The finite difference is the discrete analog of the derivative. The finite forward difference of a function is defined as
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and the finite backward difference as
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The forward finite difference is implemented in the Wolfram Language as DifferenceDelta[f, i].
If the values are tabulated at spacings , then the notation
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is used. The th forward difference would then be written as , and similarly, the th backward difference as .
However, when is viewed as a discretization of the continuous function , then the finite difference is sometimes written
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where denotes convolution and is the odd impulse pair. The finite difference operator can therefore be written
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An th power has a constant th finite difference. For example, take and make a difference table,
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The column is the constant 6.
Finite difference formulas can be very useful for extrapolating a finite amount of data in an attempt to find the general term. Specifically, if a function is known at only a few discrete values , 1, 2, ... and it is desired to determine the analytical form of , the following procedure can be used if is assumed to be a polynomial function. Denote the th value in the sequence of interest by . Then define as the forward difference , as the second forward difference , etc., constructing a table as follows
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Continue computing , , etc., until a 0 value is obtained. Then the polynomial function giving the values is given by
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When the notation , , etc., is used, this beautiful equation is called Newton's forward difference formula. To see a particular example, consider a sequence with first few values of 1, 19, 143, 607, 1789, 4211, and 8539. The difference table is then given by
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Reading off the first number in each row gives , , , , . Plugging these in gives the equation
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which indeed fits the original data exactly.
Formulas for the derivatives are given by
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(Beyer 1987, pp. 449-451; Zwillinger 1995, p. 705).
Formulas for integrals of finite differences
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are given by Beyer (1987, pp. 455-456).
Finite differences lead to difference equations, finite analogs of differential equations. In fact, umbral calculus displays many elegant analogs of well-known identities for continuous functions. Common finite difference schemes for partial differential equations include the so-called Crank-Nicolson, Du Fort-Frankel, and Laasonen methods.