Context. Adaptive optics (AO) is now a tool commonly deployed in astronomy. The real time correction of the atmospheric turbulence that AO enables allows telescopes to perform close to the diffraction limit at the core of their point spread function (PSF). Among other factors, AO-corrected PSFs depend on the ability of the wavefront corrector (WFC), generally a deformable mirror, to fit the incident wavefront corrugations. Aims. In this work, we focus on this error introduced by the WFC, the so-called fitting error. To date, analytical models only depend on the WFC cut-off frequency, and Monte Carlo simulations are the only solution for studying the impact of the WFC influence function shape on the AO-corrected PSF. We aim to develop an analytical model accounting for the influence function shape. Methods. We first obtain a general analytical model of the fitting error structure function. With additional hypotheses, we then derive an analytical model of the AO-corrected power spectral density. These two analytical solutions are compared with Monte Carlo simulations on different ideal profiles (piston, pyramid, Gaussian) as well as with real hardware (DM192 from ALPAO). Results. Our analytical predictions show a very good agreement with the Monte Carlo simulations. We show that in the image plane, the depth of the correction as well as the transition profile between the AO-corrected area and the remaining turbulent halo depend on the influence functions of the WFC. We also show that the generally assumed hypothesis of stationarity of the AO correction is actually not met. Conclusions. As the fitting error is the intrinsic optimal limit of an AO system, our analytical model allows for the assessment of the theoretical limits of extreme AO systems limited by the WFC in high-contrast imaging through a context where other errors become comparable.