Explanation Formula used: The Fourier series of a function on the interval ( − p , p ) is given as follows: f ( x ) = a 0 2 + ∑ n = 1 ∞ a n cos n x π p + b n sin n π p x Here, a 0 , a n and b n are defined as follows: a 0 = 1 p ∫ − p p f ( x ) d x a n = 1 p ∫ − p p f ( x ) ⋅ cos ( n x π p ) d x b n = 1 p ∫ − p p f ( x ) ⋅ sin ( n π p x ) d x Calculation: Use the above definition to obtain the Fourier expansion of the function f ( x ) = 2 x 2 − 1 , − 1 < x < 1 as follows: f ( x ) = a 0 2 + ∑ n = 1 ∞ a n cos n π x + b n sin n π x … … ( 1 ) Substitute f ( x ) = 2 x 2 − 1 and p = 1 in expression a 0 = 1 p ∫ − p p f ( x ) d x to obtain the value of a 0 as follows: a 0 = ∫ − 1 1 f ( x ) d x = ∫ − 1 1 ( 2 x 2 − 1 ) d x = [ 2 3 x 3 − x ] − 1 1 Further, solve the above expression: a 0 = 4 3 − 2 = 4 − 6 3 = − 2 3 Substitute f ( x ) = 2 x 2 − 1 and p = 1 in expression a n = 1 p ∫ − p p f ( x ) ⋅ cos ( n x π p ) d x to obtain the value of a n as follows: a n = ∫ − 1 1 f ( x ) ⋅ cos n x π d x = ∫ − 1 1 ( 2 x 2 − 1 ) ⋅ cos n π x d x = 2 ∫ − 1 1 x 2 ⋅ cos n π x d x − ∫ − 1 1 cos n π x d x = 2 [ x 2 n π sin n π x | − 1 1 − 2 n π ∫ − 1 1 x sin π x d x ] − [ sin n π x n π ] − 1 1 Further solve the above expression: a n = 2 [ ( 0 ) − 2 n π ∫ − 1 1 x sin π x d x ] − ( 0 ) ( sin n π x = 0 , n ∈ Z

Differential Equations with Boundary-Value Problems (MindTap Course List)

9th Edition

ISBN: 9781305965799

Author: Dennis G. Zill

Publisher: Cengage Learning

See similar textbooks

Videos

Question

Chapter 11, Problem 14RE

To determine

The expansion of function f(x)=2x2−1, −1<x<1 using Fourier series.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 11, Problem 13RE

Chapter 11, Problem 15RE

Chapter 11 Solutions

Differential Equations with Boundary-Value Problems (MindTap Course List)

Ch. 11.1 - In problem 16 show that the given functions are...Ch. 11.1 - In problem 16 show that the given functions are...Ch. 11.1 - In problem 16 show that the given functions are...Ch. 11.1 - In problem 16 show that the given functions are...Ch. 11.1 - In problem 16 show that the given functions are...Ch. 11.1 - Prob. 6E Ch. 11.1 - Prob. 7E Ch. 11.1 - In Problems 712 show that the given set of...Ch. 11.1 - In Problems 712 show that the given set of...Ch. 11.1 - In Problems 712 show that the given set of...

Knowledge Booster

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, subject and related others by exploring similar questions and additional content below.

The expansion of function f ( x ) = 2 x 2 − 1 , − 1 < x < 1 using Fourier series.

Solution Summary: The author explains the Fourier expansion of a function f(x)=2x,-11 using fourier series.

Videos

The expansion of function f(x)=2x2−1, −1<x<1 using Fourier series.

Want to see the full answer?

Chapter 11 Solutions

The expansion of function f ( x ) = 2 x 2 − 1 , − 1 &lt; x &lt; 1 using Fourier series.

Solution Summary: The author explains the Fourier expansion of a function f(x)=2x,-11 using fourier series.

Videos

The expansion of function f(x)=2x2−1, −1<x<1 using Fourier series.

Want to see the full answer?

Chapter 11 Solutions

The expansion of function f ( x ) = 2 x 2 − 1 , − 1 < x < 1 using Fourier series.