Question 12. Consider the function ø(x) = e + x. 2 12a. W down the Fourier series and the Fourier cosine series of ø(x) on the interval x E (0, 1) but leave your coefficients as integrals. 12b. Write down the full Fourier series of ø(x) on the interval x E (-1,1) but leave your coefficients as integrals. 12c. At the point x = 0, what number does the Fourier sine series in 8a converge to? In other words, if we take more and more terms in the sum and evaluate at x = 0, what number will we get close to? What number does the Fourier cosine series in 12a converge to? What number does the full Fourier series in 12b converge to? 12d. For each series, plot the function that the Fourier series will get closer to (converge to) on the interval x E (-2, 2).
Question 12. Consider the function ø(x) = e + x. 2 12a. W down the Fourier series and the Fourier cosine series of ø(x) on the interval x E (0, 1) but leave your coefficients as integrals. 12b. Write down the full Fourier series of ø(x) on the interval x E (-1,1) but leave your coefficients as integrals. 12c. At the point x = 0, what number does the Fourier sine series in 8a converge to? In other words, if we take more and more terms in the sum and evaluate at x = 0, what number will we get close to? What number does the Fourier cosine series in 12a converge to? What number does the full Fourier series in 12b converge to? 12d. For each series, plot the function that the Fourier series will get closer to (converge to) on the interval x E (-2, 2).
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:Question 12. Consider the function ø(x) = e + x.
2
12a. W
down the Fourier
series and the Fourier cosine series of ø(x) on the interval
x E (0, 1) but leave your coefficients as integrals.
12b. Write down the full Fourier series of ø(x) on the interval x E (-1,1) but leave your
coefficients as integrals.
12c. At the point x = 0, what number does the Fourier sine series in 8a converge to? In other
words, if we take more and more terms in the sum and evaluate at x = 0, what number will we get
close to? What number does the Fourier cosine series in 12a converge to? What number does
the full Fourier series in 12b converge to?
12d. For each series, plot the function that the Fourier series will get closer to (converge to)
on the interval x E (-2, 2).
Expert Solution
Step 1
Given function is
.
Solution of 12a.
Fourier sine series of on the interval is
or
where
.
Fourier cosine series of on the interval is
or
where
Step by step
Solved in 2 steps with 3 images
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