Exact Analytical Expressions for the Floor Function and the Modulo Function

Periodic discontinuous functions, such as the floor function, the modulo function, the square wave function, the delta function, and the sawtooth wave function, are ubiquitous in various fields of mathematics and engineering. Traditional representations of these functions often rely on Fourier series expansions, which can be inefficient and exhibit convergence issues like the Gibbs phenomenon. In this paper, an exact analytical expressions for these functions is introduced using a novel approach based on Cauchy's argument principle. This method provides closed-form representations that overcome the limitations of Fourier series, offering precise calculations and well-defined derivatives. Also, sample examples and illustrations using Julia code are provided to demonstrate the effectiveness of the proposed expressions.