  The quantum dilogarithm function of Faddeev is a special function that plays
a key role as the building block of quantum invariants of knots and
3-manifolds, of quantum Teichm\"uller theory and of complex Chern-Simons
theory. Motivated by conjectures on resurgence and recent interest in
wall-crossing phenomena, we prove that the Borel summation of a formal power
series solution of a linear difference equation produces Faddeev's quantum
dilogarithm. Along the way, we give an explicit formula for the meromorphic
function in Borel plane, locate its poles and residues, and describe the Stokes
phenomenon of its Laplace transforms along the Stokes rays.
