A periodic function is a function that repeats its values at regular intervals. For example, the trigonometric functions, which are used to describe waves and other repeating phenomena, are periodic. Many aspects of the natural world have periodic behavior, such as the phases of the Moon, the swinging of a pendulum, and the beating of a heart. The length of the interval over which a periodic function repeats is called its period. Any function that is not periodic is called aperiodic.
Definition
A function is defined as periodic if its values repeat at regular intervals. For example, the positions of the hands on a clock display periodic behavior as they cycle through the same positions every 12 hours. This repeating interval is known as the period. More formally, a function f {\displaystyle f} is periodic if there exists a nonzero constant P {\displaystyle P} such that
f ( x + P ) = f ( x ) {\displaystyle f(x+P)=f(x)}
for all values of x {\displaystyle x} in the domain. A nonzero constant P {\displaystyle P} for which this condition holds is called a period of the function. If a period P {\displaystyle P} exists, any integer multiple n P {\displaystyle nP} (for a positive integer n {\displaystyle n} ) is also a period. If there is a least positive period, it is called the fundamental period (also primitive period or basic period). Often, "the" period of a function is used to refer to its fundamental period. Geometrically, a periodic function's graph exhibits translational symmetry. Its graph is invariant under translation in the x {\displaystyle x} -direction by a distance of P {\displaystyle P} . This implies that the entire graph can be formed from copies of one particular portion, repeated at regular intervals.
Examples Periodic behavior can be illustrated through both common, everyday examples and more formal mathematical functions.
Real-valued functions Functions that map real numbers to real numbers can display periodicity, which is often visualized on a graph.
Sawtooth wave An example is the function f {\displaystyle f} that represents the "fractional part" of its argument. Its period is 1. For instance,
f ( 0.5 ) = f ( 1.5 ) = f ( 2.5 ) = ⋯ = 0.5 {\displaystyle f(0.5)=f(1.5)=f(2.5)=\cdots =0.5}
The graph of the function f {\displaystyle f} is a sawtooth wave.
Trigonometric functions
The trigonometric functions are common examples of periodic functions. The sine function and cosine function are periodic with a fundamental period of 2 π {\displaystyle 2\pi } , as illustrated in the figure to the right. For the sine function, this is expressed as:
sin ( x + 2 π ) = sin x {\displaystyle \sin(x+2\pi )=\sin x}
for all values of x {\displaystyle x} . The field of Fourier series investigates the concept that an arbitrary periodic function can be expressed as a sum of trigonometric functions with matching periods.
Exotic functions Some functions are periodic but possess properties that make them less intuitive. The Dirichlet function, for example, is periodic, with any nonzero rational number serving as a period. However, it does not possess a fundamental period.
Complex-valued functions Functions with a domain in the complex numbers can exhibit more complex periodic properties.
Complex exponential The complex exponential function is a periodic function with a purely imaginary period:
e i k x = cos k x + i sin k x {\displaystyle e^{ikx}=\cos kx+i\,\sin kx}
Given that the cosine and sine functions are both periodic with period 2 π {\displaystyle 2\pi } , Euler's formula demonstrates that the complex exponential function has a period L {\displaystyle L} such that
L = 2 π k {\displaystyle L={\frac {2\pi }{k}}} .
Double-periodic functions A function on the complex plane can have two distinct, incommensurate periods without being a constant function. The elliptic functions are a primary example of such functions. ("Incommensurate" in this context refers to periods that are not real multiples of each other.)
Properties Periodic functions can take on values many times. More specifically, if a function f {\displaystyle f} is periodic with period P {\displaystyle P} , then for all x {\displaystyle x} in the domain of f {\displaystyle f} and all positive integers n {\displaystyle n} ,
f ( x + n P ) = f ( x ) {\displaystyle f(x+nP)=f(x)}
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