Periodic Function

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A function f(x) is said to be periodic (or, when emphasizing the presence of a single period instead of multiple periods, singly periodic) with period p if

for n=1, 2, .... For example, the sine function sinx, illustrated above, is periodic with least period (often simply called "the" period) 2pi (as well as with period -2pi, 4pi, 6pi, etc.).

The constant function f(x)=0 is periodic with any period R for all nonzero real numbers R, so there is no concept analogous to the least period for constant functions. The following table summarizes the names given to periodic functions based on the number of independent periods they possess.

See also

Almost Periodic Function, Antiperiodic Function, Doubly Periodic Function, Least Period, Period, Periodic Point, Periodic Sequence, Singly Periodic Function, Triply Periodic Function

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References

Knopp, K. "Periodic Functions." Ch. 3 in Theory of Functions Parts I and II, Two Volumes Bound as One, Part II. New York: Dover, pp. 58-92, 1996.Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 425-427, 1953.Spanier, J. and Oldham, K. B. "Periodic Functions." Ch. 36 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 343-349, 1987.

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Periodic Function

Cite this as:

Weisstein, Eric W. "Periodic Function." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/PeriodicFunction.html

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