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Use the Cauchy-Riemann equations to show that the function $$g(z) = \sin (\bar z)$$ is not analytic at any point of $\mathbb{C}$.

Here's as far as I got -

$$\sin \left(\frac{\bar z}{1}.\frac{z}{z}\right) =\sin \left(\frac{|z|^2}{z}\right) =\sin \left(\frac{x^2 + y^2}{x+iy}\right)$$

I can't see how to separate the real and imaginary parts so that I can apply the Cauchy-Riemann equations.

  • 1
    And the complex $\sin(z)$ function is defined as.....2012-04-18
  • 3
    You could have just straight-up written $z=x-iy$ if you wanted explicit real and imaginary parts. Are you using the definition $$\sin z = \frac{e^{iz}-e^{-iz}}{2i}~?$$2012-04-18
  • 0
    I need a break, I cant believe the crap I am missing tonight :(2012-04-18
  • 0
    (That should be $\bar{z}$ in my comment, obviously.) No worries Jim.2012-04-18
  • 1
    You can prove that: CR equations $\Longleftrightarrow \frac{\partial f}{\partial \bar{z}}=0$.2012-04-18

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