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).
4
votes
1answer
222 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 ...
7
votes
3answers
235 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. ...
1
vote
2answers
74 views
Prove that: $\sup_{z \in \overline{D}} |f(z)|=\sup_{z \in \Gamma} |f(z)|$
Suppose $D=\Delta^n(a,r)=\Delta(a_1,r_1)\times \ldots \times \Delta(a_n,r_n) \subset \mathbb{C}^n$
and
$\Gamma =\partial \circ \Delta^n(a,r)=\left \{ z=(z_1, \ldots , z_n)\in ...
4
votes
2answers
194 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) ...
2
votes
1answer
47 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 ...
