 Matematica
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Trasformazione di Laplace
- $\begin{align*} \mathcal{L}[f(t)]=F(s)=\int^{\infty}_{0}{f(t)e^{-st}dt} &\\ \mathcal{L}[g(t)]=G(s)=\int^{\infty}_{0}{g(t)e^{-st}dt} \end{align*}$
- 1.
- $\mathcal{L}(1)(s)=\frac{1}{s}\quad;\quad s>0$
- 2.
- $\mathcal{L}(e^{at})(s)=\frac{1}{s-a}\quad;\quad s>a$
- 3.
- $\mathcal{L}(e^{t^n})(s)=\frac{n!}{s^{n+1}}\quad; \quad s>0 , n \;\; \text{ pozitif tam say{\i}}$
- 4.
- $\mathcal{L}(e^{t^p})(s)=\frac{\Gamma(p+1)}{s^{p+1}}\quad; \quad s>0 , p>-1$
- 5.
- $\mathcal{L}(\sin{at})(s)=\frac{a}{s^2+a^2}\quad; \quad s>0$
- 6.
- $\mathcal{L}(\cos{at})(s)=\frac{s}{s^2+a^2}\quad; \quad s>0$
- 7.
- $\mathcal{L}(e^{at}.\sin{bt})(s)=\frac{b}{(s-a)^2+b^2}\quad; \quad s>a$
- 8.
- $\mathcal{L}(e^{at}.\cos{bt})(s)=\frac{s-a}{(s-a)^2+b^2}\quad; \quad s>a$
- 9.
- $\mathcal{L}(t^{n}.e^{at})(s)=\frac{n!}{(s-a)^{n+1}}\quad; \quad s>a$
- 10.
- $\mathcal{L}(t^{n}.f(t))(s)=(-1)^n\frac{d^n}{ds^n}(\mathcal{L}(f(t)))$
- 11.
- $\mathcal{L}(f'(t))(s)=s\mathcal{L}(f(t))-f(0)=sF(s)-f(0)$
- 12.
- $\begin{align*} \mathcal{L}(f''(t))(s)=s^2\mathcal{L}(f(t))-sf(0)-f'(0) &\\=s^2F(s)-sf(0)-f'(0) \end{align*}$
- 13.
- $\begin{align*} \mathcal{L}(f^{(n)}(t))(s) &=s^n\mathcal{L}(f(t))-s^{n-1}f(0)-...-f^{(n-1)}(0)&\\ & =s^nF(s)-s^{n-1}f(0)- \cdots -f^{(n-1)}(0) \end{align*}$
- 14.
- $\mathcal{L}(af(t)+bg(t))(s)=aF(s)+bG(s)$
- 15.
- $\mathcal{L}(f(at))(s)=\frac{1}{|a|}F(\frac{s}{a})$
- 16.
- $\mathcal{L}(e^{at}f(t))(s)=F(s-a)$
- 17.
- $\mathcal{L}(f(t-a)u(t-a))(s)=e^{-as}F(s)\quad; \quad u(t-a)=\left\{\begin{matrix} 1 & t\geqslant a \\ 0 & t< a \end{matrix}\right.$
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