Period of a function
A number such that for any (or ) the numbers and also belong to and such that the following equality holds:
The numbers , where is a natural number, are also periods of . For a function on an axis or on a plane, any number is a period; for the Dirichlet function
any rational number is a period. If a function has period , then the function , where and are constants and , has period . If a real-valued function of a real argument is periodic on (and is not identically equal to a constant), then it has a least period and any other real period is a multiple of . There exist non-constant functions of a complex argument having two non-multiple periods with imaginary quotient, such as for example an elliptic function.
Similarly one defines the period of a function defined on an Abelian group.
Cf. also Periodic function.
Period of a function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Period_of_a_function&oldid=13109