# quotient of holomorphic functions at a point

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.

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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 $0$.