Evaluation and regularization of generalized Eisenstein series and application to 2D cylindrical harmonic sums

TL;DR

This paper develops a regularization method for two-dimensional generalized Eisenstein series across all Bravais lattices, providing explicit forms and applications to cylindrical harmonic sums in periodic media.

Contribution

It introduces a universal regularization identity for 2D Eisenstein series and derives explicit forms for high symmetry lattices, enabling practical evaluation of related sums.

Findings

Derived a regularization identity valid for all Bravais lattices.

Provided explicit closed-form expressions for generalized Eisenstein series.

Applied results to evaluate cylindrical harmonic sums in periodic media.

Abstract

In the study of periodic media, conditionally convergent series are frequently encountered and their regularization is crucial for applications. We derive an identity that regularizes two-dimensional generalized Eisenstein series for all Bravais lattices, yielding physically meaningful values. We also obtain explicit forms for the generalized series in terms of conventional Eisenstein series, enabling their closed-form evaluation for important high symmetry lattices. Results are then used to obtain representations for the related cylindrical harmonic sums, which are also given for all Bravais lattices. Finally, we treat displaced lattices of high symmetry, expressing them in terms of origin-centered lattices via geometric multi-set identities. These identities apply to all classes of two-dimensional sums, allowing sums to be evaluated over each constituent of a unit cell that possesses multiple inclusions.