Testing the analyticity of $2\ln z$

Question

Test whether the function $2\ln z$ is analytic.

I have tried to test this function for analyticity by letting $z=x+iy$, but I have failed to separate the real and imaginary parts. Any help would be appreciated.

Answer 1

Hint:) The logarithm is a multi-valued function and the main branch of logarithm, that is $$\log z=\ln r+i\theta\hspace{2cm}r>0,~0\leq\theta<2\pi$$ is analytic in $\mathbb{C}-\{0\}$, to verify we can let $z=e^{i\theta}$ and whether it satisfies the relation $$e^{-i\theta}\dfrac{\partial f}{\partial r}=\dfrac{1}{iz}\dfrac{\partial f}{\partial\theta}$$