Periodic function

A periodic function or cyclic function, also called a periodic waveform (or simply periodic wave), is a function that repeats its values at regular intervals or periods. The repeatable part of the function or waveform is called a cycle.^[1] For example, the trigonometric functions, which repeat at intervals of $2\pi$ radians, are periodic functions. Periodic functions are used throughout science to describe oscillations, waves, and other phenomena that exhibit periodicity. Any function that is not periodic is called aperiodic.

Not to be confused with periodic mapping.

addition, subtraction, multiplication and division of periodic functions, and

taking a power or a root of a periodic function (provided it is defined for all $x$ ).

Periodic functions can take on values many times. More specifically, if a function $f$ is periodic with period $P$ , then for all $x$ in the domain of $f$ and all positive integers $n$ ,

If $f(x)$ is a function with period $P$ , then $f(ax)$ , where $a$ is a non-zero real number such that $ax$ is within the domain of $f$ , is periodic with period ${\textstyle {\frac {P}{a}}}$ . For example, $f(x)=\sin(x)$ has period $2\pi$ and, therefore, $\sin(5x)$ will have period ${\textstyle {\frac {2\pi }{5}}}$ .

Some periodic functions can be described by Fourier series. For instance, for L² functions, Carleson's theorem states that they have a pointwise (Lebesgue) almost everywhere convergent Fourier series. Fourier series can only be used for periodic functions, or for functions on a bounded (compact) interval. If $f$ is a periodic function with period $P$ that can be described by a Fourier series, the coefficients of the series can be described by an integral over an interval of length $P$ .

Any function that consists only of periodic functions with the same period is also periodic (with period equal or smaller), including:

Generalizations[edit]

Antiperiodic functions[edit]

One subset of periodic functions is that of antiperiodic functions. This is a function $f$ such that $f(x+P)=-f(x)$ for all $x$ . For example, the sine and cosine functions are $\pi$ -antiperiodic and $2\pi$ -periodic. While a $P$ -antiperiodic function is a $2P$ -periodic function, the converse is not necessarily true.^[3]

Bloch-periodic functions[edit]

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

For set representing all notes of Western major scale: [1 9⁄8 5⁄4 4⁄3 3⁄2 5⁄3 15⁄8] the LCD is 24 therefore T = 24⁄f.

For set representing all notes of a major triad: [1 5⁄4 3⁄2] the LCD is 4 therefore T = 4⁄f.

For set representing all notes of a minor triad: [1 6⁄5 3⁄2] the LCD is 10 therefore T = 10⁄f.

Consider a real waveform consisting of superimposed frequencies, expressed in a set as ratios to a fundamental frequency, f: F = 1⁄f [f₁ f₂ f₃ ... f_N] where all non-zero elements ≥1 and at least one of the elements of the set is 1. To find the period, T, first find the least common denominator of all the elements in the set. Period can be found as T = LCD⁄f. Consider that for a simple sinusoid, T = 1⁄f. Therefore, the LCD can be seen as a periodicity multiplier.

If no least common denominator exists, for instance if one of the above elements were irrational, then the wave would not be periodic.^[4]

(1990). "One". Convexity methods in Hamiltonian mechanics. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)]. Vol. 19. Berlin: Springer-Verlag. pp. x+247. ISBN 3-540-50613-6. MR 1051888.

Periodic function

Generalizations[edit]

Antiperiodic functions[edit]

Bloch-periodic functions[edit]

Ekeland, Ivar

"Periodic function"

Weisstein, Eric W.