Formulario

Formule matematiche e trasformate

Derivate e limiti

R_f[a] = \frac{1}{(N-1)!}\lim_{x\to a}\frac{d^{N-1}}{dx^{N-1}}[(x-x_{0})^{N}f(x)]

Funzioni di variabili complesse

\frac{1}{(z^{2}+\omega^{2})^{2}}\frac{1}{2\omega}\frac{1}{z+\omega}\frac{d}{dz}\frac{z}{z^{2}+\omega^{2}}

Trasformata Z

• Z[z^{-1}u(n)] = z^{-1}
• Z[z^{l}a(n|c)] = z[za(n)](c/\lambda)
• Z_s[na(n)] = -\frac{d}{dz}Z_a[a(n)]
• Z_a(n+k) = z^{k}Z_a[a(n)] = a(0)z^{k} - a(k-1)z^{k}
• Z_a[cos n\theta] = \frac{z}{z^{2}-2z\cos\theta+1}
• Z_a[sin n\theta] = \frac{sin \theta}{z^{2}-2z \cos \theta+1}
• Z_a[a(n)] = a(0)z + a(1)z^{2} + \cdots + a(k-1)z^{k}\frac{z^{1}}{z^{k}-1}a(n) periodica di periodo k
• Z[z^{a}(n)b(n)] = Z[a(n)] - Z[b(n)]
• Z^{\lambda}(a^{n}\frac{(n-1)!}{(n-k)}) = \frac{1}{(z^{\lambda})^{k}}

Derivate e trasformate di Laplace

• Z[\Delta x(t) = x(t) - x(t-0)] = z^{-1}X(s)
• \mathscr{L}[x(t)] = \frac{1}{\alpha}\sum_{X} \big( x(s_{0}) \big)
• \mathscr{L}[x(t)e^{\alpha t}] = X(s-s_{0})
• \mathscr{L}[x(t-t_{0})] = e^{-s t_{0}}X(s)
• \mathscr{L}[x(t+u(-t))] = e^{-\alpha t_{0}}\mathscr{L}[x(t+u(t_{0}))]
• \frac{d^{k}}{ds^{k}}\ x(s) = \mathscr{L}[-t^{k}x(t)]
• \mathscr{L}[x^{(k)}(t)] = s^{k}X(s)
• \mathscr{L}[x^{(k)}(t)] = s^{k}X(s) - s^{k-1}x(0) - \cdots - \left[s^{0-1}(0)\right](s-0)
• \mathscr{L}[1] = \frac{1}{s}
• \mathscr{L}[t^{k}] = \frac{k!}{(s-s_{0})^{k+1}}
• \mathscr{L}[cos\alpha t] = \frac{3}{s^{2}+a^{2}}
• \mathscr{L}[sin\alpha t] = \frac{a}{s^{2}+a^{2}}
• \mathscr{L}[x(t)cos\omega t + \mathscr{L}[x(t) = x(\omega) + x(\omega-\omega)X(\omega->\omega)e^{st} + X(\omega+\omega)e^{-\alpha s}
• \mathscr{L}[(x)(t-\omega)] = \mathscr{L}[-x(t)^{k}x(t)]
• \mathscr{L}[x^{(k)}(t)] = (u^{k})^{s}\mathscr{L}[x(t)]
• \mathscr{L}[x^{(k)}(t)] = 2\pi x(-t)
• \mathscr{L}[\Pi(t)] = \frac{sin_{\omega}/2}{\omega/2}

Trasformata di Fourier

• \mathscr{F}[\delta] = 1
• \mathscr{F}[\xi] = 2\pi c\delta
• \mathscr{F}[\omega] = v.p.\frac{1}{\omega}+\pi\delta
• \mathscr{F}[\omega] = \frac{\omega_{\Delta}^{2}+\omega_{0}=\frac{2\pi}{\tau}}X(\omega) = \omega_{0}\sum_{m=-\infty}^{+\infty}X_{0}(\omega_{0})\delta(\omega-\omega_{0})