Laplace transforms
is a table with functions of time \(f(t)\) on the left and corresponding Laplace transforms \(L(s)\) on the right. Where applicable, \(s = \sigma + j \omega\) is the Laplace transform variable, \(T\) is the time-domain period, \(\omega_0 2\pi/T\) is the corresponding angular frequency, \(j = \sqrt{-1}\), \(a \in \mathbb{R}^+\), and \(b, t_0 \in \mathbb{R}\) are constants.
| Function of time \(t\) | Function of complex frequency \(s\) |
|---|---|
| \(a_1 f_1(t) + a_2 f_2(t)\) | \(a_1 F_1(s) + a_2 F_2(s)\) |
| \(f(t-t_0)\) | \(F(s) e^{-t_0 s}\) |
| \(f'(t)\) | \(s F(s) - f(0)\) |
| \(\dfrac{d^n f(t)}{d t^n}\) | \(s^n F(s) + s^{(n-1)} f(0) + s^{(n-2)} f'(0) + \cdots + f^{(n-1)}(0)\) |
| \(\displaystyle\int_0^t f(\tau) d\tau\) | \(\dfrac{1}{s} F(s)\) |
| \(t f(t)\) | \(-F'(s)\) |
| \(f_1(t) * f_2(t) = \displaystyle\int_{-\infty}^\infty f_1(\tau) f_2(t - \tau) d\tau\) | \(F_1(s) F_2(s)\) |
| \(\delta(t)\) | \(1\) |
| \(u_s(t)\) | \(1/s\) |
| \(u_r(t)\) | \(1/s^2\) |
| \(t^{n-1}/(n-1)!\) | \(1/s^n\) |
| \(e^{-a t}\) | \(\dfrac{1}{s + a}\) |
| \(t e^{-a t}\) | \(\dfrac{1}{(s+a)^2}\) |
| \(\dfrac{1}{(n-1)!} t^{n-1} e^{-a t}\) | \(\dfrac{1}{(s+a)^n}\) |
| \(\dfrac{1}{a-b}(e^{a t} - e^{b t})\) | \(\dfrac{1}{(s-a)(s-b)}\) (\(a\ne b\)) |
| \(\dfrac{1}{a-b}(a e^{a t} - b e^{b t})\) | \(\dfrac{s}{(s-a)(s-b)}\) (\(a\ne b\)) |
| \(\sin \omega t\) | \(\dfrac{\omega}{s^2 + \omega^2}\) |
| \(\cos \omega t\) | \(\dfrac{s}{s^2 + \omega^2}\) |
| \(e^{a t} \sin \omega t\) | \(\dfrac{\omega}{(s-a)^2 + \omega^2}\) |
| \(e^{a t} \cos \omega t\) | \(\dfrac{s-a}{(s-a)^2 + \omega^2}\) |
Online Resources for Section C.3
No online resources.