- #1
Arthur Yeh
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Homework Statement
I am only interested in 9 (a)
Determine the Fourier Cosine series of the function g(x) = x(L-x) for 0 < x < L
Homework Equations
The Answer for 9 a.
g(x) = (L^2)/6 - ∑(L^2/(nπ)^2)cos(2nπx/L)
This is the relevant equation given where ω=π/L
f(t) = a0+∑ancos(nωt)
a0=1/L ∫f(t) dt from 0 to L
an=2/L∫f(t)cos(nωt) dt from 0 to L
The Attempt at a Solution
This is my attempt at the solution
g(x) = a0 - Σancos(nωx)
where
a0=L^2/6
an= -2L^2[(cos(nπ)+1)/(nπ)^2]
I have double checked this answer both manually and through the use of an online integral calculator and i still arrive at this conclusion. As a result I believe my partial integrations are correct but my answer is in the wrong form. Initially I tried changing cos(nπ) to (-1)^n but i didnt get anywhere as i didnt couldn't get rid of the n power. I also tried working backwards from the answer using some double angle identities but did not arrive at any recognizable form.
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