Periodic Function

A Periodic Function is a real-valued function that repeats over a finite interval or time period.

• Context:
  • It can be defined as [math]\displaystyle{ f(x)=f(x+kP) }[/math], where [math]\displaystyle{ x }[/math] is a variable, [math]\displaystyle{ k }[/math] is integer number and [math]\displaystyle{ P }[/math] is the time period or the finite interval for which the function repeats.
  • It can range from being a Constant Periodic Function to being a trigonometric function.
  • It can range from being a Fixed Periodic Function to being a Parameterized Periodic Function.
• Example(s):
  • Sine Function.
  • Cosine Function.
• Counter-Example(s):
  • Real Exponential Function.
• See: Trigonometric Function, Sine Function, Cosine Function, Frequency, Period, Aperiodic Function.

---

---

References

2015

• (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/Periodic_function
  • QUOTE: In mathematics, a periodic function is a function that repeats its values in regular intervals or periods. The most important examples are the trigonometric functions, which repeat over intervals of 2π radians. Periodic functions are used throughout science to describe oscillations, waves, and other phenomena that exhibit periodicity. Any function which is not periodic is called aperiodic.

1999

• (Wolfram Mathworld , 1999) ⇒ http://mathworld.wolfram.com/PeriodicFunction.html
  • QUOTE: A function [math]\displaystyle{ f(x) }[/math] is said to be periodic (or, when emphasizing the presence of a single period instead of multiple periods, singly periodic) with period [math]\displaystyle{ p }[/math] if

[math]\displaystyle{ f(x)=f(x+np) }[/math]

for [math]\displaystyle{ n=1, 2, \dots }[/math] For example, the sine function sinx, illustrated above, is periodic with least period (often simply called "the" period) [math]\displaystyle{ 2\pi }[/math] (as well as with period [math]\displaystyle{ -2\pi, 4\pi, 6\pi, }[/math] etc.).

---