Tagged Questions

1
vote
1answer
35 views

What is the radius of convergence of a power series in two variables?

What is the radius of convergence of a power series in two real variables? If I were to fix one of the variables (i.e. make it a real constant), then would the radius of convergence simply be related ...
9
votes
1answer
242 views

Show that f is a polynomial

Suppose $f$ is an entire function on $\mathbb{C}^n$ that satisfies for every $\epsilon>0$ a growth-condition $$|f(z)|\leq C_{\epsilon}(1+|z|)^{N_{\epsilon}}e^{\epsilon | \text{Im}\,z|}$$ ...
1
vote
1answer
41 views

Zeros of function $f: \mathbb{C}^{n}\to\mathbb{C}$

I'm having trouble understanding how this condition $(*)$ comes about. I understand the proof, but do not clearly understand how the lemma follows from the proof. Any clarification or additional ...
2
votes
1answer
42 views

Holomorphic extension of a function to $\mathbb{C}^n$

I am stuck at the following question : Let $f$ be a holomorphic function on $\mathbb{C}^n \setminus \{(z_1, z_2, \ldots, z_n) | z_1=z_2=0\}$. Show that $f$ can be extended to a holomorphic function ...
1
vote
1answer
24 views

Biholomorphic Equivalence in $\mathbb{C}^n$

I am stuck at the following problem : For $n \geq 2$, are the sets $\Delta_n$ and $H_n$ biholomorphically equivalent ? Here, $ \Delta_n = \{ ( z_1, z_2, \ldots z_n) : |z_i| < 1, i = 1, 2, \ldots, ...
2
votes
1answer
54 views

Zeros set of analytic functions over complex plane with several variables

I know that the zeros of analytic function (with one variable) over complex plane are isolated. However, I am not aware about the structure of the zeros set of analytic functions over complex plane ...
4
votes
2answers
104 views

A ‘strong’ form of the Fundamental Theorem of Algebra

Let $ n \in \mathbb{N} $ and $ a_{0},\ldots,a_{n-1} \in \mathbb{C} $ be constants. By the Fundamental Theorem of Algebra, the polynomial $$ p(z) := z^{n} + \sum_{k=0}^{n-1} a_{k} z^{k} \in ...
3
votes
1answer
100 views

Weighted $L^2$ Estimates for Domains $\Omega\subseteq\mathbb{C}$

EDIT: After mrf's comment below and some discussion with my instructor for the course it was decided that the below was not really an issue. Namely, I went into reading this lecture with the notion ...
5
votes
3answers
268 views

Why isn't several complex variables as fundamental as multivariable calculus?

One typically studies analysis in $\mathbb{R}^n$ after studying analysis in $\mathbb{R}$. Why can't the same be said $\mathbb{C}$?
1
vote
1answer
49 views

Analytic extension in several variables

Let $D \subset \mathbb{C}^{n}$ be an open domain and $K \subset D$ be a compact subset such that $D - K$ is simply connected. Let furthermore $f$ be a analytic function defined on $D-K$. Is it ...
1
vote
1answer
80 views

polynomial in several variable whose maximum modulus on the ball is known exactly

I'm interested in polynomials in several variables $p(x_1,\ldots,x_n)$, with complex coefficients, such that the maximum modulus of $p$ on the unit complex $n$-ball $$ \max \{ |p(z_1,\ldots,z_n)| : ...
1
vote
1answer
61 views

smooth domain in $\mathbb{C}^2$ and smooth bounday of bounded domain in $\mathbb{C}^2$

How can we define a smooth domain in $\mathbb{C}^2$ and a smooth boundary of bounded domain in $\mathbb{C}^2$? {where $\mathbb{C}^2$ := Cartesian product of complex plane }
2
votes
1answer
69 views

Derivative of composition with holomorphic function of several variables

Let $f : \mathbb C^n \rightarrow \mathbb C^m$ be holomorphic and $g : \mathbb C^m \rightarrow \mathbb C$ be smooth. I am looking for a simple formula for the mixed partials $\partial_i \partial_{\bar ...
3
votes
3answers
122 views

How to find $(-64\mathrm{i}) ^{1/3}$?

How to find $$(-64\mathrm{i})^{\frac{1}{3}}$$ This is a complex variables question. I need help by show step by step. Thanks a lot.
2
votes
1answer
71 views

Multivariate analytic function property

Suppose function $F: \mathbb{C}^n \to \mathbb{C}$ is analytic everywhere and in every coodinate; i. e. for any $q \in \mathbb{C}^n$ and for any $j \in \{1,2,\ldots,n\}$ function $f_{q,j}: \mathbb{C} ...
1
vote
1answer
81 views

Siegel upper-half space

In dimension 1, the Siegel upper-half space $\mathbb{H}=\{\tau\in\mathbb{C}:\Im\tau>0\}$ has the property that if $\lambda\in\mathbb{C}^\times$, then $a\lambda\in\mathbb{H}$ for some ...
0
votes
0answers
108 views

Harmonic function?

