A versatile algorithm has been developed for the integer and semi-integer differentiation and the semi-integration of a discrete digital signal. A discrete differentiation function is derived from the continuous derivative theorem of Fourier transforms. Differentiation, semidifferentiation and semi-integration of any discrete digital signal can be accomplished by fast Fourier transformation using this function. In the algorithm, the order of differentiation can be a positive integer and/or a semi-integer, or a negative integer and/or a semi-integer. When it is negative, the operation corresponds to the integral or semi-integral. In the present study, the derivatives, semiderivatives and semi-integrals of theoretical and experimental voltammograms calculated using the algorithm are demonstrated. The results show that semi-integer-order derivative and semi-integral curves of theoretical cyclic voltammograms are in agreement with semidifferentiation and semi-integral electroanalysis theory. Another advantage of the algorithm is that differentiation, semidifferentiation or semi-integration with smoothing of the discrete signal can be performed simultaneously by Fourier transformation.