# Complex function of several variables

1. Jan 30, 2012

### gotjrgkr

Hi!!

While studying Global Cauchy Theorems in complex analysis, I've realized that I need to know a definition of continuity of a complex function of several variables....
Thus, I ask you the definition of continuity of a complex function of several complex variables.
What I mean is .... ; Given a complex valued function F of several complex variables(F:A$\rightarrow$C where A is a subset of C$^{n}$ for a positive integer n and C implies the complex plane), What does it mean F is continuous at a point x in A for this function??

In addition, I also ask you whether the definition of metric of a point x=(z$_{1}$,...,z$_{n}$) in the comples euclidean space C$^{n}$ is just $\sqrt{\sum^{n}_{i=1}\left|z_{i}\right|^{2}}$ or not??

2. Jan 30, 2012

### Dick

Yes, it's a metric space. So the definition of continuity is the same as in any metric space. But a metric is the distance between two points $x=(z_1,...,z_n)$ and $x'=(z'_1,...,z'_n)$ and that would be $d(x,x')=\sqrt{\sum^{n}_{i=1}\left|z_{i}-z'_{i}\right|^{2}}$.

3. Jan 30, 2012

### gotjrgkr

Oh,, I see...
I also wanted to get such an answer for my question.
1. But, I couldn't find any book dealing with a complex function of several variables.
Of course, there's a book dealing with such a kind of field. But, it seems that almost every book omit the basic part of it such as continuity, limit, and topology related with higer dimensional complex metric space... So, could you recommend a book explaining such basic parts of the complex function of several variables ( basic concepts such as continuity, definition of metric of the space, limit, etc).
I know this is a quite a different question for the main question, but
2. I also wanna ask you if in the complex euclidean space(considered as a metric space as you show) Heine-Borel Theorem still holds or not(I mean that a subset is compact in the metric space iff the subset is closed and bounded in the metric space)...

4. Jan 30, 2012

### micromass

Staff Emeritus
Yes. $\mathbb{C}$ is isomorphic with $\mathbb{R}^2$ and the Heine-Borel theorem holds in $\mathbb{R}^2$.

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