Solving functional equation $\sin f = f^2-3if+\pi$

Question

I want to find all (entire) solutions to the equation $$\sin f = f^2-3if+\pi.$$ Using the identity theorem, I was able to show that if a solution exists, it must be constant. Therefore, all that is left to do is to show that there is some $a\in\mathbb{C}$ such that $$\sin a = a^2-3ia+\pi.$$ By plugging the above equation into WolframAlpha, I know that there must exist a unique solution (namely $a\approx .19+.78i$), but I could not prove so. Any help in showing the existence (or the uniqueness) of $a$ would be appreciated.

Answer 1

$\sin a = a^2-3ia+\pi$ has infinitely many solutions $a \in \Bbb C$.

More generally, for any polynomial $p$, $\sin z - p(z)$ has infinitely many zeros in the complex plane.

This can be shown in the same way as Antonio Vargas did it in Solve $\sin(z) = z$ in complex numbers for the special case $\sin z - z$:

Assume that $\sin z - p(z)$ has only finitely many zeros. The function is entire and of order $1$, therefore the Hadamard factorization theorem gives that $$ \sin z - p(z) = q(z) e^{az+b} $$ with a polynomial $q$, and complex numbers $a, b$. For $z = iy$, $y \to +\infty$, the left-hand side is asymptotically $$ \frac{e^{-y} - e^y}{2i} + O(y^{\deg p}) \sim \frac 12 i e^y $$ and that is only possible if $\operatorname{Im} a = -1$. On the other hand, for $z = -iy$, $y \to +\infty$, the left-hand side is asymptotically $$ \frac{e^{y} - e^{-y}}{2i} + O(y^{\deg p}) \sim -\frac 12 i e^y $$ which is only possible if $\operatorname{Im} a = +1$.