Abstract:
A construction method for approximate conformal mapping of the unit
disk onto a Riemann surface (a map with a self-overlapping
image) has been described. An example has been provided to illustrate
the applicability of the method to conformal mapping of the unit
disk onto a two-sheeted covering of the domain by a Riemann surface.
The function construction is based on the approximate solution of
the second kind Fredholm integral equation by reducing it to the
finite linear equation system, so the construction is easily
programmable.
The necessary and sufficient condition for the function given on
the closed curve to be the boundary value of some function analytic
in the region on the Riemann surface bounded by the given curve is
naturally somewhat different from that for one-connected and
one-sheeted domains. We have applied this condition for a
multiply-sheeted region.
Let $z(\zeta)$ be the function that maps the unit disk onto a
multiply-sheeted region conformally. For the function
$\displaystyle \phi (z) = \ln({\zeta (z)}/{z})$, we write the
equations similar to that for one-connected and one-sheeted domains.
Note that for our example with two-sheeted domain it is necessary
to divide the right-hand side of our relations by $3$ for the points
on the contour sections bounding the nonunivalent region.
The solution then repeats the steps of the one-sheeted domain
situation for our case.
Fulltext:
PDF file (734 kB)