In Ischislenie konechnykh raznostey (1952; "Calculus of Finite Differences"), he summarized his approximation and interpolation studies.
The calculus of finite differences was formalized by George Boole but is much older.
The influential Treatise on Differential Equations appeared in 1859 and was followed the next year by its sequel, Treatise on the Calculus of Finite Differences.
Because the base was no longer close to 1, the table could not be obtained as simply as Napier's, and Briggs therefore devised techniques involving the calculus of finite differences to facilitate calculation of the entries.
The umbral calculus comes under the heading of combinatorics, the calculus of finite differences, the theory of special functions, and formal solutions to differential equations.
As the calculus has two forms, the continuous calculus with the power concept of limits, and the discrete calculus which also is called the calculus of finite differences, the calculus of variations has also both the discrete and the continuous forms.
Importantly, it shows how the study of fractional finite sums benefits from and contributes to many areas of mathematics, such as divergent series, numerical integration, approximation theory, asymptotic methods, special functions, series acceleration, Fourier analysis, the calculus of finite differences, and information theory.
The study of differential calculus is unified with the calculus of finite differences in time scale calculus.
There is, however, a generalization of the Taylor series that does converge to the value of the function itself for any bounded continuous function on (0,∞), using the calculus of finite differences.
