Analyzing the Analyticity of ln (z)

[Void123]

Homework Statement

How would I prove that $$ln (z)$$ is analytic?

Homework Equations

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The Attempt at a Solution

I rewrote it as $$ln (z) = ln (r) + i\theta$$. But, I'm not quite sure how to apply Cauchy-Riemman conditions here.

 

[clamtrox]

$$\frac{\partial f}{\partial x} = \frac{\partial f}{\partial r}\frac{\partial r}{\partial x} + \frac{\partial f}{\partial \theta}\frac{\partial \theta}{\partial x}$$

if you want to use the polar coordinates. Remember also that CR-equation can be written as

$$i \frac{\partial f}{\partial x} = \frac{\partial f}{\partial y}.$$