Geodesic
The Dirichlet problem for rectangle and new identities for elliptic integrals and functions
Žurnal Srednevolžskogo matematičeskogo obŝestva, Tome 22 (2020) no. 2, pp. 145-154

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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.
Keywords: correspondence of boundaries, kernel of integral operator, modulus of complete elliptic integral, the Weierstrass sigma-function, linear differential equation of the second order with variable coefficients.
Mots-clés : the Poisson formula 
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H. S. Alekseeva; A. È. Rassadin. The Dirichlet problem for rectangle and new identities for elliptic integrals and functions. Žurnal Srednevolžskogo matematičeskogo obŝestva, Tome 22 (2020) no. 2, pp. 145-154. http://geodesic.mathdoc.fr/item/SVMO_2020_22_2_a0/