The Integral Calculus of Finite Differences

The Integral Calculus of Finite Differences

This is a translation of "Opuscoli Matemateci sopra il Teorema della Composizione delle Forze; e sopra il Calcolo Integrale delle Differenze Finite" by Gregorio Fontana, available from www.e-rara.ch. This is a broad but challenging introduction to Finite Differences, requiring of the reader a basic knowledge of differential and integral calculus, including the solution of differential equations, plus some acquaintance with series and finite difference equations. The contents comprise the following topics: Summation of some simple series. Integration of differential fractions. Summation of reciprocal series. The general term of a series with constant finite difference of some order. Summation of the foregoing series. Integration of a particular form of linear finite difference equation with variable coefficients. Solution to a particular recurrency relation. A Chess problem: given a starting and a finishing square, in how many ways may a castle pass from start to finish in just x moves? The solution requires the simultaneous solving of three first-order linear finite difference equations with constant coefficients. Basic notions of finite differences of more than one variable. The fitting of regular observed data to a describing equation. + Comment on the composition of two Newtonian forces into a single force.