Abstract
The purpose of this work is to obtain a new type of conformal mappings of compact finite Riemann surfaces bounded by finitely many analytic Jordan curves. This is achieved by making use of Riemann-Roch theorem. As is well-known, every plane region is conformally equivalent to a parallel slit region. This theorem was carried over the case of Riemann surfaces with finite genus. The other types of conformal mappings can be found in the different literatures. It will be now deal with a different conformal mapping from those. It is a finite sheeted covering surface of the extended complex plane whose each boundary component consists of a closed interval on real axis.
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