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) ?


  • 1
    $\begingroup$ (1) is simple: let $f$ be any entire function and consider the function $g(z)=f(\overline z)$. $\endgroup$ Sep 3, 2014 at 12:08
  • 2
    $\begingroup$ @user161825 No, it's still real entire. $\endgroup$
    – user147263
    Sep 4, 2014 at 0:54

1 Answer 1

  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.

  2. No. Not only I that can't, but I'm sure there is no well-known space of functions that requires real analyticity yet excludes real-entire functions.

  3. The construction in item 1 is exhaustive: all RA\RE functions arise in this way.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.