1. Find the Fourier sine and cosine series of f(x) in each case. a. f(x) = = π − x on [0, π], [Hint : recall that we have calculated the Fourier sine and cosine series of g(x) = x on the same interval and have found the following ]. π (−1)n+1 X 1 (2k-1)² cos ((2k-1)x), x~2 sin(nx) 2 π n n=1 1 1 ~ 4 1 π 2k 1 sin ((2k-1)x) b. (*) f(x) = ². Further, sketch the periodic even and odd extensions of f corresponding to the Fourier sine and cosine series. c. (**) See if you can see any relationship between the various Fourier series of functions 1, x, and x².
1. Find the Fourier sine and cosine series of f(x) in each case. a. f(x) = = π − x on [0, π], [Hint : recall that we have calculated the Fourier sine and cosine series of g(x) = x on the same interval and have found the following ]. π (−1)n+1 X 1 (2k-1)² cos ((2k-1)x), x~2 sin(nx) 2 π n n=1 1 1 ~ 4 1 π 2k 1 sin ((2k-1)x) b. (*) f(x) = ². Further, sketch the periodic even and odd extensions of f corresponding to the Fourier sine and cosine series. c. (**) See if you can see any relationship between the various Fourier series of functions 1, x, and x².
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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![1. Find the Fourier sine and cosine series of f(x) in each case.
a. f(x) = π − x on [0, π], [Hint : recall that we have calculated the Fourier sine and cosine series of
g(x) = x on the same interval and have found the following ].
∞
∞
ㅠ
1
(−1)n+1
X
cos ((2k-1)x),
x ~
2Σ
sin(nx)
2
(2k − 1)²
n
k=1
n=1
1
1
1,
sin ((2k-1)x)
ㅠ 2k 1
k=1
b. (*) ƒ(x) = x². Further, sketch the periodic even and odd extensions of ƒ corresponding to the
Fourier sine and cosine series.
c. (**) See if you can see any relationship between the various Fourier series of functions 1, x, and x².
π](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F91f76606-a4d9-42f0-be0d-7b1366a5593f%2Fe52683a8-daba-455b-af2e-59495b6fa55f%2Flev9t6_processed.png&w=3840&q=75)
Transcribed Image Text:1. Find the Fourier sine and cosine series of f(x) in each case.
a. f(x) = π − x on [0, π], [Hint : recall that we have calculated the Fourier sine and cosine series of
g(x) = x on the same interval and have found the following ].
∞
∞
ㅠ
1
(−1)n+1
X
cos ((2k-1)x),
x ~
2Σ
sin(nx)
2
(2k − 1)²
n
k=1
n=1
1
1
1,
sin ((2k-1)x)
ㅠ 2k 1
k=1
b. (*) ƒ(x) = x². Further, sketch the periodic even and odd extensions of ƒ corresponding to the
Fourier sine and cosine series.
c. (**) See if you can see any relationship between the various Fourier series of functions 1, x, and x².
π
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