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Periodic Function: Simple Definition, Examples

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What is a Periodic Function?

periodic function

A periodic function with period “P”.



A periodic function repeats its values at set intervals, called periods. A “function” is just a type of equation where every input (e.g. the x-value) results in a unique output (e.g. the y-value).

More formally, we say that this type of function has a positive constant “k” where any input(x):


f(x + k) – f(x).


A periodic function is sometimes called fully periodic, purely periodic, or strictly periodic (Depner & Rasmussen, 2017). This broad class of functions, which can all be represented by a Fourier series, also includes (mathematically speaking) almost-periodic functions.

What is the “Period” in a Periodic Function?

The period, P, is the length of one complete cycle. It is defined as the smallest value for which the above notation holds true. The graph repeats itself after P units. You can think of a period as a repeating interval on a graph: it’s the area you can cut and paste over and over again to make a full graph of the function. To put that another way, a graph with period P stays the same if you shift it along the x-axis to the left or right.

The period (P) must be greater than zero; In other words, you can’t have a negative period.

Examples of Periodic Functions

Trigonometric functions are all periodic. The sine function and cosine function are two well known examples.

periodic function graphs

Graphs of sin(x) in red and cos(x) in blue.



The constant function is not a periodic function because—although it repeats—the periods are all equal to zero. It is an example of an aperiodic function (“aperiodic” means any function that isn’t periodic).

Real Life Examples

  • Motion of a Ferris wheel.
  • Musical sounds—it’s what makes them different them from random sounds (Hall, n.d.).
  • The number of hours of sunlight over the course of one year.
  • Flickering of a fluorescent light.

References

Desmos Graphing Calculator.
Depner, J. & Rasmussen, T. (2017). Hydrodynamics of Time-Periodic Groundwater Flow: Diffusion Waves in Porous Media, Geophysical Monograph 224. American Geophysical Union.
Hall, R. Sounding Number. Retrieved November 29, 2019 from: http://people.sju.edu/~rhall/SoundingNumber/periodicfunctions.pdf
Chapter 19: Trigonometry: Introducing Periodic Functions. Retrieved November 29, 2019 from: http://www.math.harvard.edu/archive/xb_spring_06/files/chap19-20.pdf
Periodic Functions. Article posted on the Oregon State website. Retrieved November 29, 2019 from: https://oregonstate.edu/instruct/mth251/cq/Glossary/gloss.periodic.html

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Stephanie Glen. "Periodic Function: Simple Definition, Examples" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/periodic-function/
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