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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Several Complex Analysis

Let $a \in \mathbb{C} \setminus \mathbb{R}$. Show that if $f \in O(\mathbb{C}^{∗} \times \mathbb{C}^{∗})$ such that $f(X)=0$, then $f \equiv 0$, where $X=\{(e^z,e^{az})|z \in \mathbb{C}\}.$ $f$ is ...
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2answers
102 views

Uses of Jacobian of a map on $\mathbb{R}^n$.

For a map $f:\mathbb{R}^n\to\mathbb{R}^n$, Jacobian matrix of $f$ is defined as $$\begin{bmatrix} \frac{\partial f_1}{x_1}& \frac{\partial f_1}{\partial x_2}& \ldots \frac{\partial ...
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1answer
69 views

Question regarding pluriharmonic function

A real valued function $f$ defined on an open subset $U$ of $\mathbb{C}^n$ is said to be Pluriharmonic if $$\frac{\partial^2}{\partial z_i\partial\bar{z_j}}f\equiv0,$$ for $1\leq i,j \leq n.$ I was ...
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1answer
110 views

Differential form on complex torus

Suppose $T= \mathbb{C}^n/\Gamma$ is a $n$-dim complex torus. How to prove that every exact $2$-form which has no $(0,2)$ component must be the image of a $(1,0)$ form? Is every torus Kahler? If the ...
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1answer
226 views

Formula for decomposing a form into $(p,q)$ forms

Let $L: \mathbb{C}^n \to \mathbb{C}$ be a real linear map. In other words, $L(a\vec{v}_1+b\vec{v_2}) = aL(\vec{v}_1)+bL(\vec{v}_2)$ for all $a,b \in \mathbb{R}$. Then $L$ decomposes uniquely into a ...
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1answer
94 views

Counterexample for Hartogs' Extension Theorem

I'll refer to Hartogs' Extension Theorem as it is stated in Wikipedia (https://en.wikipedia.org/wiki/Hartogs%27_extension_theorem#Formal_statement). I am trying to find a counterexample to show that ...
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22 views

doubt about definition of holomorphic polynomials

In a topic of several complex variable theory (in particular functions on $\mathbb{C}^2$), I came across a term homogeneous holomorphoic polynomial. By the word, I think it is a polynomial in complex ...
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1answer
197 views

zero set of an analytic functio of several complex variables

In one variable complex theory, we have the result that zeroes of a non-zero analytic function are isolated. In several variable theory, this result does not hold. I read it somewhere that this fact ...
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1answer
156 views

Level sets of holomorphic functions

It is a somewhat well known fact that any closed set (say in the plane) can be realized as the level set of a smooth ($C^\infty$) function, so level sets of smooth functions are as general as they can ...
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2answers
68 views

Dependence of roots on parameters

Let a function $f$ be holomorphic in a polydisk $U=U'\times U_n$,and suppose that for each fixed $z'\in U'$ it has a unique zero $z_n = \alpha(z')$ in the disk $U_n$. Then the function $\alpha(z')$ is ...
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1answer
84 views

Prove that each convex open subset of $\mathbb C^n$ is a domain of holomorphy. [closed]

Prove that each convex open subset of $\mathbb C^n$ is a domain of holomorphy. Please help. Thanks in advance!
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146 views

Why is the set of points where a complex polynomial does not vanish connected?

Let $p$ be a complex multivariate polynomial. Let $C$ be the set of those complex tuples where $p$ is nonzero. Then, $C$ is connected.
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21 views

Analytic semiconjugacy

Consider the following commutative diagram (semi-conjugacy): $$ X\;\; \stackrel{f}{\longrightarrow} \;\;X $$ $${\pi}\downarrow \;\;\;\;\;\; \;\;\;\;\downarrow {\pi}$$ ...
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1answer
82 views

Analyticity and composition of maps

Suppose $\Omega_3\subset \mathbb{C}^3$ and $\Omega_2\subset \mathbb{C}^2$ are two domains (open connected). Let $g:\Omega_3\to\Omega_2$ be a surjective analytic function and ...
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1answer
142 views

Constrained optimization with complex variables

Is there a theory of constrained optimization with complex variables, do you know any textbook on that topic? The typical textbooks on constrained optimization deal with real variables. I actually ...
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37 views

Intuitively what is it if making a modification of a torus?

It is well-known that if we have a equivalence relation in $\mathbb{R}^2$:$(z_1,z_2)\sim (z_1',z_2')$ iff $$\begin{pmatrix} z_1'\\ z_2' \\ \end{pmatrix}=\begin{pmatrix} 1&0\\ 0&1 \\ ...
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49 views

“Center of mass” of a complex hypersurface

Background: Suppose first that $f \in \mathbb{C}[z]$ and consider the "analytic hypersurfaces" $V(f)$ and $V(df)$, which are of course just sets of points with multiplicities. (That is, we think of ...
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305 views

How to imagine zeros of an analytic function of several variables

Let $f(z_1,\cdots, z_n)$ be a holomorphic function of several variables in an open subset of $\mathcal C^n$. Let $Z(f)=\{ (z_1,\cdots, z_n) \: | \: f=0\}$ be the zero set of $f$. If $n=1$, the zeros ...
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1answer
66 views

Can I switch the order of integration and “Real(z)” operation?

