# Questions tagged [cv.complex-variables]

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

**3**

votes

**1**answer

291 views

### Harmonic equivariant maps and Simpson's correspondence

Let $\Gamma\subset PSL(2,R)$ be a Fuchsian group. For which representations $\rho:\Gamma\to PSL(2,R)$ does there exist a harmonic map from the hyperbolic plane to itself satisfying
$f(\gamma z)=\rho(...

**2**

votes

**1**answer

268 views

### Is a compact subset of a Stein space admitting a fundamental system of Stein neighbourhoods necessarily holomorphically convex?

Let X be a Stein manifold and let K be a compact subset of X. Suppose that K possesses in X a fundamental system of neighbourhoods which are Stein spaces. Then, it is a result by Rossi that such a ...

**2**

votes

**1**answer

315 views

### Is the closure of an open holomorphically convex subset of a Stein space holomorphically convex?

Let X be a Stein manifold and U an open, connected, relatively compact, holomorphically convex subset of X. Is the closure of U in X holomorphically convex?
Also, if X is a Stein space with a finite ...

**14**

votes

**3**answers

3k views

### When I can safely assume that a function is a Laplace transform of other function?

If I have a function and I want to represent it as being the Laplace transform of another, that is, I want to be sure that there is $\hat{f}(s)$ such that my function $f(x)$ can be written as:
$f(x) =...

**11**

votes

**0**answers

1k views

### Dolbeault cohomology of complex tori.

Let $T=\mathbb C^n/\Lambda$ a complex torus. It is completely elementary to prove that the de Rham cohomology of $T$ in degree $q$ is isomorphic to the $q$-th exterior power of the dual of $\mathbb C^...

**12**

votes

**2**answers

549 views

### Can the unit complex 1-dimensional disc be embedded isometrically into complex euclidean space?

Let $D$ denote the unit complex 1-dimensional disc, together with the hyperbolic metric $h_D=\frac{4dz\wedge d\bar{z}}{(1-|z|^2)^2}$of curvature $-1$. By Nash's embedding theorem, we can always embed ...

**5**

votes

**3**answers

1k views

### Product of sine

For which $n\in \mathbb{N}$, can we find (reps. find explicitly) $n+1$ integers $0 < k_1 < k_2 <\cdots < k_n < q<2^{2n}$
such that
$$\prod_{i=1}^{n} \sin\left(\frac{k_i \pi}{q} \...

**3**

votes

**3**answers

999 views

### Analytic continuation via square of absolute value

Is the following fact true (I think that I can prove it but I don't trust
myself on these matters): let $f(z)$ be an analytic function defined on
some open subset $U$ of ${\mathbb C}$. Assume that the ...

**16**

votes

**1**answer

6k views

### Meaning of $\Subset$ notation

The symbol $\Subset$ (occurring in places where $\subseteq$ could occur syntactically) comes up frequently in a paper I'm reading. The paper lives at the intersection of a few areas of math, and I ...

**1**

vote

**0**answers

106 views

### Unbounded Convex domain

Take an unbounded convex domain in C^n, with n>1. Suppose that it is Kobayashi hyperbolic. Is it true that it is biholomorphic to a BOUNDED convex domain? For n=1 it is true due to the Riemann mapping ...

**8**

votes

**3**answers

845 views

### An analytic proof of the De Franchis theorem

The De Franchis theorem in its simplest form states that given two compact Riemann surfaces $\Sigma_{g_1},\Sigma_{g_2}$ where $g_1,g_2 > 1$, there are only finitely many non-constant holomorphic ...

**15**

votes

**5**answers

4k views

### Complex Analysis applications toward Number Theory

I'm an undergrad who is taking a Complex Analysis Course mainly for its applications in number theory.
So I would like to ask some guidelines about which theorems/concepts should I focus on in order ...

**5**

votes

**2**answers

702 views

### Embedding Theorem for big line bundles

Kodaira embedding theorem says that a positive line bundle is ample, i.e. high tensor powers are holomorphically embeddable into complex projective space of high dimension.
However, ampleness is not ...

**5**

votes

**2**answers

2k views

### Algorithm for Weierstrass Preparation Theorem for Formal Power Series

The Weierstrass preparation theorem for formal power series rings guarantees that if a given formal series $f(z) = \sum a_k z^k \in R[[z]]$ where $R$ is a complete local ring with maximal ideal $M$ ...

**12**

votes

**1**answer

545 views

### How can one “see” the Hopf fibration in the space of lattices in the plane?

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 ...

**6**

votes

**3**answers

1k views

### How can one express the Dedekind eta function as a sum over the lattice?

The Dedekind eta function $\eta(\tau)$ can be regarded as a formula which assigns a number to a lattice $L \subset \mathbb{C}$. The algorithm is: rotate the lattice so that one of its basis vectors ...

**1**

vote

**2**answers

304 views

### factorisation of analytic functions

If I have an analytic function in plane $F(x,y)$ that is zero on a curve $y=f(x)$, is it true that
$F=(y-f(x))^n h$, where $h$ is nonzero on the curve? More general, can be somethink said about ...

