The complex-analysis tag has no wiki summary.

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### biholomorphism complex manifold induced structure

Let $X$ be a $n$ dimensional complex manifold with complex structure $I$ and assume one has a diffeomorphism $f : \mathbb{C} \rightarrow X$ of some open set $U$ in $\mathbb{C}$ into its image $f(U)$. ...

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### Elementary Luroth theorem proof?

Hi, everyone!
I'm trying to explain the proof of Luroth theorem (every field $L$, s.t. $K\subset L\subset K(t)$, is isomorphic to $K(t)$) to the high-school audience. I'm not going to use such ...

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311 views

### Why is Mellin-inverse of Gamma periodic?

Specific Case
The periodicity is obvious from computation:
$$\cal{M}^{-1}\{\Gamma\}(x) := \frac{1}{2\pi i}\int_{c}\Gamma(s)x^{-s}d s=e^{-x}$$
However, is there a way to see directly from the integral ...

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275 views

### Morera type theorems

In Stein and Shakarchi, Complex Analysis, Princeton lectures in Analysis, Chapter 2, Problem 2 an interesting question is posed. The problem section in each chapter contains more complicated problems, ...

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408 views

### When checking if a harmonic function is continuous on its boundary, is a dense subset enough?

Let $U$ be an open connected subset of $\mathbb{C}$ and let $u:U\rightarrow \mathbb{R}$ be harmonic and bounded on $U$.
Let $f:\partial_\infty U \rightarrow \mathbb{R}$ be a continuous function, ...

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749 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 ...

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273 views

### hesse matrix under diffeomorphism

Let $u : U \rightarrow \mathbb{R}$, where $U \subset \mathbb{C}^{n}$ be a strictly plurisubharmonic smooth function and consider its complex hesse matrix $Hess^{\mathbb{C}}(u)$. Furthermore consider a ...

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### Evaluating the integral $\int_0^\infty \frac{\psi(x)-x}{x^2}dx.$

Let $\psi(x)=\sum_{n\leq x} \Lambda(n)$ be the weighted prime counting function. I am trying to evaluate the integral $$\kappa:=\int_{1}^{\infty}\frac{\psi(x)-x}{x^{2}}dx$$ in several different ways. ...

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223 views

### Are all continuous linear operators on the space of entire functions “simple”?

Let $\langle \operatorname{Ent},+,\cdot \rangle$ be the (complex) vector space of entire functions.
For all members $n$ of $\{1,2,3,...\}$, define $||\cdot ||_n : \operatorname{Ent} \to \mathbb{R}$ ...

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### Relation between partially computable function and complex function

Given a partially computable function, is there an analytic complex function which is equal to it at every point of it's domain? Or under what condition does a partially computable function correspond ...

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### Does the Euler product formula diverge for any zero of the Riemann zeta function?

Simple question (but not for me):
Does the Euler product formula diverge for any zero of the Riemann zeta function?
The reason why I ask this is that I heard we should not use the Euler product ...

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215 views

### Existence of a special holomorphic function

How can you prove the existence of a nonzero function from the subset $U= \{z| 0 \leq Re z \leq 1\}$ of $\mathbb C$ to $\mathbb C$ which is holomorphic on the interior of $U$ and vanishes on the right ...

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### 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 ?
...

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### Is there an integration free proof (or heuristic) that once differentiable implies twice differentiable for complex functions?

The title pretty much says it all. I am revisiting complex analysis for the first time since I "learned" some as an undergraduate. I am trying to wrap my head around why it should be the case that a ...

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619 views

### Quasiconformal harmonic extension of a quasi-symmetric map on $S^1$

Hello ,we know that for given $h:S^1\to S^1$, we can solve the Dirichlet problem on $\bar{D} $ with the boundary value $h$ and in fact this extension, which is the complex harmonic extension $H=E(h) $ ...

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578 views

### Holomorphic line bundles on a punctured disc

Is every holomorphic line bundle on the - say - punctured unit disc $\dot{\Delta} \subseteq \mathbf{C}$ trivial? Griffiths-Harris (p. 39) prove that $H^{p,q}_{\overline{\partial}}(\Delta) = 0$ (for $q ...

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584 views

### Hardy spaces: analysis <---> martingales

Let $H^p$ be the Hardy space of analytic functions on the open unit disk $D$: $f \in H^p$ if $f$ is analytic on $D$ and $\sup_{r < 1} \int_0^{2\pi} |f(re^{i\theta})|^p d\theta < \infty$.
...

