Lattice sums for double periodic polyanalytic functions
Authors:
- Piotr Drygaś,
- Vladimir Mityushev
Abstract
In 1892, Lord Rayleigh estimated the effective conductivity of rectangular arrays of disks and proved, employing the Eisenstein summation, that the lattice sum S2 is equal to π for the square array. Further, it became clear that such equality can be treated as a necessary condition of the macroscopic isotropy of composites governed by the Laplace equation. This yielded the description of two-dimensional conducting composites by the classic elliptic functions, including the conditionally convergent Eisenstein series. In 1935, Natanzon used a polyharmonic function to solve the plane elasticity problem. This paper is devoted to the extension of the classic lattice sums to the lattice sums for double periodic (pseudoperiodic) polyanalytic functions. The exact relations and computationally effective formulae between the polyanalytic and classic lattice sums are established. Polynomial representations of the lattice sums are obtained. They are a source of new exact formulae for the lattice sums.
- Record ID
- CUT2e8a1b19f24d4ce3bc7d1b7785ab6ea8
- Publication categories
- ;
- Author
- Journal series
- Analysis and Mathematical Physics, ISSN 1664-2368, e-ISSN 1664-235X
- Issue year
- 2023
- Vol
- 13
- No
- 5
- Pages
- [1-27]
- Article number
- 75
- Other elements of collation
- tab.; Bibliografia (na s.) - 27; Bibliografia (liczba pozycji) - 21; Oznaczenie streszczenia - Abstr.; Numeracja w czasopiśmie - Vol. 13, Iss. 5
- Keywords in English
- lattice sum, double periodic polyanalytic functions, Eisenstein summation, elliptic integrals
- ASJC Classification
- ; ;
- DOI
- DOI:10.1007/s13324-023-00838-2 Opening in a new tab
- URL
- https://link.springer.com/article/10.1007/s13324-023-00838-2 Opening in a new tab
- Related project
- Kompozyty odlewane in-situ wzmacniane nanocząstkami faz ceramicznych. . Project leader at PK: , ,
- Language
- eng (en) English
- License
- Score (nominal)
- 100
- Score source
- journalList
- Score
- Publication indicators
- Additional fields
- Indeksowana w: Web of Science, Scopus
- Uniform Resource Identifier
- https://cris.pk.edu.pl/info/article/CUT2e8a1b19f24d4ce3bc7d1b7785ab6ea8/
- URN
urn:pkr-prod:CUT2e8a1b19f24d4ce3bc7d1b7785ab6ea8
* presented citation count is obtained through Internet information analysis, and it is close to the number calculated by the Publish or PerishOpening in a new tab system.