  Green's functions of non-Hermitian systems play a fundamental role in various
dynamical processes. Because non-Hermitian systems are sensitive to boundary
conditions due to the non-Hermitian skin effect, open-boundary Green's
functions are closely related to the non-Bloch band theory. While the exact
formula of open-boundary Green's functions in single-band non-Hermitian systems
proves to be an integral along the generalized Brillouin zone (GBZ), the proper
generalization in generic multiband systems remains unclear. In this work, we
derive a formula of open-boundary Green's functions in multiband non-Hermitian
systems by viewing the multiband GBZ on the Riemann surface. This formula can
be applied to describe directional amplification in multiband systems, which
can be verified at various experimental platforms.
