"n" is a given positive integer. Find all entire functions satisfying $f^n(z)=z$ for every z in $\mathbb C$
Not getting any ideas. Can someone help please?
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is the $n$ indicating the derivative or a power? If the latter, and if $n > 1$ there are no such entire functions. $f(z) = z^{1/n} $ for $n > 1$ always has a branch point at $z=0$. – WA Don Nov 16 '20 at 16:25
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1If $f^n$ denotes the $n$-fold composition $f\circ f \circ \ldots \circ f$, there are solutions. – Daniel Fischer Nov 16 '20 at 16:33
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n denotes composition of f with itself n times. – Lawrence Mano Nov 16 '20 at 16:38
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$f$ is one-to-one, so by Picard's "great" theorem it can't have an essential singularity at $\infty$. We conclude that $f$ is a polynomial of degree $1$, and then it's easy to show $f(z) = c z + d$ where $c$ is an $n$'th root of unity (with $d=0$ in the case $c = 1$).
Robert Israel
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