Images of Maass-Poincar\'e series in the lower half-plane

TL;DR

This paper extends harmonic Maass forms to both half-planes via Poincaré series, connecting Rademacher's expansion principle with recent work on mock and partial theta functions.

Contribution

It introduces a method to define harmonic Maass forms on both half-planes using Poincaré series, expanding their analytical scope.

Findings

Extended harmonic Maass forms to both half-planes.

Linked Rademacher's expansion principle with mock theta functions.

Provided a new perspective on the structure of Maass-Poincaré series.

Abstract

In this note we extend integral weight harmonic Maass forms to functions defined on the upper and lower half-planes using the method of Poincar\'e series. This relates to Rademacher's "expansion of zero" principle, which was recently employed by Rhoades to link mock theta functions and partial theta functions.