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
votes
1answer
105 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
123 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
107 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
174 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 ...
3
votes
3answers
110 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
106 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
274 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
62 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
52 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
39 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
55 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
32 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
198 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
47 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
34 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
373 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
87 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
132 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
99 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
270 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
299 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
484 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
86 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
121 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
61 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
168 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
89 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
149 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
52 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
120 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
115 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
61 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
150 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
295 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
196 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
52 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
142 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
49 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
341 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
223 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
262 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 ...