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472 views

### Cos[Im[zetazero(n)]Log(prime)] spans a countable dense set in [-1,1]?

It is known that cos(N) spans a countable dense set in [-1,1].
(N: any natural number)
As far as I know generally, for any continuous function f defined in [a,b],
f is Riemann integrable where its ...

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324 views

### Given: $\phi(s) = \sum_p \frac{\log p}{p^s}$ Why is: $\lim_{e \rightarrow 0} e \phi(1+e)$ = 1 ?

Problem:
Given: $\phi(s) = \sum_p \frac{\log p}{p^s}$
(summation is only over primes)
Why is: $\lim_{e \rightarrow 0} e \phi(1+e)$ = 1 ?
Context: ...

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### Meromorphic 1-form and Picard's theorem

Let $D$ be the open unit disk in the complex plane and $U_1,U_2,\,\ldots\,,U_n$ be an open cover of the puntured disk $D^*= D\setminus\{0\}$. Suppose on each open set $U_j$ there is an injective ...

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### Removable sets for harmonic functions and hardy spaces of general domains

Hi,
Let $\Omega$ be a domain of the complex plane. The Hardy space $H^p(\Omega)$ is defined, for $1 \leq p<\infty$, as the class of functions $f$ that are holomorphic on $\Omega$ such that $|f|^p$ ...

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592 views

### Analysis and finitely generated groups

Dear all, this is perhaps a bit a vague question, but some references would already be very helpfull.
So let $G$ be a finitely generated group and choose some finite set of generators. This allows to ...

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808 views

### A “Cauchy integral formula” for the Poisson kernel?

The inspiration for this question is in a certain breakdown of the analogy between holomorphic and harmonic functions.
First recall the Cauchy integral formula:
Let $U$ be an open subset of ...

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251 views

### Is the complex harmonic extension of a $\mathcal{C^r}$ map from $S^1$ to $\mathbb{C}$ is smooth upto the boundary ?

Suppose we have a map $ h : S^1\to \mathbb{C} $ that we know is a $\mathcal{C^r} $ map ( in the sense of a map between 1-manifold ( or in the sense of a $2\pi$ periodic map from $\mathbb{R}\to ...

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529 views

### Non-algebraic curve visualisation

Is there any software which can automatically visualise a non-algebraic
complex curve, I mean the structure of it's ramification points and sheet?
I think a good test example would be the Lambert ...

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**1**answer

382 views

### Holomorphically Convex Hull a Subset of the convex hull of

This comes from Hörmander's "An Introduction to Complex Analysis in Several Variables".
We defined the $A(\Omega)$-hull (analytic functions in an open set $\Omega$). $\hat{K}$ of a compact set ...

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281 views

### Necessary condition for a branch point

If I have a function $f(z,\alpha)$ (let's keep it a polynomial of order $\geq 2$ in $z$, for simplicity), what would be a necessary condition for there to be branch points for this function? A friend ...

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### What holomorphic functions are limits of polynomials?

Let $\Omega$ be a connected open set in the complex plane. What is the closure of the polynomials in $\mathcal{H}(\Omega)$ the set of holomorphic functions on $\Omega$? The topology is the usual ...

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335 views

### Minimum number of polygonal lines connecting points in an annulus.

It is obvious that an open annulus in the complex plane: $S = a < |z| < b$ is connected. That is, each pair of point $z_1$ and $z_2$ in it can be joined by a polygonal line.
What is the minimum ...

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### Homology of a region of the plane

This is related to this MO question, I don't know if it's really "research-level". As in that question, let $U$ be a domain of the complex plane $\mathbb{C}$, i.e. an open connected subset. Let
$$ ...

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### Meaning of Cauchy integral theorem - the (co)homology viewpoint

I'm not sure what follows is not just a complicated way to deduce a blatant triviality, or if is even correct. Let's try.
In the elementary theory of analytic functions of $1$ complex variable, one ...

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### Is a non-compact Riemann surface an open subset of a compact one ?

Let $X$ be a non-compact holomorphic manifold of dimension $1$. Is there a compact Riemann surface $\bar{X}$ suc that $X$ is biholomorphic to an open subset of $\bar{X}$ ?
Edit: To rule out the case ...

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382 views

### Proving uniform bound

Hello
I want to prove that
$\lim_{h\rightarrow\infty}\left(\int_{0}^{\infty}\left(\cos ...

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894 views

### When are complex polynomial maps almost surjective?

Consider a complex polynomial map $f: \mathbb{C}^n \rightarrow \mathbb{C}^n$.
For $n = 1$, the fundamental theorem of algebra says that, for any $y \in \mathbb{C}$ there exists $x \in \mathbb{C}$ ...

