Editing Periodic function (section)

==Definition==
A function {{math|<var>f</var>}} is said to be '''periodic''' if, for some '''nonzero''' constant {{math|<var>P</var>}}, it is the case that

:<math>f(x+P) = f(x) </math>

for all values of {{math|<var>x</var>}} in the domain. A nonzero constant {{mvar|P}} for which this is the case is called a '''period''' of the function. If there exists a least positive<ref>For some functions, like a [[constant function]] or the [[Dirichlet function]] (the [[indicator function]] of the [[rational number]]s), a least positive period may not exist (the [[infimum]] of all positive periods {{math|<var>P</var>}} being zero).</ref> constant {{math|<var>P</var>}} with this property, it is called the '''fundamental period''' (also '''primitive period''', '''basic period''', or '''prime period'''.)  Often, "the" period of a function is used to mean its fundamental period. A function with period {{math|<var>P</var>}} will repeat on intervals of length {{math|<var>P</var>}}, and these intervals are sometimes also referred to as '''periods''' of the function.

Geometrically, a periodic function can be defined as a function whose graph exhibits [[translational symmetry]], i.e. a function {{math|<var>f</var>}} is periodic with period {{math|<var>P</var>}} if the graph of {{math|<var>f</var>}} is [[invariant (mathematics)|invariant]] under [[translation (geometry)|translation]] in the {{math|<var>x</var>}}-direction by a distance of {{math|<var>P</var>}}.  This definition of periodicity can be extended to other geometric shapes and patterns, as well as be generalized to higher dimensions, such as periodic [[tessellation]]s of the plane. A [[sequence (mathematics)|sequence]] can also be viewed as a function defined on the [[natural number]]s, and for a [[periodic sequence]] these notions are defined accordingly.