15 Laplace Derivatives, Integrals
Derivatives¶
\[ \begin{aligned} L\{ f'(t) \} &= s F(s) - f(0) \\ L\{ f''(t) \} &= s^2 F(s) - sf(0) - f'(0) \\ \Big( L\{ f(t) \} &= F(s) \Big) \end{aligned} \]
\[ \begin{aligned} f(0) &= \{ f(t) \}_{t = 0} \\ f'(0) &= \left\{ \frac{d f(t)}{dt} \right\}_{t = 0} \\ f'(t) &= \frac{df}{dx}; f''(t) = \frac{d^2f}{dx^2} \end{aligned} \]
I missed after this¶
\[ \begin{aligned} L^{-1} \left[ \frac{F(s)}{s} \right] &= \int\limits_0^t L^{-1} \Big( F(s) \Big) \ dt\\ L^{-1} \left[ \frac{F(s)}{s^2} \right] &= \int\limits_0^t \int\limits_0^t L^{-1} \Big( F(s) \Big) \ dt\\ L^{-1} \left[ \frac{F(s)}{s^n} \right] &= \text{n integrals from } 0 \to t \quad L^{-1} \Big( F(s) \Big) \ dt \end{aligned} \]