# The Laplace Transform¶

This Jupyter notebook is part of a collection of notebooks in the bachelors module Signals and Systems, Communications Engineering, Universität Rostock. Please direct questions and suggestions to Sascha.Spors@uni-rostock.de.

## Summary of Properties, Theorems and Transforms¶

The properties, theorems and transforms of the two-sided Laplace transform as derived in the previous sections are summarized in the following. The corresponding tables serve as a reference for the application of the Laplace transform in the theory of signals and systems. Please refer to the respective sections for details.

### Definition¶

The two-sided Laplace transform and its inverse are defined as

\begin{align} X(s) &= \int_{-\infty}^{\infty} x(t) \, e^{- s t} \; dt \\ x(t) &= \frac{1}{2 \pi j} \int_{\sigma - j \infty}^{\sigma + j \infty} X(s) \, e^{s t} \; ds \end{align}

where $s \in \text{ROC} \{ x(t) \}$.

### Properties and Theorems¶

The properties and theorems of the two-sided Laplace transform are given as

﻿ $x(t)$ $X(s) = \mathcal{L} \{ x(t) \}$ $\text{ROC}$
Linearity $A \, x_1(t) + B \, x_2(t)$ $A \, X_1(s) + B \, X_2(s)$ $\supseteq \text{ROC}\{x_1(t)\} \cap \text{ROC}\{x_2(t)\}$
Real-valued signal $x(t) = x^*(t)$ $X(s) = X^*(s^*)$
Scaling $x(a t)$ $\frac{1}{\lvert a \rvert} X\left( \frac{s}{a} \right)$ $s: \frac{s}{a} \in \text{ROC}\{x(t)\}$
Convolution $x(t) * h(t)$ $X(s) \cdot H(s)$ $\supseteq \text{ROC}\{x(t)\} \cap \text{ROC}\{h(t)\}$
Shift $x(t - \tau)$ $e^{-s \tau} \cdot X(s)$ $\text{ROC}\{x(t)\}$
Differentiation (causal signal) $\frac{d}{dt} x(t)$ $s \cdot X(s) - x(0+)$ $\supseteq \text{ROC}\{x(t)\}$
Integration $\int_{-\infty}^{t} x(t) \; dt$ $\frac{1}{s} \cdot X(s)$ $\supseteq \text{ROC}\{x(t)\} \cap \{s: \Re \{s\} > 0 \}$
Modulation $e^{s_0 t}\cdot x(t)$ $X(s - s_0)$ $s: s - \Re \{s_0\} \in \text{ROC}\{x(t)\}$

where $A, B, s_0 \in \mathbb{C}$, $a \in \mathbb{R} \setminus \{0\}$ and $\tau \in \mathbb{R}$.

### Selected Transforms¶

Two-sided Laplace transforms which are frequently used are given as

$x(t)$ $X(s) = \mathcal{L} \{ x(t) \}$ $\text{ROC}$
$\delta(t)$ $1$ $\mathbb{C}$
$\epsilon(t)$ $\frac{1}{s}$ $\Re \{s\} > 0$
$t \epsilon(t)$ $\frac{1}{s^2}$ $\Re \{s\} > 0$
$e^{- s_0 t} \epsilon(t)$ $\frac{1}{s + s_0}$ $\Re \{s\} > \text{Re}\{-s_0\}$
$\sin(\omega_0 t) \epsilon(t)$ $\frac{\omega_0}{s^2 + \omega_0^2}$ $\Re \{s\} > 0$
$\cos(\omega_0 t) \epsilon(t)$ $\frac{s}{s^2 + \omega_0^2}$ $\Re \{s\} > 0$
$t^n e^{-s_0 t} \epsilon(t)$ $\frac{n!}{(s+s_0)^{n+1}}$ $\Re \{s\} > \text{Re}\{-s_0\}$
$e^{-s_0 t} \cos(\omega_0 t) \epsilon(t)$ $\frac{s + s_0}{(s+s_0)^2 + \omega_0^2}$ $\Re \{s\} > \Re \{-s_0\}$
$e^{-s_0 t} \sin(\omega_0 t) \epsilon(t)$ $\frac{\omega_0}{(s+s_0)^2 + \omega_0^2}$ $\Re \{s\} > \Re \{-s_0\}$
$t \cos(\omega_0 t) \epsilon(t)$ $\frac{s^2 - \omega_0^2}{(s^2 + \omega_0^2)^2}$ $\Re \{s\} > 0$
$t \sin(\omega_0 t) \epsilon(t)$ $\frac{2 \omega_0 s}{(s^2 + \omega_0^2)^2}$ $\Re \{s\} > 0$

where $s_0 \in \mathbb{C}$, $\omega_0 \in \mathbb{R}$ and $n \in \mathbb{N}$. More one- and two-sided transforms may be found in the literature or online.