that and thus the third term of the polynomial is . Subtracting out the third term: Without any pairwise differences, it is found that the 4th and final term of the polynomial is the constant . Thus, the lowest-degree polynomial intercepting all the points in the first table is found: Arbitrarily sized kernels Using linear algebra one can construct finite difference approximations which utilize an arbitrary number of points to the left and a (possibly different) number of points to the right of the evaluation point, for any order derivative. This involves solving a linear system such that the Taylor expansion of the sum of those points around the evaluation point best approximates the Taylor expansion of the desired derivative. Such formulas can be represented graphically on a hexagonal or diamond-shaped grid. This is useful for differentiating a function on a grid, where, as one approaches the edge of the grid, one must sample fewer and fewer points on one side. The details are outlined in these notes. The Finite Difference Coefficients Calculator constructs finite difference approximations for non-standard (and even non-integer) stencils given an arbitrary stencil and a desired derivative order. Properties For all positive and Leibniz rule: In differential equations An important application of finite differences is in numerical analysis, especially in numerical differential equations, which aim at the numerical solution of ordinary and partial differential equations. The idea is to replace the derivatives appearing in the differential equation by finite differences that approximate them. The resulting methods are called finite difference methods. Common applications of the finite difference method are in computational science and engineering disciplines, such as thermal engineering, fluid mechanics, etc. Newton's series The Newton series consists of the terms of the Newton forward difference equation, named after Isaac Newton; in essence, it is the Newton interpolation formula, first published in his Principia Mathematica in 1687, namely the discrete analog of the continuous Taylor expansion, which holds for any polynomial function and for many (but not all) analytic functions. (It does not hold when is exponential type . This is easily seen, as the sine function vanishes at integer multiples of ; the corresponding Newton series is identically zero, as all finite differences are zero in this case. Yet clearly, the sine function is not zero.) Here, the expression is the binomial coefficient, and is the "falling factorial" or "lower factorial", while the empty product