# zbMATH — the first resource for mathematics

Numerical conformal mapping of multiply connected regions onto the second, third and fourth categories of Koebe’s canonical slit domains. (English) Zbl 1227.30007
In his previous papers the author presented a boundary integral method to approximate conformal mappings from a multiply connected region onto the first category of Koebe’s canonical slit domains. The present article extends the author’s approach for numerical approximations of such conformal mappings onto the second, third and fourth categories of slit domains, namely: an annulus with spiral slits, a disk with spiral slits, a plane with spiral slits, a plane with straight slits.
The numerical method is based on a boundary integral equation which is uniquely solvable. The theoretical proposals are illustrated by three examples and many figures.

##### MSC:
 30C20 Conformal mappings of special domains 30C30 Schwarz-Christoffel-type mappings
Full Text:
##### References:
 [1] Atkinson, K.E., The numerical solution of integral equations of the second kind, (1997), Cambridge Univ. Press Cambridge · Zbl 0155.47404 [2] Bergman, S., The kernel function and conformal mapping, (1970), Amer. Math. Soc. Providence · Zbl 0208.34302 [3] DeLillo, T.K.; Driscoll, T.A.; Elcrat, A.R.; Pfaltzgraff, J.A., Radial and circular slit maps of unbounded multiply connected circle domains, Proc. R. soc. lond. ser. A, 464, 1719-1737, (2008) · Zbl 1157.30020 [4] Koebe, P., Abhandlungen zur theorie der konformen abbildung, IV. abbildung mehrfach zusammenhängender schlichter bereiche auf schlitzbe-reiche, Acta math., 41, 305-344, (1918) · JFM 46.0545.02 [5] Kühnau, R., Canonical conformal and quasiconformal mappings. identities. kernel functions, (), 131-163 · Zbl 1075.30004 [6] Nasser, M.M.S., A boundary integral equation for conformal mapping of bounded multiply connected regions, Comput. methods funct. theory, 9, 127-143, (2009) · Zbl 1159.30007 [7] Nasser, M.M.S., The Riemann-Hilbert problem and the generalized Neumann kernel on unbounded multiply connected regions, The university researcher (IBB university J.), 20, 47-60, (2009) [8] Nasser, M.M.S.; Murid, A.H.M.; Zamzamir, Z., A boundary integral method for the Riemann-Hilbert problem in domains with corners, Complex var. elliptic equ., 53, 11, 989-1008, (2008) · Zbl 1159.30023 [9] Nasser, M.M.S., Numerical conformal mapping via a boundary integral equation with the generalized Neumann kernel, SIAM J. sci. comput., 31, 1695-1715, (2009) · Zbl 1198.30009 [10] Nehari, Z., Conformal mapping, (1952), Dover Publications, Inc. New York · Zbl 0048.31503 [11] Spencer, D.C., Some problems in conformal mapping, Bull. amer. math. soc., 53, 417-439, (1947) · Zbl 0054.03605 [12] Wegmann, R., Methods for numerical conformal mapping, (), 351-477 · Zbl 1131.30004 [13] Wegmann, R.; Nasser, M.M.S., The Riemann-Hilbert problem and the generalized Neumann kernel on multiply connected regions, J. comput. appl. math., 214, 36-57, (2008) · Zbl 1157.45303 [14] Wen, G.C., Conformal mapping and boundary value problems, (1992), Amer. Math. Soc. Providence, English transl. of Chinese ed. 1984
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.