Numerical conformal mapping of multiply connected regions onto the second, third and fourth categories of Koebeʼs canonical slit domains

This paper presents a boundary integral method for approximating the conformal mappings from any bounded or unbounded multiply connected region G onto the second, third and fourth categories of Koebeʼs canonical slit domains. The method can be also used for calculating the conformal mappings of simply and doubly connected regions. The method is an extension of the authorʼs method for the first category of Koebeʼs canonical slit domains (see [M.M.S. Nasser, Numerical conformal mapping via a boundary integral equation with the generalized Neumann kernel, SIAM J. Sci. Comput. 31 (2009) 1695–1715]). Three numerical examples are presented to illustrate the performance of the proposed method.