Context. The image degradation produced by atmospheric turbulence and optical aberrations is usually alleviated using post-facto image reconstruction techniques, even when observing with adaptive optics systems. Aims: These techniques rely on the development of the wavefront using Zernike functions and the non-linear optimization of a certain metric. The resulting optimization procedure is computationally heavy. Our aim is to alleviate this computational burden. Methods: We generalize the extended Zernike-Nijboer theory to carry out the analytical integration of the Fresnel integral and present a natural basis set for the development of the point spread function when the wavefront is described using Zernike functions. Results: We present a linear expansion of the point spread function in terms of analytic functions, which, in addition, takes defocusing into account in a natural way. This expansion is used to develop a very fast phase-diversity reconstruction technique, which is demonstrated in terms of some applications. Conclusions: We propose that the linear expansion of the point spread function can be applied to accelerate other reconstruction techniques in use that are based on blind deconvolution.