Say $[a,b]$ is an interval on the real line, $f$ is a continuous complex function on that interval and $c$ is any given complex number. With some elementary arguments one can show that there exists a continuous complex valued function $g$ with compact support in $(a,b)$ such that $$ \int_a^b e^{f(x) + g(x)} dx = c $$ and $$ \int_a^b g(x) e^{f(x) + g(x)} dx \neq 0 $$
I am happy to take this for granted.
Let $X$ be a Riemann surface and $\omega$ a non vanishing holomorphic 1-form on $X$. Say $\gamma$ is a simple closed curve on $X$. How can I use the above result to conclude that there exists a continuous function $u$ on the set $\gamma \subset X$ such that
$$ \int_{\gamma} \omega e^{u} = 0 $$ and $$ \int_{\gamma} u\, \omega e^{u} \neq 0 $$
I think the argument is elementary but it is not yet obvious to me. If someone could explain how one derives the above (by taking for granted the simple result about real integrals), I would be grateful.