# Can we characterize the space of functions which is real analytic but not real entire?

A complex valued function $$F,$$ defined on an open set $$E$$ in the plane $$\mathbb R^{2}$$, is said to be real-analytic in $$E$$ if to every point $$(s_{0}, t_{0})$$ in there corresponds an expansion with complex coefficients $$F(s, t)= \sum_{n,m=0}^{\infty} a_{nm}(s-s_{0})^{m} (t-t_{0})^{n},$$ which converges absolutely for all $$(s,t)$$ in some neighbourhood of $$(s_{0}, t_{0}).$$

If $$F$$ is defined in the whole plane $$\mathbb R^{2}$$ by a series $$F(s, t)= \sum_{n,m=0}^{\infty} a_{nm}s^{m} t^{n},$$ which converges absolutely for every $$(s,t),$$ the we call $$F$$ real-entire.

Let us introduce temporary notations, $$RA(\mathbb R^{2}):=\text{The space of real analytic functions on \mathbb R^{2}},$$ and $$RE(\mathbb R^{2}):=\text{The space of real entire functions on \mathbb R^{2}}$$

Note. We note that, $$RE(\mathbb R^{2}) \subset RA(\mathbb R^{2}).$$

Example. There exists $$f(s,t) = \frac{1}{(1+s^{2}) (1+t^{2})}, (s,t \in \mathbb R).$$

is real- analytic in the whole plane $$\mathbb R^{2}$$ but not real-entire; that is, $$f\in RA(\mathbb R^{2})$$ but $$f\notin RE(\mathbb R^{2}).$$

My naive questions are:

(1) How one can construct few more examples $$f$$ so that $$f\in RA(\mathbb R^{2})$$ but $$f\notin RE(\mathbb R^{2})$$ ?

(2) Can we think of some well-known function space say $$E$$, so that, $$E\subset RA(\mathbb R^{2})\setminus RE(\mathbb R^{2})$$ ?

(3) Can we expect to characterize the set $$RA(\mathbb R^{2})\setminus RE(\mathbb R^{2})$$(=The space of functions which is real analytic but not real entire) ?

Thanks,

• (1) is simple: let $f$ be any entire function and consider the function $g(z)=f(\overline z)$. – Jonas Dahlbæk Sep 3 '14 at 12:08
• @user161825 No, it's still real entire. – user147263 Sep 4 '14 at 0:54

1. Take a function $F$ of two complex variables $z,w$ such that $F$ is holomorphic on an open subset of $\mathbb C^2$ containing $\mathbb R^2$, but is not holomorphic on all of $\mathbb C^2$. Then $F$ is real analytic on $\mathbb R^2$ but is not real entire. Indeed, if the series $\sum_{n,m=0}^{\infty} a_{nm}(s-s_{0})^{m} (t-t_{0})^{n}$ converged absolutely for all $s,t\in \mathbb R^2$ it would also converge absolutely for all $(s,t)\in \mathbb C^2$, thus defining a function holomorphic on $\mathbb C^2$. Concrete examples can be obtained by taking any combination of polynomial and exponential functions in $z,w$, divided by a polynomial that does not vanish on $\mathbb R^2$ -- such as $(s^2+1)(t^2+1)$ in your example, or $s^4+t^6+1$, and so forth.