Soluzioni Prova scritta di AM4 del 13/1/2004 (Appello A e recupero Esoneri)

1)  ∫ sin^{2} t sin^{2} ωt dt = \frac{t}{4} − \frac{1}{8} sin 2t − \frac{1}{8ω} sin 2ωt

+\frac{1}{8(1 − ω^{2})}( sin 2t cos 2ωt − ω cos 2t sin 2ωt).

lim_{ω→∞} ∫_{{1 \atop 0}} sin^{2} t sin^{2} ωt dt = \frac{1}{4} − \frac{1}{8} sen 2.

2) ∫ \frac{x^{2}}{\sqrt{(x^{2} − 1)^{3}}} dx = log (x + \sqrt{x^{2} − 1}) − \frac{x}{\sqrt{x^{2} − 1}}.

3)  ∫_{S} fdσ = \sqrt{3} [\frac{1}{12}(1 + 4u^{2})^{3/2}

+\frac{1}{32}(u\sqrt{1 + 4u^{2}}(1 + 8u^{2}) − \frac{1}{2} ln (2u + \sqrt{1 + 4u^{2}}))

+(\sqrt{3} − \frac{1}{2})(u\sqrt{1 + 4u^{2}} + \frac{1}{2} ln (2u + \sqrt{1 + 4u^{2}}))]_{0}^{1} .

7) (i) α > −2 (essendo x^{α} sen x integrabile su (0, π) se e solo se α > −2). (ii) Se F̂_{α,±n} = (a_{α,n} ∓ b_{α,n})/2 si ha che a_{α,n} = 0 per ogni n ≥ 0 e

b_{1,1} = \frac{π}{2} , b_{1,n} = \frac{(−1)^{n} + 1}{π} \frac{−4n}{(n^{2} − 1)^{2}} , ∀ n ≥ 2 .

(iii) Dal lemma di Riemann-Lebesgue segue che b_{−1,n} ∼ 1/n e b_{100,n} ∼ 1/n^{3}. Più precisamente:

\frac{2}{π} b_{−1,n} = \frac{1}{n} + \frac{1}{n}∫_{{π \atop 0}} cos(nx)( \frac{sen x}{x})′ = \frac{1}{n} + o(\frac{1}{n});

\frac{2}{π} b_{100,n} = \frac{(−1)^{n}}{n^{3}} 200 π^{99} + \frac{1}{n^{3}} ∫_{{π \atop 0}} cos(nx)(x^{100} sen x)^{(3)} = \frac{(−1)^{n}}{n^{3}} 200 π^{99} + o(\frac{1}{n^{3}} )

8) (i) f̂(ξ) = \frac{−i}{\sqrt{π}} ( \frac{sen (1 − ξ)}{1 − ξ} − \frac{sen (1 + ξ)}{1 + ξ}).

(ii) u(x, t) = \frac{1}{\sqrt{π}} ∫_{{∞ \atop −∞}}e^{−ξ^{4}t} e^{ixξ} f̂(ξ)dξ.