Let $f(z,\eta)$ be an entire function. I need to calculate (numerically) the integral: $$\int\limits _{0}^{\pi}\mbox{Re}\left(\int\limits _{0}^{\pi}f\left(z,\eta\right)d\eta\right)dz$$ Can I switch ...
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30 views

Rational function on polydisc

Who has idea to prove this: Let $\mathbb{U}^n$ be a polydisc, $f\in \mathcal{O}(\mathbb{U}^n)\cap C(\bar{\mathbb{U}}^n)$, if $|f|=const$ on the skeleton of $\mathbb{U}^n$, then $f$ must be a rational ...
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2answers
118 views

Multidimensional complex integral of a holomorphic function with no poles

I am looking for a generalization of the Cauchy integral theorem. I know that there are generalizations of the Cauchy integral formula (eg the Bochner-Martinelli formula), but I do not know if this ...
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1answer
309 views

Determinant of the Jacobian of a holomorphic mapping of several complex variables

I am reading from Degree Theory by N. Lloyd, and in one section he writes about the degree of a holomorphic map of several complex variables. I am unsure about one of the steps in a proof he gives. ...
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1answer
128 views

$n$ Torus contained in the closure of the image of the unit disc under a holomorphic map?

I have the following question. Does there exists a holomorphic function $\varphi\in\mathcal{O}(\mathbb{D},\mathbb{D}^{n})$ such that $\mathbb{T}^n\subseteq\overline{\varphi(\mathbb{D})},$ where ...
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1answer
114 views

A complex problem.

We have a set $S:= \{e^{inr\pi} | n\in\Bbb N\}$. Where r is an irrational number. I wonder whether this set is dense in $\partial D(0,1)$. i.e. I want to see if $\overline S=\partial D(0,1).$ I ...
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1answer
235 views

If $f$ is $C^{\infty}$ and $f^2$ is analytic, then $f$ is analytic.

Assume that $f:\mathbb{C}^n\rightarrow \mathbb{C}$ is a $C^{\infty}$ function such that $f^2$ is (complex) analytic. Then show that $f$ is analytic. Now, when we consider the question for functions ...
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3answers
129 views

Problem related to continuous complex mapping.

We are given with a map $g:\bar D\to \Bbb C $, which is continuous on $\bar D$ and analytic on $D$. Where $D$ is a bounded domain and $\bar D=D\cup\partial D$. 1) I want to show that: ...
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30 views

determine the maximal open subset $G\subset\mathbb{C}^2$ such that the series $f$ converges in $\mathcal{O}(G)$ in the topology of Frechet space.

Consider $f(z_1,z_2)=\sum\limits_{j=0}^\infty(z_1+z_2)^j$,determine the maximal open subset $G\subset\mathbb{C}^2$ such that the series $f$ converges in $\mathcal{O}(G)$ in the topology of Frechet ...
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1answer
154 views

a corollary of Cauchy's integral formula in several complex variables

I'm learning several complex variables. There is a corollary of Cauchy's integral formula that I don't know how to prove. Let $X\subset\mathbb{C}^n$ be a domain. For each multi-index $\nu$, for each ...
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1answer
320 views

Zeros of complex function sequence (Application of Rouche's Theorem).

For a given sequence of complex functions: $\phi_n(z)= 1+\frac1n-z-e^{-z}$; here $z\in${$z| Rez>0$}. I want to prove that : (1). $\phi_n $ has a unique zero $z_n$ in the half plane. (i.e. there ...
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1answer
79 views

why can we assume that f is linear when proving open balls and open polydiscs are not biholomorphically equivalent as n>1?

I'm reading Kaup's Holomorphic functions of several variables. I have some trouble in understanding Proposition 3.11 which proves that open balls and open polydiscs are not biholomorphically ...
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1answer
59 views

how to show f is bijective?

Suppose that the set-mapping $f:X\rightarrow Y$ of one-dimensional domains of $\mathbb{C}$ induces an isomorphism $f^0:\mathcal{O}(Y)\rightarrow \mathcal{O}(X)$ defined by $g\mapsto g\circ f$ of ...
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1answer
52 views

Subharmonic logarithms

In Steven Krantz's book on several complex variables he has the following problem: Suppose $\log u_1$ and $\log u_2$ are subharmonic on an open subset of the complex plane. Show that $\log (u_1+u_2)$ ...
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1answer
82 views

Why do the Wirtinger derivatives behave like actual partial derivative operators?

