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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5
votes
1answer
53 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
31 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
176 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
46 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
27 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
30 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
41 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
342 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
86 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 ...
3
votes
1answer
128 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
94 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
253 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
278 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- ...
4
votes
2answers
452 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
85 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\} ...
1
vote
1answer
119 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.
1
vote
0answers
52 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 ...
2
votes
1answer
59 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 ...
4
votes
2answers
185 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
4
votes
1answer
162 views

Prove that $A$ has Lebesgue measure $0$.

Suppose $G$ is a connected open set of $\mathbb{C}^n$. Prove that: (1). If $f \in$ PSH(G) and $f \not \equiv \infty$ then $A=\{z \in G: f(z)=-\infty\}$ has Lebesgue measure $0$. (2). If $f \in ...
1
vote
1answer
86 views

Prove that $\log(f+g)$ be a plurisubharmonic function

Suppose $G$ is an open set of $E$,($E$ is complex Banach space) and $f,~g :G \to \left[0,\infty \right)$ such that $\log f$ and $\log g$ be two plurisubharmonic(PSH) functions in $G$. Prove that ...
3
votes
1answer
115 views

Bergman-Shilov Boundary and Peak Points

Let $\Omega$ be a domain in $\mathbb{C}^n.$ Consider the Banach algebra $A(\Omega):=\mathcal{C}({\overline{\Omega}})\cap\mathcal{O}(\Omega).$ Denote the Bergman-Shilov boundary of $A(\Omega)$ by ...
2
votes
1answer
147 views

Proper holomorphic map from the unit disc to the bidisc

Let $F\colon \mathbb{D}\rightarrow\mathbb{D}^2$ be a proper, holomorphic map. It is not very difficult to see that every proper holomorphic map from $\mathbb{D}$ to $\mathbb{D}$ is a finite "Blaschke ...
1
vote
1answer
51 views

Proper holomorphic map from the unit disc to the polydisc of dimension greater than one .

Let $F:\mathbb{D}\rightarrow\mathbb{D}^n$ be a proper holomorphic map and $n\geq2.$ I have the following questions: Does F extends in a nbhd of $\overline{\mathbb{D}}$ holomorphically? If not what ...
1
vote
2answers
118 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 ...
1
vote
1answer
110 views

Continuous extension of limiting Holomorphic function to the boundary.

Let $\Omega $ be a domain in $\mathbb{C}^n$. Consider the algebra $A(\Omega)=\mathcal{C}(\overline\Omega)\cap\mathcal{O}(\Omega).$ Let $\{F_\nu\}$ be a sequence in $A(\Omega)$ such that ...
5
votes
0answers
59 views

Branch locus along a smooth curve

Let $f : S \to X$ be a dominant morphism of smooth complex surfaces. Let $C \subset S$ be a smooth curve such that $df$ is of rank $1$ along $C$ and that in a neighborhood of $C$, $df_x$ is an ...
1
vote
1answer
146 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 ...
10
votes
1answer
293 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
188 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
50 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 ...
3
votes
1answer
135 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 ...
2
votes
1answer
44 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
42 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? ...
1
vote
1answer
48 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
320 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
31 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} ...
6
votes
2answers
214 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 ...
4
votes
1answer
244 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
88 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
62 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: ...
15
votes
3answers
1k 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
38 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
77 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 ...
7
votes
2answers
589 views

Where to learn algebraic analysis

I have been studying categories, sheaf cohomology and complex analysis (the basics since I know just a little). Then recently I tried to find out more about algebraic analysis and these microlocal ...
1
vote
1answer
105 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
86 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
107 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
138 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.
3
votes
1answer
192 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} ...