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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
Details and Options Details and Options
Examples  
Basic Examples  
Scope  
Basic Uses  
Periodic Functions over the Integers  
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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:

Wolfram Language code: FunctionPeriod[Sin[x], x]

Plot of two complete periods:

Wolfram Language code: Plot[Sin[x], {x, -2Pi, 2Pi}, Ticks -> {{-2Pi, -Pi, 0, Pi, 2Pi}, {-1, 1}}]

Find a period of a sequence:

Wolfram Language code: FunctionPeriod[Mod[n, 8], n, Integers]

Plot the sequence:

Wolfram Language code: DiscretePlot[Mod[n, 8], {n, 0, 15}]

Find periods for multidimensional functions:

Wolfram Language code: FunctionPeriod[Cos[x] + Sin[y], {x, y}]

Plot the contours:

Wolfram Language code: ContourPlot[Cos[x] + Sin[y], {x, -2Pi, 2Pi}, {y, -2Pi, 2Pi}]

Scope  (10)

Basic Uses  (5)

Find periods over integers:

Wolfram Language code: FunctionPeriod[(-1) ^ n, n, Integers]
Wolfram Language code: FunctionPeriod[Sin[n 2π / 17 + 3], n, Integers]
Wolfram Language code: FunctionPeriod[Mod[2n, 18], n, Integers]
Wolfram Language code: Table[DiscretePlot[f, {n, 35}, PlotLabel -> f, PlotTheme -> "Minimal"], {f, {(-1) ^ n, Sin[n 2π / 17 + 3], Mod[2n, 18]}}]

Find periods over reals:

Wolfram Language code: FunctionPeriod[Sin[x], x, Reals]
Wolfram Language code: FunctionPeriod[Tan[x], x, Reals]
Wolfram Language code: FunctionPeriod[SquareWave[x], x, Reals]
Wolfram Language code: Table[Plot[f, {x, 0, 4π}, PlotLabel -> f, PlotTheme -> "Minimal", Exclusions -> None], {f, {Sin[x], Tan[x], SquareWave[x]}}]

Find periods over complexes:

Wolfram Language code: FunctionPeriod[Exp[z], z, Complexes]
Wolfram Language code: FunctionPeriod[Cosh[z], z, Complexes]
Wolfram Language code: FunctionPeriod[JacobiCD[z / 2, 2], z, Complexes]
Wolfram Language code: Table[Plot3D[Re[f], {x, -4π, 4π}, {y, -4π, 4π}, PlotLabel -> Re[f], PlotTheme -> "Minimal"], {f, {Exp[x + I y], Cosh[x + I y], JacobiCD[(x + I y) / 2, 2]}}]

Periods of functions with parameters:

Wolfram Language code: Assuming[a > 0, FunctionPeriod[Sin[a x], x]]
Wolfram Language code: Plot[{Sin[x], Sin[x / 2], Sin[x / 3]}, {x, 0, 6Pi}, PlotLegends -> "Expressions"]

Periods of multivariate mappings:

Wolfram Language code: FunctionPeriod[{Tan[x + y], Cos[x - 2y]}, {x, y}]

Periodic Functions over the Integers  (5)

Basic periodic sequences include Mod:

Wolfram Language code: p = FunctionPeriod[Mod[n / 2 + 7, 11], n, Integers]
Wolfram Language code: DiscretePlot[Mod[n / 2 + 7, 11], {n, 3p}]

Mod of a polynomial:

Wolfram Language code: p = FunctionPeriod[Mod[n^3 + 7n^2 + 3, 11], n, Integers]
Wolfram Language code: DiscretePlot[Mod[n^3 + 7n^2 + 3, 11], {n, 5p}]

The function :

Wolfram Language code: FunctionPeriod[(-1)^n, n, Integers]

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

Wolfram Language code: Solve[x^7 == 1, x]
Wolfram Language code: FunctionPeriod[Power[(-1)^2 / 7, n], n, Integers]
Wolfram Language code: Table[DiscretePlot[f, {n, 20}, PlotTheme -> "Minimal"], {f, {Re[Power[(-1)^2 / 7, n]], Im[Power[(-1)^2 / 7, n]]}}]

A common way to express these are :

Wolfram Language code: FunctionPeriod[Exp[n 2π I / 7], n, Integers]

Trigonometric functions with a rational multiple of their real period:

Wolfram Language code: trigs = {Sin[n 2π / 10], Cos[n 2π / 10], Csc[n 2π / 10 + 1], Sec[n 2π / 10 + 1], Tan[n π / 10 + 1], Cot[n π / 10 + 1]};
Wolfram Language code: FunctionPeriod[#, n]& /@ trigs
Wolfram Language code: Table[DiscretePlot[f, {n, 30}, PlotTheme -> "Minimal"], {f, trigs}]

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

Wolfram Language code: τ = FunctionPeriod[TriangleWave[x / 2], x, Reals]
Wolfram Language code: FunctionPeriod[TriangleWave[n / 2 1 / 5], n, Integers]
Wolfram Language code: DiscretePlot[TriangleWave[n / 2 1 / 5], {n, 30}]

It works similarly for a function periodic over the complexes:

Wolfram Language code: τ = FunctionPeriod[Cosh[x], x, Complexes]
Wolfram Language code: FunctionPeriod[Cosh[n 2π I / 10], n, Integers]
Wolfram Language code: DiscretePlot[Re[Cosh[n 2π I / 10]], {n, 30}]

Any finite sum of periodic sequences is periodic:

Wolfram Language code: FunctionPeriod[Mod[n, 3] + 3 Sin[n 2π / 5], n, Integers]
Wolfram Language code: DiscretePlot[Mod[n, 3] + 3 Sin[n 2π / 5], {n, 60}]

Any finite product of periodic sequences is periodic:

Wolfram Language code: FunctionPeriod[Mod[n, 3]3 Sin[n 2π / 5]^2, n, Integers]
Wolfram Language code: DiscretePlot[Mod[n, 3]3 Sin[n 2π / 5]^2, {n, 60}]

Any function combination of periodic sequences is periodic:

Wolfram Language code: FunctionPeriod[Mod[n, 3]^2(Sin[n 2π / 5] + Cos[n 2π / 3]) + Mod[n ^ 2, 4], n, Integers]
Wolfram Language code: DiscretePlot[Mod[n, 3]^2(Sin[n 2π / 5] + Cos[n 2π / 3]) + Mod[n ^ 2, 4], {n, 300}, PlotRange -> All]
Wolfram Research (2014), FunctionPeriod, Wolfram Language function, https://reference.wolfram.com/language/ref/FunctionPeriod.html.

Text

Wolfram Research (2014), FunctionPeriod, Wolfram Language function, https://reference.wolfram.com/language/ref/FunctionPeriod.html.

CMS

Wolfram Language. 2014. "FunctionPeriod." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/FunctionPeriod.html.

APA

Wolfram Language. (2014). FunctionPeriod. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/FunctionPeriod.html

BibTeX

@misc{reference.wolfram_2026_functionperiod, author="Wolfram Research", title="{FunctionPeriod}", year="2014", howpublished="\url{https://reference.wolfram.com/language/ref/FunctionPeriod.html}", note=[Accessed: 13-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_functionperiod, organization={Wolfram Research}, title={FunctionPeriod}, year={2014}, url={https://reference.wolfram.com/language/ref/FunctionPeriod.html}, note=[Accessed: 13-June-2026]}