Let $f:\mathbb C \to \mathbb C$ be a function such that $(f(z))^2$ and $(f(z))^3$ are entire then is $f$ entire ?
I can conclude $f$ is entire if given $f$ is continuous ; but without continuity of $f$ , is it true ?
2026-03-26 16:56:39.1774544199
If $(f(z))^2$ and $(f(z))^3$ are entire functions ; then is $f$ entire?
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Let $g(z) = f(z)^2, h(z) = f(z)^3$, both entire. If $g(z)$ has a zero of order $k$ at $z= a$ then $h(z)$ has a zero of order $3k/2$ which means $k$ is even and $\frac{h(z)}{g(z)}$ is holomorphic at $z=a$.
$f(z) = \frac{g(z)}{h(z)}$ is the quotient of two entire functions so it is meromorphic, and it has no poles : it is entire.