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Let D ⊂ ℂ be the closed unit disc in the complex plane, and let C be a continuously embedded path in D between the points -1 and 1. The curve C splits D into two halfs $D_1$ and $D_2$. Let f : D→ℂ be …

Let $f(z)=\sum a_nz^n$ be a Taylor series that converges for $|z|<1$ and satisfies $$ |f(z)|\le \frac{1}{(1-|z|)^{k}} $$ for some fixed $k>0$. Question: What can I deduce about the growth of the Taylo …

Let $U\subset \mathbb C$ be open, bounded, simply connected, with $C^\infty$ boundary. Apply the Riemann mapping theorem to get a bilolomorphic isomorphism $$ f:U\to \mathbb D $$ between $U$ and the u …

My question is of local nature. Let $$f:\mathbb C^n\to\mathbb R$$ be a $C^\infty$ function that vanishes at $0\in \mathbb C^n$, with non-zero derivative. Then, around $0\in \mathbb C^n$, $$M:=f^{-1}(0 …

In physics intros to 2d conformal field theory, people often talk about the "group of conformal transformations". Of course, that's not a group but rather a pseudo-group... that's not what my question …

Let $E=\mathbb C/\Lambda$ be an elliptic curve, and let $D\subset E$ be a very small disc. ($D$ is round for the usual flat metric on $E$) By the main result of [1], there exists a holomorphic immers …

Does every connected, compact Riemann surface $\Sigma$ with boundary, $\partial \Sigma\not =\emptyset$, admit a holomorphic function (smooth on the boundary) $f:\Sigma\to\mathbb C$ whose derivative is …

Let $\mathbb D_r=\{z\in\mathbb C:|z|\le r\}$ be the closed unit disk of radius $r$, let $\mathring {\mathbb D}_r=\{z\in\mathbb C:|z|< r\}$ be its interior, and let $\mathbb A_r=\mathbb D_r\setminus \m …

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 $\m …

Let $X$ be a Banach space (over $\mathbb C$), and let $\mathcal L(X)$ be its algebra of bounded linear operators. Let $U\subset \mathbb C^N$ be an open subset, and $f:U\to \mathcal L(X)$ a function th …

Let $U\subset \mathbb R^n$ be an open. Let $f: S^1 \times U \to \mathbb C$ be a smooth map (i.e. $C^\infty$) s.t. for every $x\in U$ the map $f_x:=f|_{S^1\times \{x\}}:S^1\to \mathbb C$ is a smooth e …

Let $f(z)$ be a holomorphic function on the unit disc, with logarithmic growth at the boundary: $$ |f(z)| = \mathcal O\bigg(\log\Big(\frac{1}{1-|z|}\Big)\bigg). $$ Does it follow that the (distributio …