CVEL2

Find the domain of analyticity for $f(z) = \mathrm{Log}(3z-i)\,$

The principal valued $\mathrm{Log}\,$ is analytic everywhere except the ray in the complex plane $x\le 0, y=0\,$ . In this case,

$3z-i = 3(x+iy)-i = 3x + i(3y-1)\,$ so this function is not analytic when $x\le 0\,$ and $y = 1/3\,$ .

Now, in this slit plane the derivative can be computed:

$f'(z) = \frac{1}{3z-i}\frac{d}{dz}(3z-i) = \frac{3}{3z-i}\,$