What does the space $\overline{\mathbb{C}[z]}$ stands for? Does it contain all the analytic functions or there are something else? And what about the closure thing?
The space $\mathbb{C}[z]$
-1
$\begingroup$
complex-analysis
notation
-
0Don't you mean $\overline{\mathbb{C}[z]}$ in the title of the question as well? – 2011-11-21
-
0Could it be the polynomials in the variable $\overline{z}$? As in [Antiholomorphic function](http://en.wikipedia.org/wiki/Antiholomorphic_function)? – 2011-11-21
-
6It is impossible to know what that notation means if you do not tell us at least where you found it... – 2011-11-22
1 Answers
1
$\mathbb{C}[z]$ is the space of polynomial functions. I'm not sure what the bar does to it. What context does this appear in? Perhaps $\overline{\mathbb{C}[z]}$ stands for the space of entire functions because they are the limits of polynomials since they have series expansions. Other notations for the space of entire functions are $\mathcal O(\mathbb C)$ and $\mathcal H(\mathbb C)$.
-
0I was reading an article about entire functions. – 2011-11-21
-
6@Vanessa: Perhaps you could include a reference to the article in your question? – 2011-11-21
-
0So it is the biggest space of entire functions; contains all complex polynomials, and all other forms of entire functions! – 2011-11-21
-
3@Vanessa, we'd have to see that paper... – 2011-11-21
-
3I would expect that something like $1+(1/2)z+(1/4)z^2+(1/8)z^3+\dots$ would be in the set, and that's not an entire function. It would really, really be good if we could see the article. – 2011-11-22
-
0@Gerry, I meant *globally convergent* series. Perhaps, as you suggest, that space is just the space of holomorphic functions near 0. – 2011-11-22
-
2I know that's what you meant - trouble is, neither one of us knows what the author meant. All I'm saying is that if *I* were to use that symbol, I would mean for it to include anything that looked like a limit of a sequence of polynomials. I might even mean for it to include things like $1+x+2x^2+6x^3+24x^4+120x^5+\dots$, which doesn't converge anywhere but at zero. – 2011-11-22