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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
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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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).
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

ϕ(x)=ex+x.

Solution of 12a.

Fourier sine series of ϕ(x) on the interval 0,1 is

ϕ(x)=∑n=1∞bnsinnπx

or

ex+x=∑n=1∞bnsinnπx

where 

bn=2∫01ex+xsinnπxdx.

Fourier cosine series of ϕ(x) on the interval 0,1 is

ϕ(x)=a02+∑n=1∞ancosnπx

or 

ex+x=a02+∑n=1∞ancosnπx

where 

a0=2∫01ex+xdxand an=2∫01ex+xcosnπxdx

 

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