I am studying this question for my finals revision and I'm lost on how to start it, can anyone suggest something? Probably pretty simple but I've hit a dead end.
Here's the question:
If $F_i(t)$ and $f_i(s)$ are Laplace pairs, show that:
$$\mathcal L\left\{\sum_{i=1}^n F_i(t)\right\} = \sum_{i=1}^n \mathcal L\left\{F_i\right\}= \sum_{i=1}^n f_i(s)$$
Just need to be pointed in the right direction for the proof. Thank you for your time for reading this.
It comes from the linearity of the integral \begin{align} \mathcal L\left\{\sum_{i=1}^n F_i(t)\right\} &=\int_0^\infty\left(\sum_{i=1}^n F_i(t)\right)\mathrm e^{st}\mathrm d t\\ &=\int_0^\infty\left(F_1(t)+\cdots+F_n(t)\right)\mathrm e^{st}\mathrm d t\\ &=\int_0^\infty F_1(t)\mathrm e^{st}\mathrm d t+\cdots+\int_0^\infty F_n(t)\mathrm e^{st}\mathrm d t\\ &=\mathcal L\left\{F_1(t)\right\}+\cdots+\mathcal L\left\{ F_n(t)\right\}\\ &=\sum_{i=1}^n\mathcal L\left\{F_i(t)\right\}\\ &=\sum_{i=1}^n f_i(s) \end{align}