The objects of the study are the methods for calculating the shape of the reproduction signals from the inhomogeneities of a roundshaped medium by a Gaussian beam; the subjects of the study are methods for calculating the values of the convolution function of a two-dimensional Gaussian function with a generalized circle function. The goal is to improve the methods for calculating the waveform from the inhomogeneity of the medium by a Gaussian beam and similar methods of calculating the probability. In the derivation of the analytical model of the waveform generated by the interaction of the inhomogeneity of a circular medium with a Gaussian beam, the methods of the theory of special functions and probability theory are applied. An analytical model of the waveform formed during the interaction of the inhomogeneity of a circular medium with a Gaussian beam is developed, which takes into account the beam dimensions and the inhomogeneity radius, as well as the interrelation of the beam centers and the inhomogeneity. An analytic function of the waveform form from the inhomogeneity of the recording medium, which has the form of a circle, is obtained as a two-dimensional Weierstrass transform from a generalized function in the form of a series on the basis of modified Bessel functions. Recommendations for calculating the function on a computer are given. The graphics of the playback signal of the information storage device on the optical disk for some values of the variables are constructed. A proposal is made to use the obtained analytic function in probability theory.