Use this tag for questions related to the study of analytic functions of strictly more than one complex variables. For the single complex dimension case, use (complex-analysis).
1
vote
1answer
34 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
239 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 $$|f(z)|\leq C_{\epsilon}(1+|z|)^{N_{\epsilon}}e^{\epsilon |
\text{Im}\,z|}$$
...
6
votes
1answer
117 views
Convergence domain: $\{(z,w):|z|^2+|w|^2 < 1\}$
I would like a HINT for this:
Exhibit a two variable power series whose convergence domain is the unit ball $\{(z,w):|z|^2+|w|^2 < 1\}$.
($z$ and $w$ are complex numbers.) I think that it ...
1
vote
1answer
40 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, ...
1
vote
1answer
32 views
What does $\left< dz_j , \frac{\partial}{\partial z_j}\right>$ mean?
Here, in page 2 of Steven Krantz's book Function Theory of Several Complex Variables, what do those angle brackets mean? What kind of product is that between differentials and partial derivatives?
...
0
votes
0answers
42 views
How does the complex convex set look like?
The usual convex set is the real linear convex set, if we change the real linear map into complex linear map, we can get the complex convex set. A system way to do this is in the several complex ...
1
vote
1answer
30 views
What does $T_z\mathbb{R}^2\otimes\mathbb{C}$ in p. 2 of Huybrechts' book mean?
I apologize for my lack of imagination and the likely silliness of this question, but what does $T_z\mathbb{R}^2\otimes\mathbb{C}$ mean here (last paragraph)?
And how does that extension work?
Thank ...
2
votes
1answer
51 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 ...
1
vote
0answers
22 views
Rational Singularities in dimenson 2 or highter and square integrability
I was wondering if there are any results available on square integrability of quotients of polynomials $P(z_1,z_2,\ldots,z_n)$ and $Q(z_1,z_2,\ldots,z_n)$ like
$$\int_{\mathbb{T}^n} ...
4
votes
2answers
101 views
A ‘strong’ form of the Fundamental Theorem of Algebra
Let $ n \in \mathbb{N} $ and $ a_{0},\ldots,a_{n-1} \in \mathbb{C} $ be constants. By the Fundamental Theorem of Algebra, the polynomial
$$
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 ...
2
votes
2answers
47 views
Biholomorphism between an open set and $\mathbb C^n$
If $U$ is a polydisc in $\mathbb C^n$, that is, $U=\{z \in \mathbb C^n:|z_i|<1\}$, can we find a biholomorphic map from $U$ to $\mathbb C^n$?
2
votes
1answer
29 views
Biholomorphisms of the polydisk
Let $\mathbb{D}$ denote the unit disk in the complex plane, equipped with the Poincare metric.
Let us denote the group of biholomorphisms of $\mathbb{D}$ by $Aut(\mathbb{D})$.
Suppose $F: ...
5
votes
3answers
261 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
0answers
30 views
Analytic variety
Let $\Omega$ be a convex domain in $\mathbb{C^{n}}$ , $\Delta$ be unit disk in $\mathbb{C}$ and f: $\Delta \rightarrow b\Omega $ be a non-constant holomorphic map.Then the convex hull of f($\Delta$) ...
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
79 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
$$
\max \{ |p(z_1,\ldots,z_n)| : ...
1
vote
1answer
60 views
smooth domain in $\mathbb{C}^2$ and smooth bounday of bounded domain in $\mathbb{C}^2$
How can we define a smooth domain in $\mathbb{C}^2$ and a smooth boundary of bounded domain in $\mathbb{C}^2$?
{where $\mathbb{C}^2$ := Cartesian product of complex plane }
2
votes
1answer
68 views
Derivative of composition with holomorphic function of several variables
Let $f : \mathbb C^n \rightarrow \mathbb C^m$ be holomorphic and $g : \mathbb C^m \rightarrow \mathbb C$ be smooth. I am looking for a simple formula for the mixed partials $\partial_i \partial_{\bar ...
3
votes
3answers
122 views
How to find $(-64\mathrm{i}) ^{1/3}$?
How to find $$(-64\mathrm{i})^{\frac{1}{3}}$$
This is a complex variables question.
I need help by show step by step.
Thanks a lot.
2
votes
1answer
70 views
Multivariate analytic function property
Suppose function $F: \mathbb{C}^n \to \mathbb{C}$ is analytic everywhere and in every coodinate; i. e. for any $q \in \mathbb{C}^n$ and for any $j \in \{1,2,\ldots,n\}$ function $f_{q,j}: \mathbb{C} ...
1
vote
1answer
75 views
Siegel upper-half space
In dimension 1, the Siegel upper-half space $\mathbb{H}=\{\tau\in\mathbb{C}:\Im\tau>0\}$ has the property that if $\lambda\in\mathbb{C}^\times$, then $a\lambda\in\mathbb{H}$ for some ...
0
votes
0answers
104 views
Harmonic function?
Let $\varphi : U \rightarrow \mathbb{C}$ be a holomorphic function, where $U \subset \mathbb{C}^{n}$ containing $0$. Is the function $u(x_{1}, ..., x_{n}, y_{1}, ..., y_{n}) := Im( \int_{0}^{(x_{1} + ...
