Geodesic
Conformal mapping onto numerable polygon with double symmetry
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 12 (2014), pp. 37-47

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We consider simply connected domains of half-plane type with the symmetry of transfer along the real axis by $2\pi$ and symmetry with respect to vertical straight line $w=\pi+iv$, $v\in\mathbb R$, with a boundary consisting of straight line segments. Conformal mapping of the half-plane onto such domains are represented by integral of Schwarz–Christoffel integral type. The proof of the result is based on Riemann–Schwarz principle of symmetry and Schwarz–Christoffel classical formula. We found several mappings on specifically defined domain.
Keywords: numerable polygon, conformal mapping, symmetry of transfer, elliptic integrals. 
@article{IVM_2014_12_a3,
     author = {I. A. Kolesnikov and L. S. Kopaneva},
     title = {Conformal mapping onto numerable polygon with double symmetry},
     journal = {Izvesti\^a vys\v{s}ih u\v{c}ebnyh zavedenij. Matematika},
     pages = {37--47},
     publisher = {mathdoc},
     number = {12},
     year = {2014},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IVM_2014_12_a3/}
}
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I. A. Kolesnikov; L. S. Kopaneva. Conformal mapping onto numerable polygon with double symmetry. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 12 (2014), pp. 37-47. http://geodesic.mathdoc.fr/item/IVM_2014_12_a3/