(a) Find the Fourier series of the function f(x) = x²,0 ≤ x < L with period L. (b) Use the previous result to express ² as an infinite series of rational numbers.
(a) Find the Fourier series of the function f(x) = x²,0 ≤ x < L with period L. (b) Use the previous result to express ² as an infinite series of rational numbers.
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Transcribed Image Text:2. ² and rational numbers
(a) Find the Fourier series of the function f(x) = x²,0 ≤ x < L with period L.
(b) Use the previous result to express ² as an infinite series of rational numbers.
context and insightful tips
(1) Use the following relationships
ao
2πη
2πη
S(z) — 2²2 + Σ²₂ 17 (²7²²) + [³, xin (2m² - )
f(x)
=
an
X
L
L
n=1
n=1
ao
An
bn
=
[ 2²
[x
=
2²/7 [² f(x) dx,
L
2
5.² f(x) cos
2² (²
L
f(x) sin
x² cos ax dx =
x² sin ax dx
=
2πη
L
2x
a²
2x
2πη
L
(2) f(x) is neither even nor odd in this problem, so all coefficients may be non-zero.
(3) These results of integration by parts may be helpful:
-X
sin ax
X dx, n=1,2,...
cos ax +
dx, n=1,2,...
-
220 220
I
2
| |
a³
2
a³
sin ax + C,
(2)
cos ax + C.
(3)
(4)
(5)
(6)
(7)
(8)
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