Resurgent rigidity of mock theta functions: uniqueness and natural boundary crossing

Abstract

We develop a resurgent transseries approach to mock theta functions and show that the associated Mordell--Appell integrals underlie their modular and resurgent structure. From these integrals, we recover the modular transformation laws, obtain uniqueness characterizations of the mock theta vectors, provide a canonical crossing of their natural boundary, and produce new mock theta functions. After representing the Mordell--Appell integrals as Laplace transforms of elementary resurgent functions, we rotate the Laplace contour to a Stokes line. Their Stokes phenomena and their modular transformation laws reproduce those of the associated unary series and, combined with our uniqueness results, those of the mock theta functions. Conversely, we regard these modular transformation laws as functional equations governing the mock theta vectors. For Ramanujan's order-$3$ pair $(f,ω)$ and the distinguished mock theta vectors of orders $5$, $7$, and $11$, satisfying the full $SL(2,\mathbb Z)$ transformation laws, we prove that these functional equations uniquely determine a holomorphic solution in the unit disk under canonical minimal-growth normalization. This unique solution is precisely the classical mock theta vector. We show that the normalization is essential: for orders $5$, $7$, and $11$ we determine explicit nontrivial solutions of the homogeneous problem, yielding new mock theta functions with faster-growing coefficients. Finally, we give two intrinsic and natural constructions for crossing the natural boundary $|q|=1$. The first continues the modular functional equations and takes their unique holomorphic solutions on the other side. The second acts on the resurgent asymptotic series at the cusp: we replace $τ$ by $-τ$ and apply Écalle--Borel summation. We prove that the constructions agree, giving a canonical and explicit boundary crossing.