This is my problem:
Give the largest possible regions in the complex plane where the following four functions are analytic:
a) $\frac{1}{z}$
b) $\ln(z)$
c) $(z-i)^\frac{1}{2}$
d) $\text{Re}(z)$
For functions use the convention $-\pi < Arg(z) \leq \pi$.
For the first one using Cauchy-Riemann equations yields that function is analytic everywhere except at origin, for the second one i know it should be analytic everywhere except at origin and along branch cut, but i'm having trouble proving this for branch cut, for the third one it should be analytic everywhere except where $ z-i=0$ but i'm having some trouble proving that too, and for the last one, is it analytic nowhere?