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Central Difference

The central difference for a function tabulated at equal intervals $f_i$ is defined by

\delta(f_{n+1/2})=\delta_{n+1/2}=\delta^1_{n+1/2}\equiv f_{n+1}-f_n.
\end{displaymath} (1)

Higher order differences may be computed for Even and Odd powers,
$\displaystyle \delta_{n+1/2}^{2k}$ $\textstyle =$ $\displaystyle \sum_{j=0}^{2k} (-1)^j{2k\choose j} f_{n+k-j}$ (2)
$\displaystyle \delta_{n+1/2}^{2k+1}$ $\textstyle =$ $\displaystyle \sum_{j=0}^{2k+1} (-1)^j{2k+1\choose j}f_{n+k+1-j}.$ (3)

See also Backward Difference, Divided Difference, Forward Difference


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.

© 1996-9 Eric W. Weisstein