# Questions tagged [several-complex-variables]

For questions related to the study of functions of several variables, in particular the study of holomorphic functions, automorphic group actions and forms, pseudoconvexity, complex geometry, analytic spaces, analytic sheaves.

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### How will I prove that the simultaneous limits for the following exist? [closed]

\begin{align} \lim_{(x,y) \to (0,0)}f(x,y) = \lim_{(x,y) \to (0,0)} \dfrac{xy}{\sqrt{x^2+y^2}} \end{align} Actually I only know how to prove that simultaneous limits does not exist . So please give me ...
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### Is it sufficient to have a complex partial derivative for the complex partial derivative be continous?

Suppose we have $\phi(z,w)$ a function of two complex variables and that for each fixed $w$ the function $z\mapsto \phi(z,w)$ is holomorphic, that is, it exists $\frac{\partial}{\partial z}\phi(z,w)$. ...
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### Orthonormal Basis of Bergman Space

This is a problem (#39 or #40 depending on the edition) at the end of Chapter 1 in Krantz's book Function Theory of Several Complex Variables. Let $\Omega\subset\mathbb{C}^n$ be a smooth and bounded ...
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### Integral of characteristic function in three dimensions

I am having difficulty in evaluating the following integral: $\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} f(x) f(y) f(z)\chi_{{(x,x,x): x\in \mathbb{R}}} dx dy dz$. Could ...
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### Domain of convergence in several complex variables

I am trying to understand how to calculate the domain of convergence for the series [From: Tasty Bits of Several Complex Variables] : $$\sum_{j,k} c_{j,k}\; z_1^j z_2^k$$ But I don't really ...
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### PDE and Several Complex variables

I am reading SCV from Grauert and Fritzsche. I have read that as an application one can use SCV in PDE (partial differential equations). I have some general questions: Is this (SCV-PDE) different ...
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### How to determine the impact of a several factors evolution?

Let's take a simple example. Let's say I have : A population, with size N A sub-population, composed by people that goes to the hairdresser with size M (so M < N) From the two previous points, I ...
1 vote
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### Understanding arbitrary currents and currents of integration.

I've been doing some readings, and there are some notational issues with currents that I do not understand. Let $\Omega\subset \mathbb C^n$ be open. A current $T$ of bidegree $(p,q)$ is an element of ...
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### Singularity set of coherent analytic sheaf

I am trying to understand section 5.5 of the standard reference Differential geometry of complex vector bundles (S. Kobayashi). Let me set up some notation: let $X$ be a complex manifold, $x_0\in X$ ...
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### Holomorphic primitive of several complex variables function

Let $f:\Omega\subset\mathbb{C}^n\rightarrow\mathbb{C}$ a holomorphic function. For any $m\leq n$ I would like to find an $m-primitive$ of $f$, which is still holomorphic in each variable. If we ...
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### Which domains are contractible on $\mathbb{C}^n$?

Let $\mathbb{C}^n$ be the complex Euclidean space. The wiki of contractible tells us that any Euclidean space is contractible, as is any star domain on a Euclidean space. My question is: 1, Is a ...
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### Complement of zero set of holomorphic function is dense and connected

This is exercise 1.1.8 of Huybrechts' Complex Geometry. Problem: let $U$ be an open connected subset of $\mathbb{C}^n$. Let $f:U\rightarrow \mathbb{C}$ be a holomorphic function. Show that the ...
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