  This paper is the first of a series of works on the conformal bootstrap in
Liouville conformal field theory (CFT) with boundaries. We focus here on the
case of the annulus with two boundary insertions, each of which lies on the
different connected components of the boundary. In the course of proving the
bootstrap formula, we established several properties on the corresponding
annulus conformal blocks:
  1) we show that they converge everywhere on the spectral line and they are
continuous with respect to the spectrum and the primary weights.
  2) we relate them to their torus counterparts by rigorously implementing
Cardy's doubling trick for boundary CFT,
  3) we solve a conjecture of Martinec on the annulus partition function,
  4) we also extend the bootstrap formula to the one-point case.
  As an application of our bootstrap result, we give an exact formula for the
bosonic LQG partition function of the annulus when $\gamma\in (0,2)$. Our paper
serves as a key ingredient in the recent derivation of the random moduli for
the Brownian annulus by Ang, Remy, and Sun (2022). We also solve several other
conjectures relate to torus conformal blocks which arise from physics
literature.
