  Green's functions are fundamental quantities that determine the linear
responses of physical systems. The recent developments of non-Hermitian
systems, therefore, call for Green's function formulas of non-Hermitian bands.
This task is complicated by the high sensitivity of energy spectrums to
boundary conditions, which invalidates the straightforward generalization of
Hermitian formulas. Here, based on the non-Bloch band theory, we obtain simple
Green's function formulas of general one-dimensional non-Hermitian bands.
Furthermore, in the large-size limit, these formulas dramatically reduce to
finding the roots of a simple algebraic equation. As an application, our
formulation provides the desirable formulas for the defining quantities, the
gain and directionality, of directional amplification. Thus, our formulas
provide an efficient guide for designing directional amplifiers.
