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Conformal mapping of the domain bounded by a circular polygon with zero angles onto the unit disc

Vladimir Mityushev — 1998

Annales Polonici Mathematici

The conformal mapping ω(z) of a domain D onto the unit disc must satisfy the condition |ω(t)| = 1 on ∂D, the boundary of D. The last condition can be considered as a Dirichlet problem for the domain D. In the present paper this problem is reduced to a system of functional equations when ∂D is a circular polygon with zero angles. The mapping is given in terms of a Poincaré series.

O zastosowaniu sumowania według Eisensteina do pewnej sumy podwójnej

Agata MatczykVladimir MityushevZbigniew Powązka — 2009

Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia

The lattice sum S_2 for the square array conditionally converges. Having used physical arguments, Rayleigh chose an order of summation in such a way that S_2 = π. The Eisenstein summation method applied to S2 yields the same result. This paper is devoted to a rigorous proof of S_2 = π for the Eisenstein summation method. The study can be used in class for students as an interesting example which illustrates different types of convergence.

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