Let $S$ be a semigroup with no identity element and $m:S\to \Bbb C$ be given function($m\not\equiv 0$) satisfying the exponential functional equation $$ m(x+y)=m(x)m(y) $$ for all $x, y\in S$. Find all solutions $f:S\to \Bbb C$ satisfying the equation \begin{equation} f(x+y)=f(x)m(y)+f(y)m(x) \tag 1 \end{equation} for all $x, y\in S$.
Remark. If $S$ is a group, then using the fact that $m(x)\ne 0$ for all $x\in S$ and dividing $(1)$ by $m(x+y)$, we have $$ f(x)=m(x)A(x) $$ for all $x\in S$, where $A$ is an additive function.