O zastosowaniu sumowania według Eisensteina do pewnej sumy podwójnej

Agata Matczyk; Vladimir Mityushev; Zbigniew Powązka

Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia (2009)

  • Volume: 2, page 153-157
  • ISSN: 2080-9751

Abstract

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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.

How to cite

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Agata Matczyk, Vladimir Mityushev, and Zbigniew Powązka. "O zastosowaniu sumowania według Eisensteina do pewnej sumy podwójnej." Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia 2 (2009): 153-157. <http://eudml.org/doc/296269>.

@article{AgataMatczyk2009,
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.},
author = {Agata Matczyk, Vladimir Mityushev, Zbigniew Powązka},
journal = {Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia},
language = {pol},
pages = {153-157},
title = {O zastosowaniu sumowania według Eisensteina do pewnej sumy podwójnej},
url = {http://eudml.org/doc/296269},
volume = {2},
year = {2009},
}

TY - JOUR
AU - Agata Matczyk
AU - Vladimir Mityushev
AU - Zbigniew Powązka
TI - O zastosowaniu sumowania według Eisensteina do pewnej sumy podwójnej
JO - Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia
PY - 2009
VL - 2
SP - 153
EP - 157
AB - 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.
LA - pol
UR - http://eudml.org/doc/296269
ER -

References

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  1. Eisenstein, F.: 1847, Beiträge zur Theorie der elliptischen Funktionen, Crelles Journal 35, 153-247. 
  2.  
  3. Mitiuszew, W.: 1996, Zastosowanie równań funkcyjnych do wyznaczania efektywnej przewodności cieplnej materiałów kompozytowych, Wydawnictwo WSP, Słupsk. 
  4.  
  5. Mityushev, V.: 1995, Rayleigh's integral and the square array of cylinders, Arch. Mech. 47, 27-37. 
  6.  
  7. Mityushev, V.: 1997, Transport properties of finite and infinite composite materials and Rayleigh's sum, Arch. Mech. 49, 345-358. 
  8.  
  9. Rayleigh: 1892, On the influence of obstacles arranged in rectangular order upon the properties of a medium, Phil. Mag. 34, 481-502. 
  10.  
  11. Weil, A.: 1976, Elliptic functions according to Eisenstein and Kronecker, Springer-Verlag, Berlin. 

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