Solve this equation: $f(s)=P(s)\exp(Q(s))$

Question

Let $f,P,Q$ three analytic functions. Here $P$ is a polynomial.

I want to solve this equation: $$f(s)=P(s)\exp(Q(s)).$$

The unknown here are $P, Q$ and $f$ is known.

Answer 1

Assume $f$ is entire and not the zero function. Since the exponential is never zero, $P$ should capture all zeroes of $f$. That is: $(s-s_0)$ is a factor of $P$ if and only if $f(s_0)=0$. More precisely, for $s\in\mathbb C$ let $$\nu(s)=\min\{\,n\in\mathbb N_0\mid f^{(n)}(s)\ne0\,\}.$$ Then we can (and up to a constant factor: must) let $$P(s)=\prod_{z\in\mathbb C}(s-z)^{\nu(z)}$$ and $$ Q(s)=\ln\left(\frac{f(s)}{P(s)}\right).$$ Several assumptions about $f$ are needed for this to work in the first place, for example $f$ must have only finitely many zeroes.

Answer 2

$P(s)$ is an arbitrary polynomial. Then $Q(s) = \log(\frac{f(s)}{P(s)})$.