  We study holomorphic blocks in the three dimensional ${\mathcal N}=2$ gauge
theory that describes the $\mathbb{CP}^1$ model. We apply exact WKB methods to
analyze the line operator identities associated to the holomorphic blocks and
derive the analytic continuation formulae of the blocks as the twisted mass and
FI parameter are varied. The main technical result we utilize is the connection
formula for the ${}_1\phi_1$ $q$-hypergeometric function. We show in detail how
the $q$-Borel resummation methods reproduce the results obtained previously by
using block-integral methods.

High Energy Physics - Theory