O zastosowaniu sumowania według Eisensteina do pewnej sumy podwójnej

Agata Matczyk, Vladimir Mityushev, Zbigniew Powązka

Abstract


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.

References


Eisenstein, F.: 1847, Beiträge zur Theorie der elliptischen Funktionen, Crelles Journal 35, 153-247.

Mitiuszew, W.: 1996, Zastosowanie równań funkcyjnych do wyznaczania efektywnej przewodności cieplnej materiałów kompozytowych, Wydawnictwo WSP, Słupsk.

Mityushev, V.: 1995, Rayleigh's integral and the square array of cylinders, Arch. Mech. 47, 27-37.

Mityushev, V.: 1997, Transport properties of finite and infinite composite materials and Rayleigh's sum, Arch. Mech. 49, 345-358.

Rayleigh: 1892, On the influence of obstacles arranged in rectangular order upon the properties of a medium, Phil. Mag. 34, 481-502.

Weil, A.: 1976, Elliptic functions according to Eisenstein and Kronecker, Springer-Verlag, Berlin.


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