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Periodic Function

Function y=f(x)y={f{{\left({x}\right)}}} is called periodic if exists such number T≠0{T}\ne{0} that for any x{x} from domain of the function f(x+T)=f(x){f{{\left({x}+{T}\right)}}}={f{{\left({x}\right)}}}.

From the definition it follows that periodic function has infinitely many periods. If TT is a period of a function then any number of the form kT{k}{T}, where kk is an integer,is also a period of the function.

Often (but not always) among set of positive periods of the function we can find the smallest one. This period is called main period (or simply period).

For example trigonometric function y=sin⁡(x){y}={\sin{{\left({x}\right)}}} has period 2π{2}\pi because sin⁡(x+2π)=sin⁡(x){\sin{{\left({x}+{2}\pi\right)}}}={\sin{{\left({x}\right)}}} for all x{x}.