# Riemann surface intuition.

In my complex variables notes it says that the multivalued $n$-th root function $w=z^{\frac{1}{n}}$ becomes single-valued on an appropriately constructed Riemann surface. It says how to go about producing such a surface, but it's not very detailed. I'm not really into geometry and attempting to visualise what's going on makes my head hurt. I would appreciate a relatively simple exposition of this concept, not doing into overly technical details, and explaning the intuition and motivation behind the concept and how it's applied.

Algebraic intuition is to note that different values correspond to dividing the argument which is equivalent up to multiples of $2\pi$ by $n$.
Define a surface parametrized by $r>0$, the magnitude , and $0<\theta<2\pi n$, the argument.
Then map $(r,\theta) \mapsto re^{i\theta/n}$ for which we can visualize the domain as $n$ copies of the complex plane cut on the positive real axis and stitched together. The first copy corresponds to $0<\theta<2\pi$, the $j$-th copy to $2\pi(j-1)<\theta<2\pi j$. Note $(r,0)=(r,2\pi n)$ on the surface and map to the same point in the complex plane.