  Although the gravitational waves observed by advanced LIGO and Virgo are
consistent with compact binaries in a quasi-circular inspiral prior to
coalescence, eccentric inspirals are also expected to occur in Nature. Due to
their complexity, we currently lack ready-to-use, analytic waveforms in the
Fourier domain valid for sufficiently high eccentricity, and such models are
crucial to coherently extract weak signals from the noise. We here take the
first steps to derive and properly validate an analytic waveform model in the
Fourier domain that is valid for inspirals of arbitrary orbital eccentricity.
As a proof-of-concept, we build this model to leading post-Newtonian order by
combining the stationary phase approximation, a truncated sum of harmonics, and
an analytic representation of hypergeometric functions. Through comparisons
with numerical post-Newtonian waveforms, we determine how many harmonics are
required for a faithful (matches above 99%) representation of the signal up to
orbital eccentricities as large as 0.9. As a first byproduct of this analysis,
we present a novel technique to maximize the match of eccentric signals over
time of coalescence and phase at coalescence. As a second byproduct, we
determine which of the different approximations we employ leads to the largest
loss in match, which could be used to systematically improve the model because
of our analytic control. The future extension of this model to higher
post-Newtonian order will allow for an accurate and fast phenomenological
hybrid that can account for arbitrary eccentricity inspirals and mergers.
