  An optical field is described by the amplitude and phase, and thus has a
complex representation described in the complex plane. However, because the
only thing we can measure is the amplitude of the complex field on the real
axis, it is difficult to identify how the complex field behaves throughout the
complex plane. In this study, we interpreted quantitative phase imaging methods
via the Hilbert transform in terms of analytic continuation, manifesting the
behavior in the whole complex plane. Using Rouche's theorem, we proved the
imaging conditions imposed by Kramers-Kronig holographic imaging. The deviation
from the Kramers-Kronig holography conditions was examined using computational
images and experimental data. We believe that this study provides a clue for
holographic imaging using the holomorphic characteristics of a complex optical
field.
