Laplace Transform and Properties

A set of vocabulary-style flashcards covering the Laplace Transform table entries and properties based on the lecture notes from Bataan Heroes Memorial College.

$\delta(t)$ (Time Domain)

$1$ (Frequency Domain)

$u(t)$ (Time Domain)

$\frac{1}{s}$ (Frequency Domain)

$t u(t)$ (Time Domain)

$\frac{1}{s^2}$ (Frequency Domain)

$t^n u(t)$ (Time Domain)

$\frac{n!}{s^{n+1}}$ (Frequency Domain)

$e^{-at} u(t)$ (Time Domain)

$\frac{1}{s+a}$ (Frequency Domain)

$\sin(\omega t) u(t)$ (Time Domain)

$\frac{\omega}{s^2 + \omega^2}$ (Frequency Domain)

$\cos(\omega t) u(t)$ (Time Domain)

$\frac{s}{s^2 + \omega^2}$ (Frequency Domain)

Linearity (Property)

$Ax_{(1)}(t) + Bx_{(2)}(t)$ in the time domain corresponds to $AX_{(1)}(s) + BX_{(2)}(s)$ in the frequency domain.

Frequency Shifting (Property)

$e^{-at} x(t)$ in the time domain corresponds to $X(s + a)$ in the frequency domain.

Time Shifting (Property)

$x(t - a)$ in the time domain corresponds to $X(s) e^{-as}$ in the frequency domain.

Frequency Differentiation (Property)

$t^n x(t)$ in the time domain corresponds to $(-1)^n \frac{d^n X(s)}{ds^n}$ in the frequency domain.

Time Differentiation (First Derivative)

$\frac{dx(t)}{dt}$ in the time domain corresponds to $sX(s) - x(0^{-})$ in the frequency domain.

Time Differentiation (Second Derivative)

$\frac{d^2 x(t)}{dt^2}$ in the time domain corresponds to $sX(s) - sx(0) - x'(0)$ in the frequency domain.

Convolution (Property)

$x_1(t) * x_2(t)$ in the time domain corresponds to $X_1(s) X_2(s)$ in the frequency domain.

Time Integration (Property)

$\int_0^t x(\tau) d\tau$ in the time domain corresponds to $\frac{1}{s} X(s)$ in the frequency domain.

Final Value Theorem

Identifies the value of $x(\infty)$ as $\lim_{s \to 0} sX(s)$.

Initial Value Theorem

Identifies the value of $x(0)$ as $\lim_{s \to \infty} sX(s)$.