On Uniqueness of Mock Theta Functions

Abstract

We develop a resurgent approach to the problem of unique continuation of mock theta functions across their natural boundary. The starting point is the representation of the associated Mordell-Appell integrals as Laplace transforms of resurgent functions, which serve as the primary analytic objects. By rotating the Laplace contour by $\ddot{I}€$, i.e. onto the Stokes line, one obtains, in all known cases, the mock-modular relations between the Mordell-Appell integrals and the corresponding unary series in $\hat{q}=e^{-\ddot{I}€i\ddot{I}„}$ and ${\hat{q}}_1=e^{-\ddot{I}€i(-1/\ddot{I}„)}$. We then prove that these relations admit a unique solution on the $q$-side, expressed in terms of $q=e^{\ddot{I}€i\ddot{I}„}$ and $q_1=e^{\ddot{I}€i(-1/\ddot{I}„)}$, with coefficients determined by the corresponding Mordell-Appell integrals. This yields a canonical continuation across the natural boundary, given by a resurgent extension of the classical principle of permanence of relations, and singles out a distinguished family of mock theta functions in each group. We present a complete analysis for the order 3 and 5 cases (mf3 and mf5). The method extends naturally to higher orders; a general theory will appear in a separate paper.

17 pages, no figures