## On conformal maps from multiply connected domains onto lemniscatic domains

Olivier Sète and Jörg Liesen

### Abstract

We study conformal maps from multiply connected domains in the extended complex plane onto lemniscatic domains. Walsh proved the existence of such maps in 1956 and thus obtained a direct generalization of the Riemann mapping theorem to multiply connected domains. For certain polynomial preimages of simply connected sets, we derive a construction principle for Walsh's conformal map in terms of the Riemann map for the simply connected set. Moreover, we explicitly construct examples of Walsh's conformal map for certain radial slit domains and circular domains.

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### Key words

conformal mapping, multiply connected domains, lemniscatic domains

30C35, 30C20

### Links to the cited ETNA articles

 [11] Vol. 36 (2009-2010), pp. 195-223 Thomas K. DeLillo and Everett H. Kropf: Slit maps and Schwarz-Christoffel maps for multiply connected domains [20] Vol. 5 (1997), pp. 62-76 Vincent Heuveline and Miloud Sadkane: Arnoldi-Faber method for large non Hermitian eigenvalue problems

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