**3**

votes

**1**answer

435 views

### methods for interpolating a function, holomorphic in the upper halfplane

Let $n,k\colon\mathbb{R}\to\mathbb{R}$ be real functions such that function $N$ given by $N(x)=n(x)-ik(x)$ is a holomorphic function in the upper half-plane. Also I know some additional properties of ...

**7**

votes

**1**answer

1k views

### Solving the Beltrami Equation for a very simple Beltrami Coefficient

Let $\mu$ be a function on the complex plane with the property $\mu(z) = \overline{\mu(\bar{z})}$, such that $\mu(z) = \epsilon e^{-2\pi i \bar{z}}$ on the upper-half plane, where $\epsilon$ is a ...

**7**

votes

**2**answers

1k views

### Brownian Motion Winding Number

Take a simple random walk $\gamma$ in the complex plane conditioned to start at point $a$ and end at point $b$. For this random walk, we can define the winding number $W_\gamma(a,b)$ around $b$ in the ...

**0**

votes

**1**answer

614 views

### Invariance of the cylindrical Laplace equation under conformal transform

hello,
it is often said that a conformal mapping preserves the Laplace equetion in 2D.
However, if this is true for the cartesian coordinates (x,y), where the laplacian is:
$$
\frac{\partial^2 \phi}{\...

**8**

votes

**3**answers

2k views

### Complex manifolds where bounded holomorphic functions are constant

Liouville's theorem states that all bounded holomorphic functions on $\mathbb{C}^n$ are constant.
I'm wondering which connected complex manifolds have this property ?
Connected compact complex ...

**1**

vote

**2**answers

565 views

### structure of singular matrices whose entries have modulus one

Let $A$ be a $n \times n$ matrix all of whose entries has modulus 1.
Suppose the matrix $A$ is singular.
We will assume without loss of generality that all the entries in the first row and the ...

**4**

votes

**1**answer

509 views

### Factorization in the Wiener algebra on the unit disc.

Consider the Banach algebra $W^+=\ell^1(\mathbb{Z}^+)$, viewed upon as the analytic functions $f$ on the unit disc $\mathbb{D}$ such that $$\|f\|=\sum_{k\ge0}|a_k|<\infty$$ where
$$f(z)=\sum a_kz^k$...

**2**

votes

**1**answer

3k views

### Pade approximant to exponential function

Suppose:
a) $p(z)$ is an even degree polynomial (of degree $k = 2j$) with real coefficients;
b) $p(0) = 1$;
c) $p(z)$ and $p(-z)$ have no roots in common anywhere in the complex plane;
d) $f(z) = ...

**14**

votes

**2**answers

1k views

### The Cauchy–Riemann equations and analyticity

I would be interested to learn if the following generalization of the classical Looman-Menchoff theorem is true.
Assume that the function $f=u+iv$, defined on a domain $D\subset\mathbb{C}$, is such ...

**3**

votes

**5**answers

6k views

### What is the angle between two complex vectors?

Let $x, y\in R^n$ and $x, y$ are nonzero, it is well known
$\frac{x^Ty}{\parallel x\parallel_2\parallel y\parallel_2}(\parallel x\parallel_2+\parallel y\parallel_2)\le \parallel x+y\parallel_2$. How ...

**15**

votes

**2**answers

1k views

### Complete metric on the space of Jordan curves?

I was interested in putting a complete metric on the space of Jordan curves. Say, just planar Jordan curves contained in $B(\bar{0}, 2) \backslash B(\bar{0}, 1)$ which separates $\bar{0}$ and infinity....

**1**

vote

**0**answers

255 views

### Finer properties of a sequence of harmonic functions

This was a question that arose for me when I was thinking about how one proves strong unique continuation for elliptic equations. I never could come up with a satisfactory answer.
Background:
When ...

**4**

votes

**2**answers

1k views

### Are there any uses for complex sine? [sin z]

The sine function can take a complex argument. e.g. sin(x + iy)
But does it get used that way in any field? Either practical (e.g. electrical engineering) or in other fields of math? Naturally, I am ...

**23**

votes

**5**answers

2k views

### What is the naming reason of poles in complex analysis?

A function $f: \textbf{C} \to \textbf{C}$ has a pole of order $k$ if $f(z) = \frac{g(z)}{(z-z_0)^{k}}$ where $g(z)$ is a nonzero analytic function. Why do we call it poles?

**0**

votes

**3**answers

1k views

### Higher direct image of coherent sheaf

Hi.
Can any one me say if there is a simple proof of this claim which i can prove it by localization and no easy technique of nuclear spaces...
Let $f:X\rightarrow S$ be an open, surjective ...

**3**

votes

**1**answer

1k views

### Queries about the Skolem-Mahler-Lech theorem (integer zeros of exponential polynomials)

The Skolem-Mahler-Lech Theorem says that the integer zeros of an exponential polynomial are the union of complete arithmetic progressions and a finite number of exceptional zeros. http://terrytao....

**3**

votes

**1**answer

515 views

### Relation between entire function of exponential type and exponential polynomials

Is it true in general that the theory of entire function of exponential type and and that of exponential polynomials (with purely imaginary exponents) are analogous ?
Can one derive results about ...

