Let $C$ be a smooth projective curve over a field $k$ of characteristic zero and $S$ a reduced divisor on $C$ (so just a collection of points). Consider the sheaf of logarithmic differentials $\Omega^1\_C(\log S)$.


Why is it possible to find a rational section $\omega$ of $\Omega^1\_C(\log S)$ such that the residue of $\omega$ at all points of $S$ is non-zero? 


Can one give an explicit formula in the case of $\mathbb{P}^1$?



rational sections of logarithmic differentials on a curve
I think what you mean is a *regular* section of $\Omega ^1\_C(\log S)$, that is, a rational 1-form $\omega $ which is regular outside $S$ and has a simple pole at each point of $S$. Consider the exact sequence of sheaves
$$0\rightarrow \Omega ^1\_C\rightarrow \Omega ^1\_C(\log S)\rightarrow \oplus\_{s\in S}\ k(s)\rightarrow 0$$
and the associated cohomology sequence 
$$H^0(C,\Omega ^1\_C(\log S))\stackrel{\mathrm{Res}}{\longrightarrow} k^S\stackrel{\partial}{\longrightarrow} H^1(C,\Omega ^1\_C)\cong k\rightarrow 0\ .$$
It is easy to see that $\partial$ is just the sum map, so given $(r\_s)\in k^S$, there exists a form $\omega \in H^0(C,\Omega ^1\_C(\log S))$ with residue $r\_s$ at $s$ if and only if $\sum r\_s=0$. In particular there always exists such a form with all residues $\neq 0$, except when $\# S=1$.


If $C=\mathbb{P}^1$ and $S=\{s\_1,\ldots ,s\_n\} \subset \mathbb{C}$, just take $\omega =\sum\_s \dfrac{r\_s\,dz}{z-s} $; it has the required residues, and the condition $\sum r\_s=0$ ensures that $\omega $ is regular at $\infty$.


