On the Fourier coefficients of meromorphic Jacobi forms

In this paper, we describe the automorphic properties of the Fourier coefficients of meromorphic Jacobi forms. Extending results of Dabholkar, Murthy, and Zagier, and Bringmann and Folsom, we prove that the canonical Fourier coefficients of a meromorphic Jacobi form $\phi(z;\tau)$ are the holomorphic parts of some (vector-valued) almost harmonic Maass forms. We also give a precise description of their completions, which turn out to be uniquely determined by the Laurent coefficients of $\phi$ at each pole, as well as some well known real analytic functions, that appear for instance in the completion of Appell-Lerch sums.