- The Laplace transform is a mathematical tool used to transform a function of time into a function of a complex variable s, which simplifies solving differential equations and other mathematical problems
- The Laplace transform of elementary functions, such as polynomials, exponentials, trigonometric functions, and unit step functions, can be found using standard formulas and tables
- The Laplace transform has linearity, first and second shifting, and scaling properties that allow the transform of more complex functions to be computed
- The Laplace transform of derivatives and integrals can be derived from the definition of the transform, and can be used to solve differential equations with initial conditions
- Improper integrals can be evaluated using the Laplace transform by taking the limit of the transform as the upper limit of integration approaches infinity
Important Formulas to Remember in these topic
Linearity property:
$$L\{af(x)+bg(x)\} = aL\{f(x)\}+bL\{g(x)\}$$
First shifting property:
$$L\{f(x-a)u(a)\} = e^{-as}F(s)$$
Change of scale property:
$$L\{f(cx)\} = \frac{1}{c}F\left(\frac{s}{c}\right)$$
Multiplication and division by $t$:
$$L\{tf(t)\} = -\frac{d}{ds}F(s)$$
$$L\{f'(t)\} = sF(s)-f(0)$$
Unit step function:
$$u(t-a) = \begin{cases} 0, & t < a \\ 1, & t \geq a \end{cases}$$
LT of unit step function:
$$L\{u(t-a)\} = \frac{e^{-as}}{s}$$
Second shifting property:
$$L\{f(t-a)u(t-a)\} = e^{-as}L\{f(t)\}$$
Evaluation of improper integrals:
$$L\left\{\int_{0}^{\infty} f(t) dt\right\} = \int_{0}^{\infty} e^{-st}f(t) dt$$