  Solving the holography equation has long been a numerical task. While
effective, the numeric approach has its own set of limitations. Relying solely
on numerical approaches often obscures the intricate interplay and influence of
the individual terms within the equation. This not only hampers a deeper
understanding of the underlying physics but also makes it challenging to
predict or control specific outcomes. In this study, we address these
challenges by leveraging our recently published updated Fraunhofer diffraction
expression. This approach allows us to derive an analytic solution for
complex-valued phase disks in on-axis holography. This solution facilitates the
direct computation of each term's influence within the holographic equation,
paving the way for a more profound comprehension and application of the
holographic process. When compared to experimental results and the numeric
Fresnel diffraction solution, our analytic approach shows impressive accuracy,
considering the inherent approximations. Notably, it remains precise for
Fresnel numbers that extend well beyond the traditionally accepted boundaries
of the Fraunhofer regime.
