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        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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        18 views

        Dense Stein subset in complex manifold

        Let $X$ be a smooth proper algebraic variety. Then $X$ has a dense affine open subset. In particular, any smooth proper algebraic variety has a dense Stein open subset as the complement of a divisor. ...
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        25 views

        Holomorphic maps from upper half plane to itself (or equivalently Poincare disc to itself)

        Suppose I parametrize complex plane by coordinates,$$z = x+i y,\ \bar z=x-i y$$ then the upper half plane, $\mathbb H_+$ is given by $y>0$. I am looking for chiral coordinate transformations, $f(z)$...
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        Does there exist a Runge Fatou-Bieberbach in each Fatou-Bieberbach domain?

        A Fatou-Bieberbach domain $\Omega \subseteq \mathbb{C}^n$ is a domain that is a proper subset of $\mathbb{C}^n$ and is biholomorphic to $\mathbb{C}^n$. A domain is said to be Runge if for each ...
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        87 views

        On Schwarz Lemma [on hold]

        If $f$ is a holomorphic function on the open unit disc $D$ with $f(0)=0, \;\;|f|<1,$ then we can obtain an improved version of Schwarz Lemma as $$|f(z)|\leq |z|\frac{|f'(0)|+\lambda}{1+\lambda|f'...
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        31 views

        Under what conditions is this family normal?

        Let $\mathcal{S} = \{s \in \mathbb{C}\,\mid\,|\Im(s)| < 1\}$ be a strip of the complex plane. Let $q(s,z)$ be a holomorphic function on $\mathcal{S} \times \mathbb{C}$. Letting $\mathcal{K}$ be a ...
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        2answers
        707 views

        Roots of $x^n-x^{n-1}-\cdots-x-1$

        It is easy to see that $f(x)=x^n-x^{n-1}-\cdots-x-1$ has only one positive root $\alpha$ which lies in the interval $(1,2)$. But it is claimed that this root is a Pisot number, i.e., the other roots ...
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        2answers
        496 views

        What is a really good book for complex variables? [closed]

        I'm an engineering student but I self-study pure mathematics. I am looking for a Complex Variables Introduction book (to study before complex analysis). I have the Brown and Churchill book but I was ...
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        100 views

        Padé Approximants of Power Series with Natural Boundaries

        Consider a power series $\sum_{n=0}^{\infty}c_{n}z^{n}$ for which $c_{n}\in\left\{ 0,1\right\}$ for all $n$. One can write this as: $$\varsigma_{V}\left(z\right)\overset{\textrm{def}}{=}\sum_{v\in V}...
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        52 views

        Ratio of exponentially weighted Selberg integrals

        I'm interested in bounding the following ratio of integral: $$\frac{\int_{0<x_k<...<x_1<1}\prod_{i=1}^kx_i^{m-\frac{k+1}{2}}\prod_{i<j}(x_i-x_j)\exp(-\sum_{i=1}^kw_ix_i)}{\int_{0<x_k&...
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        0answers
        89 views

        Korovkin subset of $C(\mathbb{T})$

        Let $K$ be a compact Hausdorff space and $A$ be a subset of $C(K)$. $A$ is said to be a Korovkin set if for every sequence $(T_n)$ of positive linear operators on $C(K)$, the condition $\|T_n(f)- f\|_\...
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        1answer
        257 views

        Almost complex manifold of dimension 2… locally isomorphic to ??

        I know that this is supposed to be standard, but I don't know how to search for it... hence the question: Let $J$ be an almost complex structure on $M:=\mathbb R^2$, i.e., a $C^\infty$ section of $\...
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        1answer
        120 views

        Conformal mappings and its singularity

        I have a question about singularities of conformal mappings. Let $\mathbb{H} \subset \mathbb{C} \cong \mathbb{R}^2$ be the upper half-place and let $D$ be a Jordan domain. Let $\varphi:\mathbb{H} \to ...
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        First Cousin Problem for Bergman spaces

        I recall (an easy case of) the first Cousin problem : Let $\Omega_1, \Omega_2$ be two open subsets of the complex plane $\mathbb{C}$ with non-empty intersection and $f$ be holomorphic on $\...
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        0answers
        47 views

        Criteria for a limit to be a proper function

        This question is obviously broad; turning this broadness into something sharp is part of the problem. Given a sequence of functions defined on a Riemann Surface $R$, valued in $\Bbb C^2$, what ...
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        0answers
        48 views

        Positivity and zeros of Heun's function

        I am interested in understanding where in the complex plane a Heun function might vanish, or where it (or its real part) is positive. Consider the case where the Frobenius indices at $0$ are $(0,1- \...

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