**2**

votes

**1**answer

248 views

### Linear independence in the algebraic closure of $\mathbb{C}(z)$

Fix $N>0$. Let $b_i=(b_{i,1}, b_{i,2}, b_{i,3}, b_{i,4})$, $i=1,\ldots, m$, be distinct 4-tuples of integers with with all $0\leq b_{i,j}< N$. (The zero tuple is disallowed.)
Define $w_i=(\...

**1**

vote

**3**answers

555 views

### How to find the almost period of an exponential polynomial

Let $u(t) = \Sigma_{k=1}^n c_k e^{i \lambda_k t} (c_k \in \mathbb C, \lambda_k \in \mathbb R) $ be an exponential polynomial of order $n$ with purely imaginary exponents. We can assume that the ...

**119**

votes

**2**answers

12k 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 ...

**34**

votes

**1**answer

9k views

### What, exactly, has Louis de Branges proved about the Riemann Hypothesis?

Hi,
I know this is a dangerous topic which could attract many cranks and nutters, but:
According to Wikipedia [and probably his own website, but I have a hard time seeing exactly what he's claiming] ...

**3**

votes

**0**answers

436 views

### Stability by flat base change of certain properties

Hi.
Let $f:X\rightarrow S$ be a surjective proper, open morphism of reduced or without embedded component complex spaces (or, in alg.geom, surjective proper, universally open morphism of excellent, ...

**4**

votes

**1**answer

5k views

### Inverse of a function defined by an integral

Hi, I have a function defined by an integral as follows.
$$
z=f(w) = \int_0^w \frac{(\zeta-a_1)^{\alpha_1}(\zeta-a_2)^{\alpha_2}...}{(\zeta-b_1)^{\beta_1}(\zeta-b_2)^{\beta_2}...}\ d\zeta
$$
where $w$ ...

**4**

votes

**0**answers

294 views

### Transforming a multivariable integral to make it separable

In the following I will omit requirements of smoothness, extent of domain, finiteness, etc, both to simplify the exposition and because I don't know exactly what the requirements are. Please imagine ...

**3**

votes

**0**answers

253 views

### Quotient of manifolds by groups and embeddings

Let $f:X_1\to X_2$ be a closed submanifold. Let $\rho:G_1\to G_2$ be a closed Lie subgroup. Let $G_1$ acts on $X_1$ and $G_2$ on $X_2$ and suppose $f$ is $\rho$-equivariant. I would like to get a ...

**4**

votes

**2**answers

606 views

### Are there compact analogues of Cartan's theorems A and B?

Cartan's theorem A says that on for a coherent sheaf ${\mathcal{F}}$ on a Stein manifold X, the fibres ${\mathcal{F}}_x$ over each point x in X are generated by global sections.
I'm wondering if ...

**2**

votes

**1**answer

1k views

### The normal derivative of the Green's function

I was wondering if anything was known about the following:
Let $\mathbb{D}^2=\lbrace x^2+y^2< 1 \rbrace \subset \mathbb{R}^2$ be the open unit disk.
Consider now the Green's functions $G(z; p)$ ...

**4**

votes

**1**answer

627 views

### Converse of Picard's Big Theorem?

The celebrated Big Theorem of Picard's is that, in every open set containing an essential singularity of a function $f(z)$, $f(z)$ takes on every value (except for at most one) of $\mathbb{C}$ ...

**0**

votes

**1**answer

230 views

### Flat locus of $S_{1}$-morphism

Hi, everybody.
Consider an ${\rm S}_{1}$- morphism $f:X\rightarrow S$ of reduced complex spaces. Assume that $f$ is open (universally open in Alg.geom), equidimensional with $n$-pure dimensional ...

**7**

votes

**2**answers

841 views

### Contour integration problem from probability

Can integrals of the form
$$
\int_{-\infty}^{\infty}{\exp\left(-\left[x - c\right]^{2}\right) \over 1 + x^{2}}\, {\rm d}x
$$
be computed in closed form using contour integration (or any other ...

**3**

votes

**1**answer

408 views

### Infinite-dimensional complex polynomial or rational Lie algebras and their pseudogroups

In studying the transformation groups generated by holomorphic vector fields V(z) d/dz on ℂ, I've noticed the (surely well-known) fact that the complex quadratic vector fields:
&...

**15**

votes

**2**answers

1k views

### Getting a differential equation for a function from a functional equation of its Mellin transform

If $f$ is a locally integrable function then its Mellin transform
$\mathcal{M}[f]$ is defined by
$$ \mathcal{M}[f] (s) = \int_0^{\infty} x^{s - 1} f (x) dx . $$
This integral usually converges in a ...

**5**

votes

**3**answers

688 views

### Square of an elliptic curve and projective plane

Let's assume one takes $E = \mathbb{C}^* / \langle p \rangle$ an elliptic (Tate) curve over the complex field ($p = e^{2 \pi i \tau}$ where $1, \tau$ are the 2 periods in additive notation; $\Im \tau &...