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Laplace Transforms

Laplace transforms convert a function f(t) in the time domain into function in the Laplace domain F(s).

$$F(s) = \mathcal{L}\left(f(t)\right) = \int_0^\infty f(t)e^{-s\,t}dt$$

As an example of the Laplace transform, consider a constant c. The function f(t) = c and the following expression is integrated.

$$\mathcal{L}(c)=\int_0^\infty c \, e^{-s\,t} dt = -\frac{c}{s}e^{-s\,t} \biggr\rvert_0^\infty = 0 - \left(-\frac{c}{s} \right) = \frac{c}{s}$$

Mathematicians have developed tables of commonly used Laplace transforms. Below is a summary table with a few of the entries that will be most common for analysis of linear differential equations in this course. Notice that the derived value for a constant c is the unit step function with c=1 where a signal output changes from 0 to 1 at time=0.

f(t) in Time Domain F(s) in Laplace Domain
$$\delta(t)\quad \mathrm{unit \; impulse}$$1
$$S(t) \quad \mathrm{unit \; step}$$$$\frac{1}{s}$$
$$t \quad \mathrm{ramp \; with \; slope = 1}$$$$\frac{1}{s^2}$$
$$t^{n-1}$$$$\frac{(n-1)!}{s^n}$$
$$e^{-b\,t}$$$$\frac{1}{s+b}$$
$$1-e^{-t/\tau}$$$$\frac{1}{s(\tau s + 1)}$$
$$\sin(\omega t)$$$$\frac{\omega}{s^2+\omega^2}$$
$$\cos(\omega t)$$$$\frac{s}{s^2+\omega^2}$$
$$\frac{df}{dt}$$$$sF(s)-f(0)$$
$$\frac{d^nf}{dt^n}$$$$s^n F(s) - s^{n-1} f(0) - s^{n-2}f^{(1)}(0) - \ldots \\ - sf^{(n-2)}(0) - f^{(n-1)}(0)$$
$$\int f(t)$$$$\frac{F(s)}{s}$$
$$f\left(t-t_0\right)S\left(t-t_0\right)$$$$e^{-t_0s}F(s)$$

Note that the functions f(t) and F(s) are defined for time greater than or equal to zero. The next step of transforming a linear differential equation into a transfer function is to reposition the variables to create an input to output representation of a differential equation.