  We propose the complex-plane generalization of a powerful algebraic sequence
acceleration algorithm, the Method of Weighted Averages (MWA), to guarantee
exponential-cum-algebraic convergence of Fourier and Fourier-Hankel (F-H)
integral transforms. This "complex-plane" MWA, effected via a linear-path
detour in the complex plane, results in rapid, absolute convergence of
field/potential solutions in multi-layered environments regardless of the
source-observer geometry and anisotropy/loss of the media present. In this
work, we first introduce a new integration path used to evaluate the field
contribution arising from the radiation spectra. Subsequently, we (1) exhibit
the foundational relations behind the complex-plane extension to a general
Levin-type sequence convergence accelerator, (2) specialize this analysis to
one member of the Levin transform family (the MWA), (3) address and circumvent
restrictions, arising for two-dimensional integrals associated with wave
dynamics problems, through minimal complex-plane detour restrictions and a
novel partition of the integration domain, (4) develop and compare two
formulations based on standard/real-axis MWA variants, and (5) present
validation results and convergence characteristics for one of these two
formulations.
