We present results for the first observed outburst from the transient X-ray binary source MAXI J0637-430. This study is based on eight observations from the Nuclear Spectroscopic Telescope Array (NuSTAR) and six observations from the Neil Gehrels Swift Observatory X-Ray Telescope (Swift/XRT) collected from 2019 November 19 to 2020 April 26 as the 3-79 keV source flux declined from 8.2e-10 to 1.4e-12 erg/cm^2/s. We see the source transition from a soft state with a strong disk-blackbody component to a hard state dominated by a power-law or thermal Comptonization component. NuSTAR provides the first reported coverage of MAXI J0637-430 above 10 keV, and these broadband spectra show that a two-component model does not provide an adequate description of the soft state spectrum. As such, we test whether blackbody emission from the plunging region could explain the excess emission. As an alternative, we test a reflection model that includes a physical Comptonization continuum. Finally, we also invoke a spectral component based on reflection of blackbody returning radiation due to the bending of light by the strong gravity of the black hole. We discuss the physical implications of each scenario and demonstrate the value of constraining the source distance.
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