Finite difference

The calculus of finite differences is a branch of mathematics which constitutes the discrete counterpart to analysis ( differential and integral calculus ). While the analysis deals with functions that are defined on continuous spaces ( to be able to establish a limit concept ), in particular with functions on the real numbers, we are interested in the differences calculus of functions on the integers ℤ. The difference calculation can be used to calculate of rows.

Differences and sums

The known continuous differential calculus is based on the differential operator, which is defined as follows:

The calculus of finite differences, however, uses a so-called difference operator:

The reverse operation is not achieved as in the continuous differential calculus with the indefinite integral, but with an indefinite sum is related to the difference operator as follows:

Behaves on here as to the continuous differential calculus. stands for the value of any function which is constant for a whole number ().

The counterpart to definite integrals are certain sums. These correspond to ordinary sums without the value at the highest index:

Properties

Invariant function

A invariant under the differential operator function is the exponential function of base e in the calculus of finite differences, the exponential function to the base 2 is invariant, as can be determined easily:

Falling faculties

A simple calculation rule there for falling faculties that are defined for every integer as follows:

This expression behaves in the calculus of finite differences as follows:

Where the- th harmonic number. The harmonic number is thus the opposite of the natural logarithm. The agreement goes so far that also applies.

Falling faculties and powers can always be converted by means of Stirling numbers of the first and second kind in one another:

Moreover, the binomial theorem is also true for falling faculties.

Example of the calculation of the sum of the first square numbers:

Product rule and partial summation

The product rule of continuous differential calculus is valid in the following form:

This rule can be through the introduction of a shift operator defined, compact expressed as, :

The change of the terms leads to the formula of partial summation similar integration by parts:

Example for calculating the sum:

Here is and, so, and.

The formula for partial summation gives:.

This eventually leads to the solution: