Is $\cosh(z)\exp(iz)+ \sin(i\overline{z})\exp(z)$ complex differentiable.

Question

Is $$f: \mathbb{C} \to \mathbb{C}, \quad z \mapsto f(z) := \cosh(z)\exp(iz)+ \sin(i\overline{z})\exp(z)$$ complex differentiable?

I found this question in a sample exam for complex analysis and couldn't figure out a solution yet. I first tried to split the complex trigonometric functions into real and imag. part, and use the Cauchy-Riemann equations. This ended up in a huge term, so I guess theres a much faster way, which I fail to see.

Thanks in advance.

Answer 1

If it were, so would be $\sinh \bar{z}$. But nontrivial nice functions of $\bar{z}$ like that aren't analytic. If you want to complete that step of the proof rigourously, show it fails the Cauchy-Riemann equations.