If $h$ is an arbitrary holomorphic function then there exist $\phi_1, \phi_2$ holomorphic such that $h = f\phi_1 + g\phi_2$ I don't need this statement to be true globally - even if it's locally true I'm fine with that. I'm on a noncompact Riemann surface but that doesn't matter because I only want this to hold locally anyway, so we can just look at little subsets of $\mathbb{C}$. Say $h$ is an arbitrary holomorphic function.  Say $f,g$ are holomorphic functions with no common zero. Then I want to show that there exist $\phi_1, \phi_2$ holomorphic such that $h = f\phi_1 + g\phi_2$ locally (so on some neighbourhood of some fixed point). I'm happy to just work with Taylor series of all these functions (which is what I was doing) because as I said, I'm fine with this holding just locally. But I can't quite figure out how it works. I'd appreciate some help, please!
 A: Here is an advanced level but very general answer.
Let $X$ be a complex manifold (or even  complex space with singularities ) and let $f_1,\cdots, f_n\in \mathcal O(X)$ be globally defined holomorphic functions .
Then we have a an $\mathcal O_X$-linear map of coherent sheaves $$f=(f_1,\cdots, f_n):  \mathcal O_X^n\to  \mathcal O_X                 $$ sending $(\phi_1,\cdots, \phi_n)$ to $\sum f_i\phi_i$.
The hypothesis that the $f_i$'s have no common zero at any point $x\in X$ implies that this morphism of sheaves is surjective,  so that denoting by $\mathcal K$ its kernel we get the exact sequence of coherent sheaves $$0\to\mathcal K  \to\   \mathcal O_X^n\to  \mathcal O_X \to 0                $$ The long exact sequence of cohomology then yields the fragment $$\cdots              \to \mathcal O_X(X)^n\to  \mathcal O_X(X) \to H^1(X,\mathcal K)\to \cdots      $$ 
If $X$ is Stein, as is for example every non-compact Riemann surface, Cartan-Serre's Théorème B for coherent sheaves  implies that $H^1(X,\mathcal K)=0$, so that $$\mathcal O_X(X)^n\to  \mathcal O_X(X)\to 0$$ is surjective.
 This means exactly that every $h\in \mathcal O_X(X)$ can be written as $$h=\sum f_i\Phi_i$$ for some suitable global holomorphic functions $\Phi_i\in \mathcal O_X(X)$.
A: Hint: In Rudin Real and Complex Analysis there's a proof that while $H(\Omega)$ is not a PID, nonetheless every finitely generated ideal is principal.
