The finite difference is the discrete analog of the Derivative. The finite Forward Difference of a function
is defined as

(1) |

(2) |

(3) |

However, when is viewed as a discretization of the continuous function , then the finite difference is
sometimes written

(4) |

(5) |

An th Power has a constant th finite difference. For example, take and make a Difference Table,

(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

(7) |

(8) |

(9) |

Beyer (1987) gives formulas for the derivatives

(10) |

(11) |

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-Nicholson, Du Fort-Frankel, and Laasonen methods.

**References**

Abramowitz, M. and Stegun, C. A. (Eds.). ``Differences.'' §25.1 in
*Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing.*
New York: Dover, pp. 877-878, 1972.

Beyer, W. H. *CRC Standard Mathematical Tables, 28th ed.* Boca Raton, FL: CRC Press, pp. 429-515, 1987.

Boole, G. and Moulton, J. F. *A Treatise on the Calculus of Finite Differences, 2nd rev. ed.* New York: Dover, 1960.

Conway, J. H. and Guy, R. K. ``Newton's Useful Little Formula.'' In *The Book of Numbers.* New York:
Springer-Verlag, pp. 81-83, 1996.

Iyanaga, S. and Kawada, Y. (Eds.). ``Interpolation.'' Appendix A, Table 21 in
*Encyclopedic Dictionary of Mathematics.* Cambridge, MA: MIT Press, pp. 1482-1483, 1980.

Jordan, K. *Calculus of Finite Differences, 2nd ed.* New York: Chelsea, 1950.

Levy, H. and Lessman, F. *Finite Difference Equations.*
New York: Dover, 1992.

Milne-Thomson, L. M. *The Calculus of Finite Differences.* London: Macmillan, 1951.

Richardson, C. H. *An Introduction to the Calculus of Finite Differences.* New York: Van Nostrand, 1954.

Spiegel, M. *Calculus of Finite Differences and Differential Equations.* New York: McGraw-Hill, 1971.

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1999-05-26