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Periodic function

A mathematical function that repeats values at regular intervals. Covers definition, fundamental period, examples (trigonometric, waves), key properties, history, and applications.

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Overview

In mathematics, a periodic function is one whose values recur in a regular cycle. Concretely, a function f is called periodic if there exists a nonzero number T (the period) such that f(x + T) = f(x) for every x in the domain where both sides are defined. The repeated motif or pattern may be described as a repeated pattern that continues indefinitely. When a least positive T exists, it is called the fundamental period.

Familiar examples are the trigonometric functions: sin(x) and cos(x) both repeat every 2π, so 2π is a period and indeed their fundamental period on the real line.

Key properties

Periodic functions enjoy several useful analytic and algebraic features. Typical properties include:

  • The derivative of a periodic differentiable function is periodic with the same period. Integrals over a full period often simplify many calculations in analysis and physics.
  • The sum, difference or product of periodic functions can be periodic. If two functions have commensurate periods (their ratio is rational) a common period exists; if their periods are incommensurate the combination is generally not periodic and may instead be quasi-periodic.
  • Scaling the input (e.g., f(ax)) changes the period by a reciprocal factor; multiplying the output by a constant preserves periodicity.

Examples and variations

Beyond sine and cosine, common periodic signals include square waves, triangular waves and sawtooth waves used in electronics and signal processing. Complex exponentials e^{iωt} represent pure oscillations and form the basis of Fourier series, which express many periodic functions as sums of harmonics. Periodicity also appears in discrete sequences when values repeat after a fixed number of steps.

History and development

Human observation of periodic phenomena goes back to celestial cycles and seasonal patterns. In mathematical analysis, the systematic study of periodic functions advanced with the development of Fourier series, which showed how a wide class of periodic behaviours can be decomposed into sines and cosines. This insight underpins harmonic analysis and many practical methods in engineering.

Applications and distinctions

Periodic functions model repeating processes across science and engineering: mechanical oscillations, alternating electrical currents, acoustic waves, and biological rhythms. Important distinctions include periodic versus quasi-periodic (superposition of incommensurate frequencies) and almost-periodic (functions that recur approximately). The concept of a fundamental period and its integer multiples is central when classifying and analyzing repeating behavior.

Illustration of a periodic function with period P

Function graph of the sine function
Function graph of the sine function
function graph of the tangent function
function graph of the tangent function

Real periodic functions

Definition

A real number Tis a period of a function defined in \mathcal{D}_f \subseteq \mathbb{R}defined if for each xfrom {\displaystyle {\mathcal {D}}_{f}}holds:

  • {\displaystyle x+T}is in {\displaystyle {\mathcal {D}}_{f}}and
  • {\displaystyle f(x+T)=f(x)\;.}

The function fis periodic if it T \neq 0admits at least one period One then also says that fis " T-periodic".

For {\displaystyle {\mathcal {D}}_{f}=\mathbb {R} }, which is often the case, the first property is always satisfied.

Properties of the periods

The following properties apply to the period:

  • If Ta period of fthen also -Ta period of f;
  • If T_{1}and T_{2}two periods of f, then {\displaystyle k_{1}T_{1}+k_{2}T_{2}}with {\displaystyle k_{1},k_{2}\in \mathbb {Z} }a period of f.

Mostly one is interested in the smallest positive period. This exists for any nonconstant continuous periodic function. (A constant function is periodic with any period other than 0.) If fhas a least positive period, then the periods of are fthe multiples of T. In the other case, the set of periods of is fdense in \mathbb {R} .

Examples

Trigonometric functions

The standard examples of periodic functions are the trigonometric functions. For example, the sine function defined on all of \mathbb {R} defined sine function is periodic. Its function values repeat at intervals of 2\pi (\pi is the circular number Pi); thus, it has period 2\pi .
The tangent function with domain of definition {\displaystyle \mathbb {R} \setminus \{k\pi +{\tfrac {\pi }{2}}\;|\;k\in \mathbb {Z} \}}is also a trigonometric function; it has period\pi rather than 2\pi although it can be represented as the quotient of two 2\pi -periodic functions: {\displaystyle \tan x={\tfrac {\sin x}{\cos x}}}.

Sum of cos and sin functions

Sums of cos and sin functions with a common (not necessarily smallest) period Tare again periodic. (In the figure, the common period is 2\pi .) This property of cos and sin functions is the basis of Fourier series. If two functions do not have a common period, the sum is not periodic. Example: {\displaystyle f(x)=\sin x+\sin(\pi x)}is not periodic.

Periodic continuation

In the definition example at the top of the figure, a function given on a half-open interval (a,b]given on a half-open interval has been b-a{\displaystyle T=b-a}continued to a periodic function of period by simply shifting by integer multiples of This type is called direct periodic continuation, to distinguish even and odd periodic continuation.

The following formal definition also provides a way to evaluate a periodically continued function with a computer, since the rounding function used is directly or indirectly realized in many mathematics systems.