Despite the fact that they're not partial derivative operators, the Wirtinger derivatives obey things like the chain rule. Of course I can prove such things by manipulating formulas, but this gives ...
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1answer
36 views

request for reference

I am currently trying to grasp basic theorems of complex analysis in higher dimensions for different classes of functions. e.g. analytic, harmonic, subharmonic and pluriharmonic. Out of these a lot of ...
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365 views

What is pluripotential theory?

My tutor for electromagnetism showed me a problem about point charges in a disk and their equilibria. He referred me to a subject called "pluripotential theory". I googled it and I did not find what I ...
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1answer
52 views

Prove that $u \circ f $ is plurisubharmonic on $\Omega_1$

I'm trying to show that the theorem in my book: Let $f: \Omega_1 \to \Omega_2$ be a holomorphic map between open sub - sets $\Omega_1, \Omega_2$ of $\Bbb C^n$. If $u$ is plurisubharmonic on ...
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2answers
49 views

Show that a $C^2$-function $u$ is plurisubharmonic if and only if the Hessian matrix $H_u(z)(\omega, \omega)>0$

I'm trying to show that the theorem in my book: A $C^2$-function $u$ is plurisubharmonic if and only if the matrix (the complex Hessian) $$H_u(z)=\left( \dfrac{\partial^2 u}{\partial z_j ...
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1answer
44 views

Show that $\widetilde{u} \in PSH(\Omega)$

EDITED I'm reading the book: (Oxford science publications._ London Mathematical Society monographs, new ser., no. 6) Maciej Klimek -Pluripotential theory -OUP (1992). I don't understand ...
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2answers
670 views

Generalization of Cauchy Residue theorem to Multi-dimensional holomorphic functions

We know Cauchy Residue theorem from the Complex analysis. however I wonder if there is a kind of Generalization of Cauchy integral and Residue theorem to the complex multidimensional holomorphic ...
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145 views

The Wirtinger theorem proof

Let $M$ be a complex hermitian manifold with symplectic form $\Omega$ and $N \subseteq M$ be its smooth $2n$-dimensional (real dimensions, $2n \geq 2$) compact submanifold. I want to show the ...
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1answer
199 views

Why holomorphic injection on $C^n$must be biholomorphic?

This result is certainly right in the 1-dim'l case. But I don't know how to show the general case by induction. Can anyone tell me the detail please?
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Properties of plurisubharmonic functions

In book: (Oxford science publications._ London Mathematical Society monographs, new ser., no. 6) Maciej Klimek -Pluripotential theory -OUP (1992) Theorem 2.9.14 (ii): Let $\Omega$ be an open ...
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506 views

Implicit function theorem for several complex variables.

This is the statement, in case you're not familiar with it. Let $ f_j(w,z), \; j=1, \ldots, m $ be analytic functions of $ (w,z) = (w_1, \ldots, w_m,z_1,\ldots,z_n) $ in a neighborhood of $w^0,z^0$ ...
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510 views

Which book on complex analysis is good for self study?

Which book on complex analysis is good for self study? I am an average student and have just a very basic knowledge of this subject.I want to cover up to Runge's Theorem. I heard about few books- ...
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2answers
919 views

Multivariate Residue Theorem?

Is there an extension of the residue theorem to multivariate complex functions? Say you have a function of $n$ complex variables $s_{n}$ and you wish to integrate it over some region in ...
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92 views

Prove that $\log(\max(\left | z_1 \right |,\left | z_2 \right |,\ldots,\left | z_n \right |)) \in MPSH(\Omega)$

This's an example: For $u(z_1,z_1,\ldots,z_n)=\log(\max(\left | z_1 \right |,\left | z_2 \right |,\ldots,\left | z_n \right |))$, where $z=(z_1,z_1,\ldots,z_n) \in \Omega=\mathbb{C}^n \setminus\{0\} ...
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171 views

Subharmonic, Plurisubharmonic

Can you give me two examples of Subharmonic, Plurisubharmonic? (and not Subharmonic, not Plurisubharmonic) . Then prove that your examples. I'm looking forward to your help. Thanks.
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65 views

A functional power series equation

I am interested in solving the following functional equation: $$(1-w-zw)F(z,w)+zw^nF(z,w^2)=z-zw\,.$$ Here $n\geq2$ is a fixed integer and $F(z,w)$ is a power series in two variables with complex ...
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1answer
85 views

Book searching in Pluripotential theory

Can anyone recommend me a book on pluripotential theory with an intuitive approach? I have some course notes on that subject, but it's really abstract and theoretical. I want to understand why ...
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2answers
225 views

Is every holomorphic function in a neighborhood of the origin of $C^n$ necessarily bounded?

Is every holomorphic function in a neighborhood of the origin of $C^n$ necessarily bounded? In which case the statement can be true? Thanks in advance