Consider the function from the complex plane to itself given by
w = f(z) = z^{n}, where
n is at least 2. The
zplane
may be divided into
n sectors given by
arg z lying between
(k  1)(2π/n) and
k(2π/n) for
k = 1, ..., n.
There is a onetoone correspondence between each sector and the whole
wplane, except for the positive real axis. The image of each sector
is obtained by performing a cut along the positive real axis; this cut
has an upper and a lower edge. Corresponding to the n sectors in the
zplane,
take
n identical copies of the
wplane with the cut. These
will be the
sheets of the Riemann surface and are distinguished
by a label
k which serves to identify the corresponding sector.
For
k = 1, ...,
n 1 attach the lower edge of the sheet labeled
k with the upper edge of the sheet labeled
k + 1. To complete
the cycle, attach the lower edge of the sheet labeled
n to the upper
edge of the sheet labeled 1. In a physical sense, this is not possible
without selfintersection but the idealized model shall be free of this
discrepancy. The result of the construction is a
Riemann surface
whose points are in onetoone correspondence with the points of the
zplane.

Figure 1. Riemann surface of two sheets

This correspondence is continuous in the following sense. When z
moves in its plane the corresponding point w is free to move on
the Riemann surface. The point w = 0 connects all the sheets and
is called the branch point. A curve must wind n times around the
branch point before it closes.

Figure 2. Riemann surface of three sheets

Now consider the
n  valued relation
z = n^{th} root(w). To each nonzero
w, there
correspond
n values of
z. If the
wplane is replaced
by the Riemann surface just constructed, then each complex nonzero
w
is represented by
n points of the Riemann surface at superposed
positions. Let the point on the uppermost sheet represent the principal
value and the other
n  1 points represent the other values. Then
z = n^{th} root(w) becomes a singlevalued, continuous,
onetoone correspondence of the points of the Riemann surface with the
points of the
zplane. The Riemann surface is orientable, since
every orientation of a sheet is carried over to the sheet next to it.