At which points if any does the function

$$f(z) = z\operatorname{Re}(z) + \bar{z}\operatorname{Im}(z)$$ satisfy the Cauchy-Riemann equations?


At which points, if any is this function analytic. Justify your answer.


A. I applied the Cauchy Riemann equations and found that they are satisfied at x = 1, y = -1.

B. As they are not differentiable anywhere else in C, particularly in some neighbourhood of (1, -1), they function is analytic nowhere.

Are my answers for A and B correct?

  • 1
    $\begingroup$ Dear Jim, A. is false. And as to B., I don't see what you mean with "they are not differentiable" . $\endgroup$ Apr 18, 2012 at 15:09
  • $\begingroup$ Carelessness during differentiation cost me again. $\endgroup$
    – Jim_CS
    Apr 19, 2012 at 10:55

2 Answers 2


A. Only at 0 B. Analytic Nowhere


cauchy-riemann equation is satisfied at all the points which lie on the line x+y=0.

  • $\begingroup$ Incorrect. Write the function as $z^2/2+(1+i)z\bar z/2-(\bar z)^2$ and you will see that the $\bar z$-derivative is $(1+i)z/2-2\bar z$. This can't be zero unless $z=0$, because at all other points the two terms are of different absolute values. $\endgroup$
    – user31373
    Sep 27, 2012 at 0:19

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.