Abstract: For Seifert homology spheres, we derive a holomorphic function of K whose value at integer K is the sl 2 Witten–Reshetikhin–Turaev invariant, Z K , at q= exp 2π i / K . This function is expressed as a sum of terms, which can be naturally corresponded to the contributions of flat connections in the stationary phase expansion of the Witten–Chern–Simons path integral. The trivial connection contribution is found to have an asymptotic expansion in powers of K −1 which, for K an odd prime power, converges K -adically to the exact total value of the invariant Z K at that root of unity. Evaluations at rational $K$ are also discussed. Using similar techniques, an expression for the coloured Jones polynomial of a torus knot is obtained, providing a trivialconnection contribution which is an analytic function of the colour. This demonstrates that the stationary phase expansion of the Chern–Simons–Witten theory is exact for Seifert manifolds and for torus knots in S 3 . The possibility of generalising such results is also discussed.