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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19
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3answers
2k 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 $\mathbb{C}$?
10
votes
1answer
220 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 ...
10
votes
1answer
312 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 $$|f(z)|\leq C_{\epsilon}(1+|z|)^{N_{\epsilon}}e^{\epsilon | \text{Im}\,z|}$$ ...
9
votes
2answers
772 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 ...
9
votes
1answer
127 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 ...
8
votes
2answers
364 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
votes
2answers
348 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 ...
7
votes
3answers
634 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. ...
7
votes
1answer
371 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 ...
6
votes
2answers
254 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 ...
6
votes
2answers
271 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 ...
6
votes
1answer
144 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 ...
6
votes
2answers
850 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 ...
6
votes
1answer
226 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 ...
6
votes
1answer
152 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 ...
6
votes
1answer
279 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 ...
6
votes
0answers
55 views

Methods for “recognizing” a polynomial of several variables.

If $f(z)$ is an entire function of a single complex variable, then the following are indirect methods for recognizing that $f$ is a polynomial. 1) Show that $f^{(n)}\equiv0$ for some $n\geq0$. 2) ...
5
votes
1answer
252 views

The germ induced by an irreducible polynomial

Let $P\in\mathbb{C}[z_1,z_2,\ldots,z_n]$ be an irreducible polynomial. Let $a\in\mathbb{C}^n$ be such that $P(a)=0.$ Consider the germ of holomorphic functions at the point $a,$ denoted by ...
5
votes
1answer
312 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 ...
5
votes
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 ...
5
votes
1answer
74 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 ...
5
votes
0answers
71 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 ...
4
votes
3answers
478 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
1answer
636 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 ...
4
votes
2answers
216 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
193 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?
4
votes
1answer
28 views

Notation for the set of holomorphic differential forms

I used to write $\Omega^p(M)$ for the set of $p$-forms on $M$ and $\Omega^{p,q}(M)$ for the set of $(p,q)$-forms on a complex manifold $M$. Now some authors use $\Omega^p(M)$ to denote the set ...
4
votes
2answers
363 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) ...
4
votes
1answer
356 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?
4
votes
1answer
229 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
1answer
177 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 ...
4
votes
1answer
34 views

Calculating integral using Cauchy integral formula in two variables

I want to compute the integral: $\iint_{\partial_0P}\frac{1}{1-4zw}dzdw$ (or any similar integral) using Cauchy integral formula for two complex variables over polydiscs. The distinguished boundary is ...
4
votes
1answer
99 views

Polydisc is not biholomorphic to any strictly pseudoconvex domain

I want to prove the poly disc $P=\left\{z\in \mathbb{C}^2 : |z_1|<1,|z_2|<1\right\}$ is not biholomorphic to any strictly pseudo convex domain in $\mathbb{C}^2.$ Can any one provide a hint?
4
votes
1answer
487 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 $w^0,z^0$ ...
4
votes
1answer
199 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?
4
votes
1answer
221 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 ...
4
votes
1answer
56 views

Does a left-invariant vector field on a complex Lie group preserve holomorphic functions?

Let $G$ be a (finite-dimensional) complex Lie group, and suppose $f : G \to \mathbb{C}$ is holomorphic. Let $X$ be a left-invariant vector field on $G$. Must $Xf$ be holomorphic? I think I have a ...
4
votes
0answers
65 views

Openness of a subset in complex 2-plane

Let $U$ be a subset of $\mathbb{C}^{2}$ containing the origin $0$. Assume that for any curve $C$ (an affine variety of dimension 1, maybe singular) passing through $0$ we have $U \cap C$ is ...
4
votes
0answers
204 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 ...
3
votes
3answers
142 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
307 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 ...
3
votes
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: ...
3
votes
1answer
132 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 $j$ ...
3
votes
1answer
70 views

Interpretation of the Weierstrass Preparation Theorem

I am reading Griffiths and Harris' Principles of Algebraic Geometry but I am having trouble making sense of a statement following the Weierstrass Preparation Theorem (p.9 in my edition). The ...
3
votes
2answers
117 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 ...
3
votes
1answer
202 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 ...
3
votes
1answer
297 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} ...
3
votes
1answer
112 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
votes
1answer
66 views

Irreducible polynomial and the zero set of its derivative

Let $P$ be a polynomial in $\mathbb{C}[z_1,z_2,...,z_n].$ Consider the derivative of $P,$ $D_{\mathbb{C}}P$, as a holomorphic map from $\mathbb{C}^n$ to $\mathbb{C}^n.$ I have the following question: ...
3
votes
1answer
105 views

Is the ring of entire functions coherent?

Call a commutative ring $R$ coherent if for each $n\in \{1,2,3,\cdots\}$ and each $n$-tuple $(r_1, ..., r_n)$ in $R^n$, the kernel of the map $R^n\owns (s_1, \cdots, s_n) \mapsto r_1 s_1 +\cdots + ...