votes
3
answers
1k
views
Uniformization theorem in higher dimensions
Let $M$ be a 4-manifold with a complex structure.
Does there exist a finite list of simply connected complex 4-manifolds $M_1, ... , M_n$ such that M is the quotient of some $M_i$ by the action of a ...
- 7,062
12
votes
2
answers
3k
views
What are conditions on real coefficients for zeros of a polynomial to be on the unit circle?
My complex analysis is decades in the rear view mirror. Perhaps someone here can help. I am looking for necessary and sufficient conditions on the coefficients of of a real polynomial of one complex ...
- 299
12
votes
4
answers
3k
views
Elliptic Curves, Lattices, Lie Algebras
I've recently started to look at elliptic curves and have three basic questions:
Is it correct to say that elliptic curves $E$ in the projective plane are in bijective correspondence with lattices $...
- 1,462
6
votes
4
answers
2k
views
Space of $(1,0)$-holomorphic forms on a Riemann surface
In a complex analysis course I have been given the following definition:
Let $X$ be a Riemann surface, denote by $H(1,0)$ the space of all $(1,0)$-holomorphic forms on $X$ and consider the quotient ...
- 143