Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. 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

Is it true that a function is analytic iff it satisfies the Cauchy-Riemann equations? I am reading Freitag's Complex Analysis and I am asked to show that ${\partial f\over \partial \bar{z}}=0$ iff $f$ is analytic. Is this because $f$ is analytic iff it satisfies the CR equations iff ${\partial f\over \partial \bar{z}}=0$? (It is obvious that $f$ satisfies CR $\implies {\partial f\over \partial \bar{z}}=0$ but what about the other relations? Are they true? I know that if a function is analytic, it must satisfy CR equations, but I don't know if the other direction is true or if ${\partial f\over \partial \bar{z}}=0$ necessarily mean that CR equations are satisfied.)

share|cite|improve this question

To conclude from the Cauchy-Riemann equations or from $\frac{\partial f}{\partial \overline{z}}=0$ that $f$ is analytic, you probably want to assume at least that the partial derivatives of $f$ are continuous. It's possible to get along without that, but not at all easy: look up the Looman-Menchoff theorem.

share|cite|improve this answer

If $\frac{\partial f}{\partial \overline{z}}=0$, we can write $f=u(x,y)+iv(x,y)$ to get the following equations: $$ \frac{1}{2}\left(\frac{\partial}{\partial x}-\frac{1}{i}\frac{\partial}{\partial y}\right)f=0$$ and substituting our decomposition of $f$, $$ \frac{\partial u}{\partial x}+i\frac{\partial v}{\partial x} +i\frac{\partial u}{\partial y}-\frac{\partial v}{\partial y}=0$$ From here, can you see that the Cauchy-Riemann equations must be satisfied?

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.