Answer to Question #1907 in Complex Analysis for SAVVAS SAVVAN

Question

Could someone shine some light on how to find the inverse,the domain and the rule of the
following function?

f(z)= ln [(z+1+i)/3]

Answer 1

Similar to the functions of real variable. The inverse function to logariphm is exponenta:
ln [(z+1+i)/3] = ln (z+a) - ln3, where a = 1+i.
ln(f(z)+a) - ln3 = z
f(z)+a = e(z+ln3)
f(z) = e^(z+ln3) - a = 3e^z -a
Thus inverse is 3e^z-1-i.
The domain is z+a > 0, z > -a, z > (1+i).

The domain is