Chapter 6.3, Problem 133AYU

Describe your understanding of the meaning of a periodic function.

To determine

To find: Meaning of a periodic function.

Answer to Problem 133AYU

Solution:

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 $2\pi$ radians. Periodic functions are used throughout science to describe oscillations, wanes, and other phenomena that exhibit periodicity. Any function that is not periodic is called aperiodic.

A function $f$ is said to be periodic if, for some nonzero constant $P$, we have $f\left(x+p\right)=f\left(x\right)$ for all values of $x$ in the domain. A nonzero constant $P$ for which this is the case is called a period of the function. If there exists a least positive constant $P$ with this property, it is called the fundamental period (also primitive period, basic period, or prime period.) Often, “the” period of a function is used to mean its fundamental period. A function with period $P$ will repeat on intervals of length $P$, and these intervals are sometimes also referred to as periods of the function.

Explanation of Solution

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 $2\pi$ radians. Periodic functions are used throughout science to describe oscillations, wanes, and other phenomena that exhibit periodicity. Any function that is not periodic is called aperiodic.

A function $f$ is said to be periodic if, for some nonzero constant $P$, we have $f\left(x+p\right)=f\left(x\right)$ for all values of $x$ in the domain. A nonzero constant $P$ for which this is the case is called a period of the function. If there exists a least positive constant $P$ with this property, it is called the fundamental period (also primitive period, basic period, or prime period.) Often, “the” period of a function is used to mean its fundamental period. A function with period $P$ will repeat on intervals of length $P$, and these intervals are sometimes also referred to as periods of the function.