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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2
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1answer
16 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 ...
2
votes
1answer
43 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 ...
6
votes
0answers
70 views
+250

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 ...
1
vote
1answer
35 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 ...
1
vote
2answers
33 views

Geometrical Meaning of derivative of complex function

What's the geometrical meaning of f'(z) in complex analysis, as we know in real analysis f'(x) has meaning ie. Slope of curve or gives max/ min. But what does derivative f'(z) has geometrical meaning ...
1
vote
0answers
15 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 ...
0
votes
0answers
58 views

Complex Quasi linear Differential Equation

So, I was reading some papers when I found a several complex variable quasi-linear differential equation. Let $f(z_1,z_2,z_3,z_4)$ be a complex holomorphic function, with $z_i \in ...
1
vote
1answer
27 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 ...
6
votes
1answer
69 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 ...
3
votes
2answers
55 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 ...
1
vote
1answer
54 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!
0
votes
1answer
99 views

Why is the set of points where a complex polynomial does not vanish is 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.
0
votes
1answer
48 views

use example to prove the sum of two nonanalytic functions can be analytic [closed]

Find two functions, each of which is nowhere analytic, but whose sum is an entire function.
1
vote
0answers
17 views

Analytic semiconjugacy

Consider the following commutative diagram (semi-conjugacy): $$ X\;\; \stackrel{f}{\longrightarrow} \;\;X $$ $${\pi}\downarrow \;\;\;\;\;\; \;\;\;\;\downarrow {\pi}$$ ...
1
vote
1answer
54 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 ...
0
votes
0answers
19 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 ...
0
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0answers
35 views

holomorphic Lie group automorphisms of the complex Heisenberg group preserving the lattice

I don't understand Lie group,but I need an elementary result about it,so I ask for help for a simple question. Take a complex Heisenberg group $G$ $$ \left( \begin{array}{ccc} 1 & z_1 & ...
1
vote
0answers
31 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 \\ ...
1
vote
0answers
34 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 ...
0
votes
0answers
15 views

inequality on integrals (2)

Is it possible find a constant $C>0$ such that $\displaystyle\sum_{i,j=1}^n\int_\Omega |u_{x_i}||v_{x_j}|dx\leq C \left(\sum_{i=1}^n\int_\Omega ...
2
votes
1answer
51 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 ...
1
vote
0answers
19 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 ...
3
votes
2answers
63 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 ...
2
votes
1answer
72 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. ...
9
votes
1answer
121 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 ...
5
votes
1answer
104 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 ...
3
votes
1answer
160 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 ...
4
votes
0answers
36 views

Continuous complex functions.

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 $\bar D=D\cup\partial D$. Then $\partial(g(D))\subseteq g(\partial ...
3
votes
3answers
106 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: ...
0
votes
1answer
26 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 ...
2
votes
1answer
93 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 ...
3
votes
1answer
227 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 ...
1
vote
1answer
56 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 ...
2
votes
1answer
51 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 ...
2
votes
1answer
37 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 $\log (u_1+u_2)$ ...
5
votes
1answer
49 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 ...
1
vote
1answer
28 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 ...
7
votes
2answers
132 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 ...
1
vote
1answer
44 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 ...
0
votes
0answers
26 views

Prove that $\int_{{B(0,\epsilon)}\setminus \{z_1=0\}}\det\left(\text{Hessian}_u(z)\right)\mathrm{d}V=\infty$

I have a problem: For $u(z_1,z_2)=\left (-\log\left | z_1 \right | \right )^\alpha\cdot \left ( \left | z_2 \right |^2-1 \right )$, where $\alpha \in \left (0,1 \right )$. Prove that if $\alpha ...
0
votes
2answers
23 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 ...
1
vote
1answer
38 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 ...
2
votes
2answers
268 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 ...
1
vote
0answers
62 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 ...
2
votes
1answer
103 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?
3
votes
1answer
83 views

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 ...
3
votes
1answer
192 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 $w^0,z^0$ ...
2
votes
3answers
220 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- ...
3
votes
2answers
350 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 ...
0
votes
2answers
82 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\} ...