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You might find the lab on Fourier Series useful in
answering this question. The Matlab lab sheet is here. Find the Fourier series of the 2π-periodic function
In entering your answers, we will denote the coefficients in the
series by
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f(x) = a0 + |
∞
∑
m=1
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( a2m−1 cos((2m−1) x) + b2m−1 sin((2m−1) x) + a2m cos(2m x) + b2m sin(2m x) ) |
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In this sum, m goes from 1 to infinity.
The various coefficients for which you are asked are
rational functions of m, and should be entered as such.
Use the facts that sin(mπ) = 0, cos(2mπ) = 1,
cos((2m−1)π) = −1 to remove trig functions from your
expressions for the coefficients.
Enter π as Pi.
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a0 =
Your answer is correct.
The mark for your last attempt was 1.00
The teacher's answer was:
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For the remainder of the question, m ≥ 1:
a2m−1 =
Your answer is correct.
The mark for your last attempt was 1.00
The teacher's answer was:
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a2m =
Your answer is correct.
The mark for your last attempt was 1.00
The teacher's answer was:
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b2m−1 =
Your answer is correct.
The mark for your last attempt was 1.00
The teacher's answer was:
This can be entered as:
2/(2*m-1)
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b2m =
Your answer is correct.
The mark for your last attempt was 1.00
The teacher's answer was:
This can be entered as:
-1/m
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Solution:
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While there are general formulae for Fourier coefficients, it is usual,
and saves effort, to check, at the beginning, whether the function is even
or odd. Why? ...
- Even functions necessarily have all their
sine series coefficients equal to zero.
- Odd functions necessarily have all their
cosine series coefficients equal to zero.
In this problem, the function f is odd, and
hence all an = 0.
To find the sine coefficients, use the
formulae:
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bn = |
2
π
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| ⌠
⌡
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π
0
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f(x) sin(n x) dx, n ≥ 1. |
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Matlab can be used to evaluate the integrals. While it is possible to give
Matlab the information that m, n or whatever are integers and hence that
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sin(n π) = 0, cos(2 m π) = 1, cos((2 m − 1)π) = −1, etc. |
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it is probably easier in this problem to find a few coefficients and look
for a pattern.
If the problem arose in a real engineering situation, one would do both
this first step - numeric values of m as examples - and the
general m case. In any event, Matlab such as the following should find
the integrals:
syms x f s0 s1 s2 Pi c sm
s0 = sym(0); s1 = sym(1); s2 = sym(2); Pi = sym(pi); c = s2/Pi;
f = x
for m = 1:5
sm = sym(m)
b2mMinus1 = simplify(c*int(f*sin((s2*sm-s1)*x), x, s0, Pi))
b2m = simplify(c*int(f*sin(s2*sm*x), x, s0, Pi))
end |
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Mark New version
| Question note: |
[["m"], ["x"], ["n"], ["f_", "x"], ["nr_", "2"], ["s_", "\"syms x f
s0 s1 s2 Pi c sm \\ns0 = sym(0); s1 = sym(1); s2 = sym(2); Pi =
sym(pi); c = s2/Pi; \\nf = x \\nfor m = 1:5 \\n sm = sym(m) \\n
b2mMinus1 = simplify(c*int(f*sin((s2*sm-s1)*x), x, s0, Pi)) \\n b2m =
simplify(c*int(f*sin(s2*sm*x), x, s0, Pi)) \\nend\""], ["a2m_", "0"],
["a2mm1_", "0"], ["b2mm1_", "2/(2*m-1)"], ["b2m_", "-1/m"], ["a0Lis_",
"[1/2*Pi, 1/3*Pi^2, 1/6*Pi^2, 2/Pi]"], ["a2mLis_", "[0, 1/m^2, -1/m^2,
-4/Pi/(2*m-1)/(1+2*m)]"], ["b2mLis_", "[0, -1/m,
8*m/Pi/(2*m-1)/(1+2*m), -3/2/m^3, 0]"], ["d_", "\"discontinuous at
integer multiples of $\\\\pi$\""], ["de_", "\"discontinuous at even
multiples of $\\\\pi$\""], ["do_", "\"discontinuous at odd multiples of
$\\\\pi$\""], ["c_", "\"continuous on the whole real line\""], ["i_",
"2"], ["nicef_", "x"], ["na_", "4"], ["a0_", "0"], ["evnFnLis_",
"[abs(x), x^2, abs(x)*(Pi-abs(x)), abs(sin(x))]"], ["oddFnLis_",
"[signum(x), x, signum(x)*cos(x), x*(Pi^2-x^2), x*(Pi-abs(x))]"],
["noeFnLis_", "[x, max(0,sin(x))]"], ["a2mm1Lis_", "[-4/Pi/(2*m-1)^2,
-4/(2*m-1)^2, 0, 0]"], ["b2mm1Lis_", "[4/Pi/(2*m-1), 2/(2*m-1), 0,
12/(2*m-1)^3, 8/Pi/(2*m-1)^3]"], ["oddcttyLis_", "[\"discontinuous at
integer multiples of $\\\\pi$\", \"discontinuous at odd multiples of
$\\\\pi$\", \"discontinuous at integer multiples of $\\\\pi$\",
\"continuous on the whole real line\", \"continuous on the whole real
line\"]"], ["evncttyLis_", "[\"continuous on the whole real line\",
\"continuous on the whole real line\", \"continuous on the whole real
line\", \"continuous on the whole real line\"]"], ["noecttyLis_",
"[\"discontinuous at even multiples of $\\\\pi$\", \"continuous on the
whole real line\"]"], ["niceEvnFns_", "[x, x^2, x*(Pi-x), sin(x)]"],
["niceOddFns_", "[1, x, cos(x), x*(Pi^2-x^2), x*(Pi-x)]"], ["nb_",
"5"], ["nn_"]] |
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