(a) Find the Fourier series of the function f(x) = x²,0 ≤ x < L with period L. In the following, consider L = t. (b) Provide a plot of the function f(x) = x²,0 ≤ x < π, and the Fourier series up to n = 3 included. Hint: you can use Desmos or an equivalent plotting software, though you can not use any software to obtain the result in (a) or confirm your results algebraically or numerically. Only for plotting your result from (a). (c) Obtain an expression of ² as an infinite series of rational numbers. That is, π² = 1 Cn, where cn is a rational number that depends on n. Hint: Set L = π and x = π/2. (d) Show that the sum of the first 3 terms of the series of T² found in the previous answer is 31/3. Just FYI, π² is ~ 5% smaller than 31/3.
(a) Find the Fourier series of the function f(x) = x²,0 ≤ x < L with period L. In the following, consider L = t. (b) Provide a plot of the function f(x) = x²,0 ≤ x < π, and the Fourier series up to n = 3 included. Hint: you can use Desmos or an equivalent plotting software, though you can not use any software to obtain the result in (a) or confirm your results algebraically or numerically. Only for plotting your result from (a). (c) Obtain an expression of ² as an infinite series of rational numbers. That is, π² = 1 Cn, where cn is a rational number that depends on n. Hint: Set L = π and x = π/2. (d) Show that the sum of the first 3 terms of the series of T² found in the previous answer is 31/3. Just FYI, π² is ~ 5% smaller than 31/3.
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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no need to plot.
Transcribed Image Text:(a) Find the Fourier series of the function f(x) = x²,0 ≤ x < L with period L.
In the following, consider L = T.
(b) Provide a plot of the function f(x) = x²,0 ≤ x <T, and the Fourier series
up to n = 3 included. Hint: you can use Desmos or an equivalent plotting
software, though you can not use any software to obtain the result in (a) or
confirm your results algebraically or numerically. Only for plotting your result
from (a).
(c) Obtain an expression of T2 as an infinite series of rational numbers. That
is, π² = Σ¤-1 Cn, where cʼn is a rational number that depends on n. Hint: Set
L = π and x = π/2..
n=1
(d) Show that the sum of the first 3 terms of the series of n² found in the previous
answer is 31/3. Just FYI, ² is ~ 5% smaller than 31/3.
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