Open AccessThe Dirichlet problem for rectangle and new identities for elliptic integrals and functions
Author(s) -
Helen S. Alekseeva,
A. E. Rassadin
Publication year - 2020
Publication title -
žurnal srednevolžskogo matematičeskogo obŝestva
Language(s) - English
Resource type - Journals
eISSN - 2587-7496
pISSN - 2079-6900
DOI - 10.15507/2079-6900.22.202002.145-154
Subject(s) - rectangle , mathematics , elliptic function , conformal map , mathematical analysis , dirichlet problem , elliptic integral , domain (mathematical analysis) , jacobian matrix and determinant , jacobi elliptic functions , function (biology) , dirichlet distribution , elliptic rational functions , elliptic curve , pure mathematics , quarter period , geometry , boundary value problem , evolutionary biology , biology
In the paper, results of comparison of two different methods of exact solution of the Dirichlet problem for rectangle are presented, namely, method of conformal mapping and method of variables’ separation. By means of this procedure normal derivative of Green’s function for rectangular domain was expressed via Jacobian elliptic functions. Under approaching to rectangle’s boundaries these formulas give new representations of the Dirac delta function. Moreover in the framework of suggested ideology a number of identities for the complete elliptic integral of the first kind were obtained. These formulas may be applied to summation of both numerical and functional series; also they may be useful for analytic number theory.