Question

$f(t)=t, \quad 0<t<2,$ and $f(t)$ has period 2

   $f(t)=t, \quad 0<t<2,$ and $f(t)$ has period 2

Fundamentals of Differential Equations

Fundamentals of Differential Equations

R. Kent Nagle 9th Edition

Chapter 7, Problem 1

Instant Answer

Solved by Expert Tanishq Gupta

08/05/2020

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So, $T=2$. Show more…

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$f(t)=t, \quad 0<t<2,$ and $f(t)$ has period 2

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Key Concepts

Periodic Functions

A periodic function is one that repeats its values at regular intervals or periods. This concept is crucial because it allows a function defined on a finite interval to be extended to the entire real line by repeating its behavior after a specified interval, known as the period.

Fundamental Period

The fundamental period of a periodic function is the smallest positive interval over which the function completes one full cycle of its behavior. Identifying the fundamental period is essential for understanding the repeating nature of the function and for applications such as Fourier analysis.

Periodic Extension

Periodic extension involves taking a function that is initially defined on a finite interval and extending it to a function defined on the entire real line by repeating the behavior of the function over each interval of length equal to the period. This concept is key when working with functions that naturally model repeating phenomena.

Piecewise-Defined Function

A piecewise-defined function is one that is defined by different expressions over different intervals. In the context of periodic functions, the function can be defined on one fundamental period and then periodically extended to cover the entire domain, requiring careful handling at the boundaries of the intervals.

Fourier Series

The Fourier series is a method for representing a periodic function as a sum of sine and cosine functions (or complex exponentials). This concept is central in analyzing periodic functions, as it decomposes the function into its frequency components, which is particularly useful in fields like signal processing and differential equations.

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