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.


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 $0$.

| cite | improve this answer | |

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.