Matematik

Fourierrække

14. april 2016 af NobelPrize (Slettet) - Niveau: Universitet/Videregående

Hello,

I need help with the following exercise:

I need to show that the function:

$f(t)=\frac{\pi}{8}(\pi t-t^2), 0\leq t\leq \pi$

has the Fourier serie:

$\sum_{0}^{\infty}\ \frac{(-1)^{2+n}}{(2n-1)^3}= \frac{\pi^3}{32}$ .

I got as suggestion that

$\int t \textrm { sin} (nt) dt = \frac{\textrm{sin}(nt)}{n^2}- t \frac{\textrm{cos}(nt)}{n}$

and

$\int t^2 \textrm { sin} (nt) dt = 2 \frac{\textrm{cos}(nt)}{n^3}+2t \frac{\textrm{sin}(nt)}{n^2}-t^2\frac{\textup{\textrm{sin}(nt)}}{n}$

So I start looking for the coefficient b_nof the Fourier series:

$b_n= \frac{2}{\pi}\int_{0}^{\pi} \frac{\pi}{8}(\pi t-t^2) \textrm{sin}(nt)dt=\frac{1}{4} \int_{0}^{\pi}t(\pi-t)\textrm{sin}(nt)dt$

Am I doing it right? I do not know hot to integrate this, since it is not the common intebral by part, having 3 terms.

I can see that it looks similar to the suggestion that I got, but still I have a $(\pi-t)$ that I don't know how to treat.

Any idea or suggestion?

Thanks

Brugbart svar (0)

Svar #1
14. april 2016 af AskTheAfghan

Please show the whole problem.

Svar #2
14. april 2016 af NobelPrize (Slettet)

Here there is the whole text of the exercise.

Vedhæftet fil:14606544608131571957757.jpg

Brugbart svar (0)

Svar #3
14. april 2016 af VandalS

I suspect you're overthinking the problem - it should just be a matter of using the sum rule for integrals to split your final integral into the two integrals mentioned in the hints.

Brugbart svar (0)

Svar #4
14. april 2016 af AskTheAfghan

I might be wrong. Since f is odd, you can use the fact, in this case where t lies in [0, π], that a_n(f) = 0 and b_n(f) = (1/π)∫₀^π f(t) sin(n t) dt, for all n = 0, 1, 2, .... Note that

(1/π) f(t) sin(n t) = (π/8) t sin(n t) + (1/8) t² sin(n t).

Then use the hints. The result would be something like

a₀(f)/2 + Σ_n≥1 [ a_n(f) cos(n t) + b_n(f) sin(n t)] = Σ_n≥1 [ b_n(f) sin(n t) ].

You will need to rewrite this sum a lot. (I didn't check it myself).

Svar #5
17. april 2016 af NobelPrize (Slettet)

I really wanted to make it harder then what it was needed! Thanks for the tips!

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