- The Laplace transform has several properties, such as the shifting theorem, scaling property, and multiplication and division by s, that can be used to simplify computation of the transform of more complex functions
- The shifting theorem allows the Laplace transform of a function to be shifted in time by a fixed amount, while the scaling property allows the transform to be scaled by a constant factor
- Partial fraction decomposition can be used to compute the Laplace transform of rational functions, which can then be inverted using the inverse Laplace transform
- The convolution theorem states that the Laplace transform of the convolution of two functions is equal to the product of their Laplace transforms, which can be used to solve differential equations
Important Formulas to Remember in these topic
Shifting theorem:
$$L\{f(t-a)u(t-a)\} = e^{-as}F(s)$$
Change of scale property:
$$L\{f(cx)\} = \frac{1}{c}F\left(\frac{s}{c}\right)$$
Multiplication and division by $s$:
$$L\{t f(t)\} = -\frac{d}{ds}F(s)$$
$$L\{f'(t)\} = sF(s)-f(0)$$
ILT by using partial fractions:
$$F(s) = \frac{P(s)}{Q(s)} = \frac{A_1}{s-a_1} + \frac{A_2}{s-a_2} + \dots + \frac{A_n}{s-a_n}$$
$$f(t) = \sum_{i=1}^{n} A_i e^{a_it}$$
Convolution theorem:
$$L\{f(t) * g(t)\} = F(s)G(s)$$