# FunctionPeriod

FunctionPeriod[f,x]

gives a period p of the function f over the reals such that .

FunctionPeriod[f,x,dom]

gives a period with x restricted to the domain dom.

FunctionPeriod[{f1,f2,},{x1,x2,},]

gives periods {p1,p2,} for {x1,x2,} such that .

# Details • A period p is taken to be zero if no period can be found.
• Possible domains dom are Reals, Integers, and Complexes.
• Periods over the Complexes are given in a list and can consist of one or two complex periods.

# Examples

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## Basic Examples(3)

Find a period of the sine function:

Plot of two complete periods:

Find a period of a sequence:

Plot the sequence:

Find periods for multidimensional functions:

Plot the contours:

## Scope(9)

### Basic Uses(4)

Find periods over integers:

Find periods over reals:

Find periods over complexes:

Periods of functions with parameters:

### Periodic Functions over the Integers(5)

Basic periodic sequences include Mod:

Mod of a polynomial:

The function :

And in general powers of roots of unity, i.e. roots of the polynomial :

A common way to express these are :

Trigonometric functions with a rational multiple of their real period:

A function where is periodic over the reals with period and rational:

It works similarly for a function periodic over the complexes:

Any finite sum of periodic sequences is periodic:

Any finite product of periodic sequences is periodic:

Any function combination of periodic sequences is periodic: