One way this could be written more in keeping with the "when successively different values of the logw are specified" is perhaps:
and so on similarly for n=3, this function is 3-fold period and 3 values are bounced around ad infinitum.
Hence we only need assume k=0,....,n-1 to arrive at all the distinct values.
![z=w^{1/n}=\left(re^{i\theta+2k\pi i}\right)^{1/n}=\sqrt[n]{r}\cdot e^{\frac{i\theta+2k\pi i}{n}}=\sqrt[n]{r}\cdot e^{\frac{i\theta}{n}}\cdot e^{\frac{2k\pi i}{n}}=z_0\cdot e^{\frac{2k\pi i}{n}}](latexrender/pictures/d16f69a794ae1a1d7cdb3c5695781814.png)
solutions to the equation will be given by 
for 
.
corrected. Thanks to vasco for noticing.
