  The particle production in a de Sitter space provides an interesting model to
understand the curvature effect on Schwinger pair production by a constant
electric field or Schwinger mechanism on the de Sitter radiation. For that
purpose, we employ the recently introduced complex analysis method, in which
the quantum evolution in the complex time explains the pair production via the
geometric transition amplitude and gives the pair-production rate as the
contour integral. We compare the result by the contour integral with that of
the phase-integral method.
