# Tagged Questions

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### 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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### 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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### 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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### 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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### 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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### 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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### Conformal Mappings for hyperbolic polygon

I am searching for a conformal mapping from the upper halfplane onto a hyperbolic polygon, i.e. the sides of the polygon have to be geodesics. The classical Schwarz Christoffel theorem does the job ...
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### Harmonic level sets and boundary data

This is probably a classic problem, so a good reference book or paper to get me started on this type of question would be great: Let $\mathbb{D} \subset \mathbb{C}$ be the unit disk with boundary ...
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### Is there a “classical” proof of this $j$-value congruence?

Let $j: \mathbf{C} - \mathbf{R} \rightarrow \mathbf{C}$ denote the classical $j$-function from the theory of elliptic functions. That is, $j(\tau)$ is the $j$-invariant of the elliptic curve ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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 ...