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Let $z$ be a complex number and let $f(z)$ be any transcendental entire function.

Is it true that $f(z) = C + \exp a + \exp b + \exp c $ where $ a,b,c $ are entire functions of $z$ and $C$ is a constant ?

If so, for a given $f$ in Taylor form is there an easy way to compute $a,b,c,C$ ?

Note that $C$ is not neccessarily equal to $f(0)$.

If the conjecture is true is there a unique solution for every $f(z)$ ?

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  • $\begingroup$ Strange but the question marks seem to move around and end Up at wrong places when rotating my phone. The rest is fine. $\endgroup$ – mick Jul 15 '15 at 7:30
  • $\begingroup$ @gerry transcendental !! $\endgroup$ – mick Jul 15 '15 at 7:31
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    $\begingroup$ OK, $e^z+z$, then. $\endgroup$ – Gerry Myerson Jul 15 '15 at 7:33
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    $\begingroup$ What motivates this conjecture? Why 3 functions? $\endgroup$ – lhf Jul 15 '15 at 11:21
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    $\begingroup$ It's certainly not unique, e.g., $e^z=e^z+e^t+e^{t+\pi i}$ for any $t$. $\endgroup$ – Gerry Myerson Jul 15 '15 at 12:57

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