Differentiability of $\ f(z) = e^{\bar z}$

Question

Hey so I've just started trying to teach myself complex analysis and Ijudt came accross the following question:

Find the real and imaginary parts of the following function and decide where the function is differentiable and where it is holomorphic. Determine the derivative at the points where the function is differentiable.

$$\ f(z) = e^{\bar z}$$

I have no idea how to go about this question any help would be greatly appreciated, just bare in mind I'm new to complex analysis

Answer 1

"I have no idea how to go about this question"

How about asking yourself simpler questions:

• What are the real and imaginary parts of $f$?
• How do the real and imaginary parts of $f$ relate to the differentiability of it? What theorems do you know about complex differentiability?
• Can you find out at what point $f$ is differentiable?

Answer 2

Hint: $$e^{\bar{z}}=e^{x-yi}=e^x(\cos(y)-i\sin(y)) $$ and then use the Cauchy Riemann equations.