Analisi 2 - Schemi riassuntivi

Serie di Fourier

c₀ + ∑_k=1^∞ (a_k cos kx + b_k sin kx) = g(x)

dove g(x) = f(x + T) = f(2π/T x)

Teorema: g(x + T) = g(x + T)

dim.:

g(x + T) = f(2π/T (x + T)) = f(2π/T + 2π)

= f(2π/T) = g(x)

Convergenza:

1. Assoluta convergenza

∑_k=1^∞ |a_k cos kx + b_k sin kx| ≤ |a_k cos kx| + |b_k sin kx| = (|a_k| |cos kx| + |b_k| |sin kx|)

≤ (|a_k| + |b_k|)

se ∑_k=1^∞ |a_k|, ∑_k=1^∞ |b_k| converge. ass. => serie Fourier converge

2. 2^o criterio di Dirichlet

∑_k=1^∞ a_k cos kx, ∑_k=1^∞ b_k sin kx

a_k ≠ 0, b_k ≠ 0 => serie di Fourier converge in (0,2π)

3) 2^o criterio di Dirichlet

\[\sum_{k=1}^{\infty}(-1)^{k}a_k\sin kx \quad \sum_{k=1}^{\infty}(-3)^{k}b_k\sin kx\]

\[a_k \to 0 \quad b_k \to 0\]

\[serie \ di \ fourier \ converge \ in \ \forall x \neq \frac{\pi}{\pi}\]

\[\sum_{k=1}^{\infty} (-1)^{k} (-3)^{k} a_k = \sum_{k=1}^{\infty} a_k\]

Trova la somma:

1)

\[\int_{-\pi}^{\pi} f(x)dx = \int_{-\pi}^{\pi} \Big[\frac{a_0}{2} + \sum_{k=1}^{\infty} [a_k \cos kx + b_k \sin kx]\Big]dx = \frac{a_0}{2} 2\pi + \sum_{k=1}^{\infty} a_k\]

\[\int_{-\pi}^{\pi} \cos kx dx + b_k \int_{-\pi}^{\pi} \sin kx dx = a_0 = \frac{a_0}{\pi}\]

\[\Rightarrow\]

\[a_0 = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) dx\]

2)

\[\int_{-\pi}^{\pi} f(x) \cos mx\ dx = \int_{-\pi}^{\pi} \Big[\frac{a_0}{2} \cos mx + \sum_{k=1}^{\infty} a_k\Big] \int_{-\pi}^{\pi} \cos kx \cos mx \ dx + b_k\]

\[\int_{-\pi}^{\pi} \sin kx \cos mx = a_k \int_{-\pi}^{\pi} \frac{1}{2} [\cos(kx+mx) + \cos(kx-mx)] dx =\]

\[= a_k \frac{1}{2} \cdot 2\pi \quad , \ per \ k = m \]

\[\Rightarrow\]

\[a_k = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \cos kx dx\]