# Is Every transcendental entire function $f(z) = C + \exp a + \exp b + \exp c$?

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 $f(z)$ ?

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