A very accurate method to approximate discontinuous functions with a finite number of discontinuities

Abstract: A simple and very accurate method to approximate a function with a finite number of discontinuities is presented. This method relies on hyperbolic tangent functions of rational arguments as connecting functions at the discontinuities, each argument being the reciprocal of Newton binomials that depend on the abscissae that define the domain of the discontinuous function and upon the abscissae of discontinuities. Our approximants take the form of linear combinations of such hyperbolic tangent functions with coefficients that are obtained by solving a linear system of inhomogeneous equations whose righthand sides are the partition functions that define the given discontinuous function. These approximants are analytic, and being free from the Gibbs phenomenon certainly converge at the discontinuity points much better than other known approximants to discontinuous functions, typical relative errors being of the order of 10-14 even when as close as 10-12 to the discontinuity points. Moreover, they can be readily scaled to larger intervals. Our method is here illustrated with a representative set of discontinuous mathematical physics functions, and by studying the dynamics of an oscillator subjected to a discontinuous force, but it can be applied to important cases of discontinuous functions in physics, mathematics, engineering and physical chemistry.