Let $\varphi : U \rightarrow \mathbb{C}$ be a holomorphic function, where $U \subset \mathbb{C}^{n}$ containing $0$. Is the function $u(x_{1}, ..., x_{n}, y_{1}, ..., y_{n}) := Im( \int_{0}^{(x_{1} + ...
0
votes
1answer
81 views

formal partial complex derivatives and symmetry

This is probably a silly question but let me ask. As it is well known for a general function $f:\mathbb R^2\to \mathbb R$ which posesses partial derivatives of second order it is not necessarily true ...
1
vote
1answer
44 views

Sheaf on a Stein variety such that $H^{1}(X, \mathcal{F}) \neq 0$

I would like to find a non-coherent sheaf on a Stein variety $X$ such that $H^{1}(X, \mathcal{F}) \neq 0$. Does anyone know any example? Thank you!
6
votes
2answers
164 views

A complex valued continuous function which is holomorphic outside of its zeros

Let $D$ be a non-empty connected open subset of $\mathbb{C}^n$. Let $f$ be a complex valued continuous function on $D$. Let $Z$ be the set of zeros of $f$. Suppose $f$ is holomoprphic on $D - Z$. Is ...
2
votes
1answer
123 views

$n$-sheeted branched covering

Michael Artin's algebra let $f(x,y)$ be an irreducible polynomial in $\mathbb{C}[x,y]$ which has degree $ n>0$ in the variable $y$. The Riemann surface of $f(x,y)$ is an $n$-sheeted branched ...
2
votes
0answers
54 views

Holomorphic function vanishing in a real hyperplane

A real hyperplane of $\mathbb{C}^{n}$ is given by a equation $\textrm{Re}( \ell(Z)) = c$, where $c \in \mathbb{R}$ and $\ell(Z) = \sum_{j=1}^{n} a_{j}z_{j}, a_{j} \in \mathbb{C}$. How to prove that if ...
4
votes
2answers
189 views

Uniform convergence of the Bergman kernel's orthonormal basis representation on compact subsets

Consider the Bergman kernel $K_\Omega$ associated to a domain $\Omega \subseteq \mathbb C^n$. By the reproducing property, it is easy to show that $$K_\Omega(z,\zeta) = \sum_{n=1}^\infty \varphi_k(z) ...
5
votes
1answer
90 views

A sequence that tell us if a holomorphic function of several variables is identically zero

Is there any sequence $\{ Z_{\nu} \}_{\nu \in \mathbb{N}}$ in $\mathbb{C}^{n}$, $Z_{\nu} \rightarrow 0$, such that any holomorphic function in $\mathbb{C}^{n}$ which vanishes in $Z_{\nu}$ for all $\nu ...
2
votes
1answer
103 views

Easy way to calculate $(dd^c u)^n$

Let $u$ be a $C^2$ function from $\mathbb{C}^n$ to $\mathbb{C}$. Define $$ \partial u = \sum\limits_{i=1}^n \frac{\partial u}{\partial z_i}dz_i, \\ \overline{\partial} u = \sum\limits_{i=1}^n ...
4
votes
1answer
171 views

Explicit counter-example to corona problem

The corona problem is known to fail for the complex polydisk, for dimension greater than 2. Does anyone has an explicit example of such functions?
1
vote
1answer
54 views

understanding the definition of domains of holomorphy

can anyone give me an example and explain why any open set in $\mathbb{C}$ is a domain of holomorphy? I have understood the fact from here but not able to understand their explanation for $n=1$
3
votes
1answer
146 views

Uniform convergent series

Let $\Omega$ be domain in $\mathbb C^2$. For each compact set $K_j$ define the holomorphic function $f_j$ on $\Omega$, such that $$\sup_{k_j}|f_j|<2^{-j}.$$ Define $$f= ...
1
vote
1answer
144 views

To show given function is smooth

Let $\phi$ be upper semi-continuous function defined on $\Omega\subset \mathbb C^2$. Let $\Omega_n= \{z \in \Omega: d(z,\partial \Omega) > \frac1n \}$. Let $\chi\in C_c^\infty$ of $|z_1|$, ...
3
votes
1answer
168 views

holomorphic function is real analytic?

$f$ is a holomorphic function on $\mathbb C^n$. If we regard $f$ as a function $F$ from $\mathbb R^{2n} \to \mathbb R^2$, is it necessarily that $F$ is real analytic?
1
vote
1answer
102 views

Domain of convergence of Mulitvariable series

What is region of convergence $(D\subset \mathbb C^2)$ of $$\sum_{n=0}^\infty(z_1^kz_2^l)^n$$ for fixed $k$ and $l$ integers. $z_1$ and $z_2$ are elements of complex plane. What is method of ...
2
votes
0answers
82 views

Uniform Convergence of Bergman Kernel

In my complex analysis class we've shown that if $\Omega \subset \Bbb C^n$ is compact and if $H^2(\Omega)$ is the set of square integrable holomorphic functions on $\Omega$, then the Bergman kernel is ...
2
votes
1answer
72 views

Connected Reinhardt Domain which is not complete

Can i have an example of connected Reinhardt domain in $C^n$ which contains zero. But it is not complete. Complete means: For $w= (w_1,..w_n)\in D$, if $z$ is such that $|z_j|\leq |w_j$ for all $j$ ...
8
votes
2answers
238 views

Characterizing holomorphic functions in $L^2(\Bbb C^n)$

One of my homework problems this week is to "characterize all holomorphic functions in $L^2(\Bbb C^n)$". I'm sorry for not being able to provide much work on my progress, but that is because I really ...
7
votes
3answers
229 views

Sources on Several Complex Variables

I have searched the past entries about sources on SCV but couldn't find about this topic. If I am not careful enough, sorry for this! We are using Hörmander's book which is really hard to follow. ...