Finite difference

If h has a fixed (non-zero) value instead of approaching zero, then the right-hand side of the above equation would be written

However, the central (also called centered) difference yields a more accurate approximation. If f is three times differentiable,

Similarly we can apply other differencing formulas in a recursive manner.

The relationship of these higher-order differences with the respective derivatives is straightforward,

Higher-order differences can also be used to construct better approximations. As mentioned above, the first-order difference approximates the first-order derivative up to a term of order h. However, the combination

If necessary, the finite difference can be centered about any point by mixing forward, backward, and central differences.

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 constructs finite difference approximations for non-standard (and even non-integer) stencils given an arbitrary stencil and a desired derivative order.

Carlson's theorem provides necessary and sufficient conditions for a Newton series to be unique, if it exists. However, a Newton series does not, in general, exist.

In a compressed and slightly more general form and equidistant nodes the formula reads

This formula holds in the sense that both operators give the same result when applied to a polynomial.

The analogous formulas for the backward and central difference operators are

Fourier sums of continuum functions are readily mapped to umbral Fourier sums faithfully

The inverse operator of the forward difference operator, so then the umbral integral, is the indefinite sum or antidifference operator.

Finite differences can be considered in more than one variable. They are analogous to partial derivatives in several variables.

Alternatively, for applications in which the computation of f is the most costly step, and both first and second derivatives must be computed, a more efficient formula for the last case is