MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question
up vote 2 down vote accepted

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 $0$.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

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