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From old qualifying exam: Let $E$ be the union of the two coordinate axes, i.e. $E = \{z=x+iy : xy=0\}$. Describe all entire functions satisfying $f(E) \subset E$.

I feel like the best approach is to consider the power series of $f$. My first approach was to write down constraints by considering the function applied to the real or imaginary axis. When I didn't get anywhere with this, I began thinking of the function geometrically: on $E$ it's allowed to scale by a real constant, and rotate by $k\pi/2$. But again, I couldn't see how to usefully translate this to produce information about the power series. Thanks!

As an example, $z^2$ has this property. In fact, so does $az^2+bz^4$ (with $a,b \in \mathbb{R}$) since each term maps the imaginary axis to the real axis, which ends up back on the real axis when added. A similar argument shows real odd polynomials work, too.

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    So a guess would be: any entire function whose power series 1) has strictly even (and no constant term) or strictly odd terms and 2) has strictly real or strictly imaginary coefficients. Can't see why this would capture all such functions.2012-01-27

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If $f(E)\subset E$, then $(f^2)(\mathbb{R})\subset\mathbb{R}$. Thus, the coefficients of the power series of $(f)^2$ (around $z=0$) are real. If $\{c_n\}$ are the Taylor coefficients of $f$ around $z=0$, then $ \sum_{j=0}^nc_jc_{n-j}\in\mathbb{R},\quad n=0,1,2,\dots\tag1 $ If $f(E)\subset E$, the same is true of $g(z)=f(i\,z)$. Since the coefficients of $g$ are $i^n\,c_n$, it folows that $ i^n\,\sum_{j=0}^nc_jc_{n-j}\in\mathbb{R},\quad n=0,1,2,\dots\tag2 $ From (1) and (2) we get $ \sum_{j=0}^nc_jc_{n-j}=0\quad\text{if $n$ is odd.}\tag3 $ Now it is not to difficult to see that the power series of $f$ must be one of $ \sum_{n=0}^\infty c_nz^{2n},\quad i\,\sum_{n=0}^\infty c_nz^{2n},\quad \sum_{n=0}^\infty c_nz^{2n+1},\quad i\,\sum_{n=0}^\infty c_nz^{2n+1},\text{ with } c_n\in\mathbb{R}. $

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    @NielsDiepeveen Doh! For some reason I was thinking $z\mapsto1+z$ in my brain... which makes no sense.2012-01-28