1
vote
1answer
65 views
How to calculate $\Re\psi(\mathrm{i}y)=$? and $\Im\psi(\mathrm{i}y)=\frac{1}{2}y^{-1}+\frac{1}{2}\pi\coth{\pi y}$?
How to calculate $$\Re\psi(\mathrm{i}y)= ?$$ and how to proof $$\Im\psi(\mathrm{i}y)=\frac{1}{2}y^{-1}+\frac{1}{2}\pi\coth{\pi y}.$$
Here $\mathrm{i}^2=-1.\psi(s)$ is digamma function.
Can you help ...
0
votes
1answer
80 views
formal partial complex derivatives and symmetry
This is probably a silly question but let me ask.
As it is well known for a general function $f:\mathbb R^2\to \mathbb R$ which posesses partial derivatives of second order it is not necessarily true ...
1
vote
1answer
44 views
Sheaf on a Stein variety such that $H^{1}(X, \mathcal{F}) \neq 0$
I would like to find a non-coherent sheaf on a Stein variety $X$ such that $H^{1}(X, \mathcal{F}) \neq 0$. Does anyone know any example?
Thank you!
6
votes
2answers
161 views
A complex valued continuous function which is holomorphic outside of its zeros
Let $D$ be a non-empty connected open subset of $\mathbb{C}^n$.
Let $f$ be a complex valued continuous function on $D$.
Let $Z$ be the set of zeros of $f$.
Suppose $f$ is holomoprphic on $D - Z$.
Is ...
2
votes
1answer
120 views
$n$-sheeted branched covering
Michael Artin's algebra
let $f(x,y)$ be an irreducible polynomial in $\mathbb{C}[x,y]$ which has degree $ n>0$ in the variable $y$. The Riemann surface of $f(x,y)$ is an $n$-sheeted branched ...
2
votes
0answers
54 views
Holomorphic function vanishing in a real hyperplane
A real hyperplane of $\mathbb{C}^{n}$ is given by a equation $\textrm{Re}( \ell(Z)) = c$, where $c \in \mathbb{R}$ and $\ell(Z) = \sum_{j=1}^{n} a_{j}z_{j}, a_{j} \in \mathbb{C}$. How to prove that if ...
4
votes
2answers
186 views
Uniform convergence of the Bergman kernel's orthonormal basis representation on compact subsets
Consider the Bergman kernel $K_\Omega$ associated to a domain $\Omega \subseteq \mathbb C^n$. By the reproducing property, it is easy to show that
$$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
$$
\partial u = \sum\limits_{i=1}^n \frac{\partial u}{\partial z_i}dz_i, \\
\overline{\partial} u = \sum\limits_{i=1}^n ...
4
votes
1answer
169 views
Explicit counter-example to corona problem
The corona problem is known to fail for the complex polydisk, for dimension greater than 2. Does anyone has an explicit example of such functions?
1
vote
1answer
54 views
understanding the definition of domains of holomorphy
can anyone give me an example and explain why any open set in $\mathbb{C}$ is a domain of holomorphy?
I have understood the fact from here but not able to understand their explanation for $n=1$
1
vote
2answers
159 views
a suggestion for several complex variable book [duplicate]
Could any one tell me name of some books on several complex variable for some one who will start reading the subject for the first time in his life. he has back ground on Differential geometry,complex ...
2
votes
1answer
68 views
Extension of biholomorphic map to the boundary in higher dimension
Well I was thinking on this problem, any idea to progress will be gladly accepted, Let $f:\Omega\rightarrow V$ be a biholomorphic map where $\Omega$ and $V$ are bounded open set in $\mathbb{C}^n$ with ...
3
votes
1answer
144 views
Uniform convergent series
Let $\Omega$ be domain in $\mathbb C^2$. For each compact set $K_j$ define the holomorphic function $f_j$ on $\Omega$, such that $$\sup_{k_j}|f_j|<2^{-j}.$$
Define $$f= ...
2
votes
0answers
65 views
Eigenvalue of a form
I came across the following matrix while reading an article..Can you please help me to understand the following.
We are defining following form:
...
1
vote
1answer
142 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
166 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
53 views
5-variable polynomial, constant in 1 variable
I have a polynomial function $f(x_1,x_2,x_3,x_4): \mathbb{C}^4 \to \mathbb{C}$, which obeys the equality $f(x_1+tx_3,x_2+tx_4,x_3,x_4) = f(x_1,x_2,x_3,x_4)$ for all $t \in \mathbb{C}$.
My question ...
3
votes
0answers
130 views
What's the difference between analytic singularity and algebraic singularity?
Let $f$ be a holomorphic function defined in $U\in \mathbb{C}^n$, and $f=0$ gives an analytic variety. Suppose $f$ is irreducible as a germ of holomorphic function in $\mathcal{A}_z$, here $z$ is a ...
4
votes
1answer
124 views
Domain of Holomorphy
How to show that $D=\{ |z_1|<1\} \cup \{ |z_2|<1\} \subset \mathbb{C}^2$ is not a domain of holomorphy. What is the smallest domain of holomorphy $ S\supset D $ ?
I think we cannot just add the ...
1
vote
1answer
101 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 ...
4
votes
1answer
219 views
Holomorphic function of a matrix
A statement is made below. The questions are:
(a) Is the statement true?
(b) If it is, does it appear in the literature?
Here is the statement.
For any matrix $A$ in $M_n(\mathbb C)$, write ...
