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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?

  • 0
    Carelessness during differentiation cost me again.2012-04-19

2 Answers 2

0

A. Only at 0 B. Analytic Nowhere

0

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

  • 0
    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.2012-09-27