The physical theory of the Foucault test has been investigated to represent the complex amplitude and irradiance of the shadowgram in terms of the wavefront error; however, most of the studies have limited the treatment for the particular case of nearly diffraction-limited optical devices (i.e., aberrations smaller than the wavelength). In this paper we discard this restriction, and in order to show a more precise interpretation from the physical theory we derive expressions for the complex amplitude and the irradiance over an optical device with larger aberrations. To the best of our knowledge, it is the first time an expression is obtained in closed form. As will be seen, the result of this derivation is obtained using some properties of the Hilbert transform that permit representing the irradiance in a simple form in terms of the partial derivatives of the wavefront error. Additionally, we briefly describe from this point of view a methodology for the quantitative analysis of the test.