Period of a function

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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.

How to Cite This Entry:
Period of a function. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by A.A. Konyushkov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article