what does it mean for a function to be periodic

Types of Functions >

Contents:

• Periodic Function
• Aperiodic Part
• Well-nigh Periodic

What is a Periodic Role?

A periodic function with catamenia "P".

A periodic function repeats its values at gear up intervals, chosen periods. A "function" is just a blazon of equation where every input (e.one thousand. the 10-value) results in a unique output (e.one thousand. the y-value).

More formally, nosotros say that this type of function has a positive abiding "one thousand" where any input(x):

f(x + thousand) – f(x).

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

What is the "Catamenia" in a Periodic Part?

The period, P, is the length of one complete cycle. Information technology is divers as the smallest value for which the above notation holds truthful. The graph repeats itself after P units. You tin can call back of a period as a repeating interval on a graph: it's the area yous tin cut and paste over and over again to brand a full graph of the role. To put that another way, a graph with period P stays the aforementioned if you shift it along the x-axis to the left or right.

The catamenia (P) must be greater than zero; In other words, you lot tin 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.

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

The constant function is not a periodic function considering—although information technology repeats—the periods are all equal to aught. It is an example of an aperiodic function ("aperiodic" ways any role that isn't periodic).

Existent Life Examples

• Move of a Ferris wheel.
• Musical sounds—information technology's what makes them different them from random sounds (Hall, n.d.).
• The number of hours of sunlight over the course of 1 year.
• Flickering of a fluorescent light.

Aperiodic Function (Non periodic Role) Definition

The form of aperiodic (non periodic) functions includes the subtypes of nigh periodic and quasiperiodic functions.

An aperiodic function (or non periodic part) is whatsoever office that isn't periodic (Depner & Rasmussen, 2017). As periodic functions repeat their values at set periods, you could also call back of a non periodic function equally ane that doesn't have repeating intervals.

Although an aperiodic function isn't not periodic in nature, at that place is a very close relationship: mathematically, you can think of them as periodic functions with a menstruum of infinity (Adams, 2020).

"The transition from a periodic function to an aperiodic function is accomplished past assuasive the fundamental flow T to increase without limit. In other words, if T becomes infinite, the office never repeats itself and, therefore, the function is aperiodic" ~ Caggiano (1996)

Aperiodic Function Subclasses

Two important subclasses of aperiodic functions are about periodic and quasiperiodic functions.

At commencement, it might seem that subclasses of "non periodic functions" isn't useful at all. Merely the reverse is true: many of these functions have very close relationships with periodic functions, mathematically speaking. What is considered "shut" differs from author to author, but in general they are continued by their periodic nature:

• Nearly-periodic function, although not periodic themselves, can be represented by a sum of two or more periodic functions.
• Quasiperiodic functions are a combination of periodic functions of dissimilar frequencies that never completely match up.

Most Periodic (Quasiperiodic) Function

Almost periodic functions are a subtype of aperiodic functions.

Almost-periodic functions, are an important grade of aperiodic functions. They tin can be represented by a sum of two or more periodic functions. In other words, they tin can be formed by summing ii or more harmonic parts. Two of the parts must exist frequencies that aren't rational multiples of one another (Depner & Rasmussen, 2017).

As an example, the post-obit almost periodic part has two distinct harmonic parts:
f(t) = half-dozen sin(4t) + xiv cos(six√fourt).

Quasi-Periodic Office

Quasi-periodic functions are a special case of almost periodic functions. They are a non periodic; They are a combination of periodic functions of dissimilar frequencies that never match exactly.

Perhaps the simplest style to create one is just to add together two periodic functions: ane with a rational menstruum and one with an irrational period (Ong, 2020). Fourier transforms of quasi-periodic functions are discrete sets of delta functions; they can always exist expressed every bit a series of sines and cosines with non matching lengths—or with an amount of arithmetically independent footing vectors that exceed the number of independent variables (Cahn, 2001).

There are several ways to define quasiperiodic functions mathematically. I fairly straightforward way (Jorba & Simo, 1984):

"A function f is a quasiperiodic role with basic frequencies ω_i, …, ω_r if f(t) = F(θ_one,…, θ_r) where F is 2π periodic in all its arguments and θ_j = ω_jt for j = 1, …,r"

References

Desmos Graphing Calculator.
Adams, Grand. (2020). Continuous-Time Signals and Systems (Edition two.0).
Caggiano, D. (1996). Comparison of Different Betoken Processing Algorithms to Extract the Respiration Waveform from the ECG. Retrieved Nov thirteen, 2020 from: http://athenaeum.njit.edu/vol01/etd/1990s/1996/njit-etd1996-014/njit-etd1996-014.pdf
Cahn, J. (2001). Quasicrystals. Journal of Research of the National Institute of Standards and Applied science. 106, 975–982.
Depner, J. & Rasmussen, T. (2017). Hydrodynamics of Time-Periodic Groundwater Menstruation: Diffusion Waves in Porous Media, Geophysical Monograph 224. American Geophysical Marriage.
Dua, R. (2014). Experimentation of Transforms. Retrieved November thirteen, 2020 from: http://web.mst.edu/~rdua/Digital%20Signal%20Processing_files/Sample%202.pdf
Hall, R. Sounding Number. Retrieved November 29, 2019 from: http://people.sju.edu/~rhall/SoundingNumber/periodicfunctions.pdf
Jorba, A. & Simo, C. (1984). On Quasiperiodic Perturbations of Elliptic Equilibrium Points. Retrieved Nov thirteen, 2020 from: https://upcommons.upc.edu/bitstream/handle/2117/901/9501jorba.pdf
Affiliate nineteen: Trigonometry: Introducing Periodic Functions. Retrieved November 29, 2019 from: http://world wide web.math.harvard.edu/archive/xb_spring_06/files/chap19-20.pdf
Ong, D. (2020). Quasiperiodic music. Retrieved November 13, 2020 from: https://export.arxiv.org/ftp/arxiv/papers/2009/2009.04667.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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