In the Fourier series expansion of the even function f(x), the cosine coefficients
𝑎
𝑛
are equal to zero.

Question

In the Fourier series expansion of the even function f(x), the cosine coefficients 
𝑎
𝑛
 are equal to zero.

BlackTom AI · Accepted Answer

The statement concerns the Fourier series expansion of an even function f(x). In such a case, the Fourier series contains only cosine terms, because cosine is an even function and sine terms are odd and thus integrate to zero over a symmetric interval. Therefore, the sine coefficients b_n are zero for an even function.

The given claim says that the cosine coefficients a_n are equal to zero. This is incorrect in general. The a_n coefficients determine the weights of the cosine terms in the series; for many even functions these coefficients are nonzero, reflecting the presence and shape of the even component. Only in the special case where the function itself is identically zero would all a_n be zero.

To summarize, for an even function f(x), we expect b_n = 0 while a_n are typically nonzero (unless f is the zero function). The assertion that a_n must be zero is therefore false.

AMME2000 BMET2960 BMET9960 (ND) Week 6 Pre-work

In the Fourier series expansion of the even function f(x), the cosine coefficients 𝑎 𝑛 are equal to zero.

View Explanation

Log in for full answers

Similar Questions

In the Fourier series expansion of the even function f(x), the cosine coefficients are equal to zero.

Open the Initial Condition Demo and select any of the curves defined on the interval [ 0 , 𝐿 ] . Now toggle between 'Even' and 'Odd' to extend this curve as an even or odd function, respectively, on the interval [ − 𝐿 , 𝐿 ] . Which of the following statements is true?

Open the Fourier Series Demo and select the "Sinusoid" function. The reason the Fourier series representation (shown in green) and analytical representation (shown in red) overlay exactly when the the number of Fourier terms is ≥ 2 is because:

Open the Fourier Series Demo and select the "Arbitrary" function. Which term in the Fourier series representation is responsible for the vertical offset of the Fourier-approximated curve?

The Fourier series expansion of the square wave shown below contains:

In a consumer society, many adults channel creativity into buying things

More Practical Tools for Students Powered by AI Study Helper

Homework AI Solver

Stylized AI Paper Writer

Plagiarism Checker Assistant

Citation AI Academic Writing Tool

In-Class Translation Assistant

AI Note Generator

AI Quiz Answers

Past Exam Questions from University Test Bank

Smart Practice Assistant

Adaptive Practice

Making Your Study Simpler