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### how to prove the relationship between pseudoconvexity and the monge-ampere matrix?

In several complex variables , to determine the pseudoconvexity of a domain in $C^n$ is very important . There are various criterion to decide whether a domain is pseudoconvex . In particular ,if ...

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### Orbits for homogenous complex polynomials under unitary rotation of variables

Let's have two complex homogeneous polynomials of degree $k$: $f(z_1,\cdots,z_n)$ and $g(z_1,\cdots,z_n)$. We consider rotations of variables in the form of $\vec{z}' = U \vec{z}$, where $U\in SU(n)$.
...

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608 views

### Irreducible non-singular M-matrices and complex numbers

It is well known that a non-singular M-matrix that is irreducible has a strictly positive inverse (all entries $>0$).
An M-matrix is a matrix that has eigenvalues with positive real part, and the ...

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376 views

### Topology and convergence for conformal maps of disk

Let $\mathbb{D}$ be the open unit disk in $\mathbb{C}$, centre $0$. Write $\mathcal{S}$ for the holomorphic injective maps $\{ f : \mathbb{D} \to \mathbb{D} | f(z) = e^{-\lambda} z + O(z^2) \}$ i.e. ...

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### Polylogarithm inequality

Recall the polylogarithm $Li_s(z)=\sum_{n=1}^\infty \frac{z^n}{n^s}.$
For $|z|<1,$ is $\Re \left( \frac{Li_1(z)}{Li_2(z)} - \frac{2}{3}\frac{Li_2(z)}{Li_3(z)} \right)>0?$
The numerics suggest ...

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### Proper holomorphic map from unit disk to punctured unit disk

It is easy to see that there cannot be a proper holomorphic map from the punctured unit disk to the unit disk in the complex plane. What about the other direction; does there exist a proper ...

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### Is there a “Riemann mapping theorem” for a circle in C^2 ?

The Riemann mapping theorem says that if you have a simple closed curve in $\mathbb{C}$, then there is an essentially unique way to map a holomorphic disc to the interior. Is there any reasonable ...

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### Is there a manifold structure on a space of conformal maps?

I would be very grateful for any information or pointers for the following:
1) Fix an open subset $U$ of $\mathbb{CP}^1$. a) Does the set of all holomorphic maps from $U$ to $\mathbb{C}$ (with the ...

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### magic square in the complex plane with equal integrals along every horizontal, vertical and diagonal

What can you say about a function defined on a square region of the complex plane, if the integral of the function along any horizontal, vertical or diagonal of the square is equal ? - an analytic ...

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235 views

### Question in complex analysis arising from large $N$ gauge theory

This is a question in complex analysis that comes up in the treatment of large
$N$ gauge theory with gauge group $SU(N)$ in the 't Hooft limit where $N$ is taken to infinity with $\lambda=g^2 N$ fixed ...

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### Pole data of meromorphic matrix function

Let $T(z)$ be a meromorphic square matrix function, that is - a matrix whose entries are complex meromorphic function of one variable.
Recall that such a $T$ is said to have a right pole of order $r$ ...

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### Relation between complex analysis and harmonic function theory [closed]

There are some theorems in harmonic function theory that resemble results in complex analysis, like:
Holomorphic functions and complex functions are analytic;
Cauchy's integral formula in complex ...

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### Does Riemann map depend continuously on the domain?

I've always taken this for granted until recently:
In the simplest case, given Jordan curve $C \subseteq \mathbb{C}$ containing a neighborhood of $\bar{0}$ in its interior. Given parametrizations ...

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### Conjugate Groups of (quasi) Fuchsian Groups

I apologize in advance if this question is so trivial or too low level.
Let $\Gamma$ be a Fuchsian group. Let $\mathcal{F}$ be the set of pairs $(\mu,f)$, where $\mu \in L^\infty(\mathbb{C})$ such ...

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### How manifold-like is Aut(C^n) in the holomorphic category?

This question is similar to, but not the same as this one. Take the space of automorphisms of $\mathbb{C}^n$ in the holomorphic category, with the compact-open topology. For $n=1$ this is just ...

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### What $Re(f(z))=c$ can be if $f$ is a holomorphic function ?

Suppose that $f:U\subset\mathbb{C}\to\mathbb{C}$, where $U$ is a region in the complex plane, is a holomorphic function.
Of course, if $c\in\mathbb{R}$ is a regular value for $\text{Re}(f(z))$ then ...