Let $0\in U \subset \mathbb{C}$ 
be a small neighborhood of origin in the complex plane and $f\_1,f\_2\colon U\to \mathbb{C}$ be two complex valued functions such that
$$f\_1(0)=f\_2(0)=0$$
$$\bar\partial{f}\_1(0)= \bar\partial f\_2(0) =0$$
$$df\_1 ,df\_2\neq 0$$.


Let $f\_2/f\_1\colon U\setminus \{0\} \to \mathbb{C}$ be the quotient function.


Is it true that $f\_2/f\_1$ extend to the origin and $\bar\partial (f\_2/f\_1)(0)=0$?


Note that if $\bar\partial f\_i $ vanishes over entire $U$, i.e. if they are holomorphic, then above assumption imply that they have a simple zero and thus the quotient is well-defined.



quotient of holomorphic functions at a point
No. Writing as usual $z=x+iy$, take $f\_1=z$ and $f\_2=z+x^2$. Then your conditions are satisfied but $\dfrac{f\_2}{f\_1}=1+\dfrac{x^2}{z} $ does not extend to $0$.


