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:ID:       e73baa24-1a29-4f35-9d3d-0fad4a3a8e59
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#+title: Laplace Transform
#+author: Preston Pan
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* Introduction
The dual-edge Laplace Transform is defined as:
\begin{align}
\label{Laplace Transform}
F(s) = \int_{-\infty}^{\infty}f(t)e^{-st}dt
\end{align}
when $s$ is complex (which it usually is), the [[id:262ca511-432f-404f-8320-09a2afe1dfb7][Fourier Transform]] can be taken to be a special case of the
dual-edge Laplace Transform. One can think of this as analyzing the complex exponential domain, rather than just
the frequency domain (imaginary exponential domain). Now, multiplying the signal by the [[id:53dade38-21e1-4fa9-a552-6ceab8a75f82][Heaviside Step Function]]:
\begin{align}
\label{Step Function}
F(s) = \int_{-\infty}^{\infty}H(t)f(t)e^{-st}dt = \int_{0}^{\infty}f(t)e^{-st}dt
\end{align}
gives you the conventional Laplace Transform.
** Usage
The Laplace Transform is primarily used for analyzing [[id:32a116d9-b813-4b5a-a2e8-6dd7b767ec16][linear differential equations]] as it converts these equations into
algebraic equations. The inverse Laplace Transform is complicated, and is therefore not used often. Instead, Laplace
Transform tables are used in order to convert back into the time-domain. Taking the Laplace transform of all terms in
a linear differential equation will yield this result. One of the simplest differential equations that the Laplace
Transform can solve is the [[id:6dbe2931-cc18-48fc-8cc1-6c71935a6be3][mass-spring system]], and it also generally has applications in [[id:a7d6d6e9-9f7a-446f-b6af-255c802f86b1][circuit analysis]].