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).

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1answer
167 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 ...
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1answer
87 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 ...
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1answer
209 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 ...
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1answer
51 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!
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2answers
230 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 ...
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1answer
470 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 ...
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0answers
75 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 ...
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2answers
276 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) ...
6
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1answer
131 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 ...
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1answer
120 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 ...
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1answer
193 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?
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1answer
73 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 $n=1$
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2answers
469 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 ...
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1answer
89 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
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1answer
168 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= ...
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0answers
71 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: ...
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1answer
351 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 $|z_1|$, ...
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1answer
298 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?
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1answer
56 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 ...
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0answers
180 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 ...
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1answer
177 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 ...
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1answer
118 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 ...
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0answers
114 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 ...
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1answer
105 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 $j$ ...
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2answers
311 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
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1answer
326 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
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3answers
454 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. ...
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1answer
256 views

Consequence of Cauchy Integral Formula for Several Complex Variables in Gunning's book

I am reading Gunning's book Introduction to Holomorphic Functions of Several Variables, Vol. I, and I am stuck in the proof of Maximum modulus theorem: if $f$ is holomorphic in a connected open subset ...