DefinitionIf
a function f_{0}on the interval [a, b]with {\displaystyle f_{0}(a)=f_{0}(b)}is given, then the function fwith

{\displaystyle f(x)=f_{0}{\big (}x-\left\lfloor {\frac {x-a}{T}}\right\rfloor \cdot T{\big )},\;x\in \mathbb {R} }

the (direct) periodic continuation of f_{0}on all of \mathbb {R} and {\displaystyle \;T=b-a\;}its period.
⌊{\displaystyle \lfloor \cdot \rfloor }is the rounding function. Using the rounding function ensures that the function is f_{0}only evaluated for x values from its definition range (see figure).

Example: periodic continuation of parabolic arc {\displaystyle f_{0}(x)=(x-1)(4-x),\;a=1,b=4}with period {\displaystyle \;T=b-a=3\;}. The function value at the point (for example) {\displaystyle x=8}

{\displaystyle f(8)=f_{0}{\big (}8-\lfloor {\frac {7}{3}}\rfloor \cdot 3{\big )}=f_{0}(8-2\cdot 3)=f_{0}(2)=2\;.}

Since periodic functions are often developed in Fourier series and an even/odd periodic function can only be represented with cosine/sine terms, the following continuations are of particular interest:

Odd continuation:
In this case, one starts from a function {\displaystyle [0,b]}defined on
the interval f_{0}with {\displaystyle f_{0}(0)=f_{0}(b)=0}. In a first step, one continues the function by mirroring it at the zero point on the interval : {\displaystyle [-b,0]}

{\displaystyle f_{u}(x)={\begin{cases}\quad f_{0}(x)\;,\ \quad 0\leq x\leq b\\-f_{0}(-x)\;,\;-b\leq x<0\ .\end{cases}}}

The function [-b,b]defined on the interval f_{u}is now directly periodically continued (as described above). This gives rise to an odd periodic function defined on \mathbb {R} defined odd periodic function fof period {\displaystyle T=2b}.

Just continued:
The analog procedure with

{\displaystyle f_{g}(x)={\begin{cases}f_{0}(x),\quad \ \ 0\leq x\leq b\\f_{0}(-x),\;-b\leq x<0\end{cases}}}

yields an even periodic function of period {\displaystyle T=2b}.

Fourier series: Example

→ Main article: Fourier series

The Fourier series of a 2\pi -periodic odd function fhas the form

{\displaystyle \;\sum _{k=1}^{\infty }b_{k}\sin \left(kt\right)\;}

with

{\displaystyle b_{k}={\frac {1}{\pi }}\int _{-\pi }^{\pi }f(t)\cdot \sin \left(kt\right)\mathrm {d} t\quad {\text{für }}k\geq 1\;.}

The goal of a Fourier series expansion is to approximate a periodic function on (all of !) \mathbb {R} by sums of simple periodic functions. Ideally, the Fourier series represents the given function on \mathbb {R} . (A power series expansion approximates a function, which is not a polynomial, with its partial sums on a bounded (!) interval by polynomials).

In the figure, a function [0,\pi ]given on the interval f_{0}(two straight line segments, red) fcontinued odd to a 2\pi -periodic function and then developed into a Fourier series (with sin terms only). One can see how well/badly partial sums of the Fourier series (of lengths n= 3,6,12) f_{0}approximate the function While f_{0}is discontinuous (it has jump points), the partial sums as sums of sin terms are all continuous.

In the example

{\displaystyle f_{0}(x)={\begin{cases}x\;,\ \quad 0\leq x\leq {\frac {\pi }{2}}\\2.5\;,\;{\frac {\pi }{2}}<x<\pi \\0\;,\qquad x=\pi \ .\end{cases}}}

and the partial sum for n=3:

{\displaystyle \;\sum _{k=1}^{3}b_{k}\sin \left(kt\right)={\frac {1}{\pi }}{\big (}7\sin x+({\frac {\pi }{2}}-5)\sin 2x+{\frac {13}{9}}\sin 3x{\big )}\ .}

Periodic function
above: {\displaystyle {\mathcal {D}}_{f}=\mathbb {R} }. (blue),
bottom: {\displaystyle {\mathcal {D}}_{f}}subset of \mathbb {R} (blue),
purple: period

Sum of cos and sin functions
Sum of cos and sin functions
Periodic continuation of a parabolic arc
Periodic continuation of a parabolic arc
Periodic continuation of the function in the pink area: top: odd, bottom: even
Periodic continuation of the function in the pink area: top: odd, bottom: even
Fourier series: various Partial sums (blue)
Fourier series: various Partial sums (blue)

More general definition

The notion of periodic function is not limited to real functions. It can be defined more generally for functions on whose source set an addition is declared.

Let Gbe an (additive) semigroup, Ma set, and {\displaystyle f\colon G\to M}a function. If there exists a {\displaystyle T\in G}with

{\displaystyle f(g+T)=f(g)}

for all g\in Gthen the function is called fperiodic with period T.

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URL: https://en.alegsaonline.com/art/75830

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