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Inspired by this question about conjugation of reql analytic maps to a holomorphic function and with a group action view point we ask the following question. The complex Lie group $H=\math …

It is well known that a holomorphic vector field $z'=f(z), z\in \mathbb{C}$ does not have any limit cycle.See the last paragraph of this post Orbits space of real-analytic planar foliations One can …

Let ${\mathbb{R}^2}^+=\{(x,y)\in \mathbb{R}^2\mid x>0, y>0\}$. Let $P:{\mathbb{R}^2}^+\to \mathbb{R}$ be the function with $P(a,b)=$ $\text{The perimeter of ellipse}\;\; \frac{x^2}{a^2}+\frac{y^2}{b^ …

Let $M,N$ be two disjoint closed holomorphic submanifolds of $\mathbb{C}^n$. Is there a holomorphic map $f:\mathbb{C}^n\to \mathbb{C}$ with $f(M)=0,\;f(N)=1$.

Let $X=P\partial_x+Q\partial_y$ be a vector field on the plane $\mathbb{R}^2$. Assume that we have :$$P_xP_y+Q_xQ_y=0$$ Does this imply that the vector field $X$ is a divergence-free vector field w …

I learned from this paper about the Verblunsky theorem. My question is that: What kind of generalizations of this theorem is availlable? In particular I am interested in the following two possible …

Let $H$ be an infinite dimensional seperable Hilbert space. Is there an Irreducible involutive sub algebra $D$ of $B(H)$ with the following properties?: 1)For every open set $U\subset H$ and every Fr …

A real half space in $\mathbb{C}^2$ is $$\{(z,w)\in \mathbb{C}^2\mid \phi(z,w) > \lambda\}$$ where $\lambda$ is a real number and $\phi$ is a $\mathbb{R}$- linear functional from $\mathbb{C}^2$ to $\m …

For every positive real number $x$ we define $$E(x)= \int_0^{\infty} x^t/t!\,\mathrm dt$$ where $t!=\Gamma(t+1)$. This is motivated by classical exponential function. Is this function well defined (t …

Let $M$ be a $1$-dimensional complex manifold. Let $A$ be the space of all holomorphic functions $f:M\to \mathbb{C}$ such that either $f$ is a constant function or every level set $f^{-1}(c)$ is a fin …

The space of smooth vector fields on $\mathbb{R}^2$ and open unit disc $\mathbb{D}$ are denoted by $\chi^{\infty} (\mathbb{R}^2)$ and $\chi^{\infty}(\mathbb{D})$, respectively. A vector field on $\mat …

Edit: I revise the question according to the comment of Gabe Conant. What is an example of a non constant entire function $f:\mathbb{C}\to \mathbb{C}$ which satisfy the following?: For every …

Is every algebraic curve in $\mathbb{R}^2$($\mathbb{C}^2$), the set of critical points of a polynomial in $\mathbb{R}[x,y]$($\mathbb{C}[x,y]$)? In particular what is a real (complex) polynomial whose …

In this question the disc algebra $\mathcal{A}(\mathbb{D})$ is the Banach algebra of all holomorphic functions on the unit open disc $\mathbb{D} \subset \mathbb{C}$ which have a continuou …

Let $D=\{z\in \mathbb{C}\mid |z|<1\}$. Is there a holomorphic function $f:D\to \mathbb{C}$ such that for every $n\in \mathbb{N} \cup \{0\},\;f^{(n)}$ has a continuous extension to $\bar D$ but $f$ has …