Periodic function

The sine function is periodic with period ${\displaystyle 2\pi }$, since

${\displaystyle \sin(x+2\pi )=\sin x}$

for all values of ${\displaystyle x}$. This function repeats on intervals of length ${\displaystyle 2\pi }$ (see the graph to the right).

Everyday examples are seen when the variable is time; for instance the hands of a clock or the phases of the moon show periodic behaviour. Periodic motion is motion in which the position(s) of the system are expressible as periodic functions, all with the same period.

For a function on the real numbers or on the integers, that means that the entire graph can be formed from copies of one particular portion, repeated at regular intervals.

A simple example of a periodic function is the function ${\displaystyle f}$ that gives the "fractional part" of its argument. Its period is 1. In particular,

${\displaystyle f(0.5)=f(1.5)=f(2.5)=\cdots =0.5}$

The graph of the function ${\displaystyle f}$ is the sawtooth wave.

A plot of ${\displaystyle f(x)=\sin(x)}$ and ${\displaystyle g(x)=\cos(x)}$; both functions are periodic with period ${\displaystyle 2\pi }$.

The trigonometric functions sine and cosine are common periodic functions, with period ${\displaystyle 2\pi }$ (see the figure on the right). The subject of Fourier series investigates the idea that an 'arbitrary' periodic function is a sum of trigonometric functions with matching periods.

According to the definition above, some exotic functions, for example the Dirichlet function, are also periodic; in the case of Dirichlet function, any nonzero rational number is a period.

Using complex variables we have the common period function:

${\displaystyle e^{ikx}=\cos kx+i\,\sin kx.}$

Since the cosine and sine functions are both periodic with period ${\displaystyle 2\pi }$, the complex exponential is made up of cosine and sine waves. This means that Euler's formula (above) has the property such that if ${\displaystyle L}$ is the period of the function, then

${\displaystyle L={\frac {2\pi }{k}}.}$

A function whose domain is the complex numbers can have two incommensurate periods without being constant. The elliptic functions are such functions. ("Incommensurate" in this context means not real multiples of each other.)

One subset of periodic functions is that of antiperiodic functions. This is a function ${\displaystyle f}$ such that ${\displaystyle f(x+P)=-f(x)}$ for all ${\displaystyle x}$. For example, the sine and cosine functions are ${\displaystyle \pi }$-antiperiodic and ${\displaystyle 2\pi }$-periodic. While a ${\displaystyle P}$-antiperiodic function is a ${\displaystyle 2P}$-periodic function, the converse is not necessarily true.

A further generalization appears in the context of Bloch's theorems and Floquet theory, which govern the solution of various periodic differential equations. In this context, the solution (in one dimension) is typically a function of the form

${\displaystyle f(x+P)=e^{ikP}f(x)~,}$

where ${\displaystyle k}$ is a real or complex number (the Bloch wavevector or Floquet exponent). Functions of this form are sometimes called Bloch-periodic in this context. A periodic function is the special case ${\displaystyle k=0}$, and an antiperiodic function is the special case ${\displaystyle k=\pi /P}$. Whenever ${\displaystyle kP/\pi }$ is rational, the function is also periodic.

In signal processing you encounter the problem, that Fourier series represent periodic functions and that Fourier series satisfy convolution theorems (i.e. convolution of Fourier series corresponds to multiplication of represented periodic function and vice versa), but periodic functions cannot be convolved with the usual definition, since the involved integrals diverge. A possible way out is to define a periodic function on a bounded but periodic domain. To this end you can use the notion of a quotient space:

${\displaystyle {\mathbb {R} /\mathbb {Z} }=\{x+\mathbb {Z} :x\in \mathbb {R} \}=\{\{y:y\in \mathbb {R} \land y-x\in \mathbb {Z} \}:x\in \mathbb {R} \}}$.

That is, each element in ${\displaystyle {\mathbb {R} /\mathbb {Z} }}$ is an equivalence class of real numbers that share the same fractional part. Thus a function like ${\displaystyle f:{\mathbb {R} /\mathbb {Z} }\to \mathbb {R} }$ is a representation of a 1-periodic function.