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A

At which points if any does the function

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

B

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

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?

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    $\begingroup$ Dear Jim, A. is false. And as to B., I don't see what you mean with "they are not differentiable" . $\endgroup$ – Georges Elencwajg Apr 18 '12 at 15:09
  • $\begingroup$ Carelessness during differentiation cost me again. $\endgroup$ – Jim_CS Apr 19 '12 at 10:55
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A. Only at 0 B. Analytic Nowhere

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

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  • $\begingroup$ 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 '12 at 0:19

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