MATH 223
Lecture 01
Laplace Transformation
Laplace Transformation is an integral transform. French mathematician Pierre-Simon Laplace developed this method while solving an improper integral in conventional methods. The notation for Laplace transformation is \(\mathcal{L} \{F(t)\} = f(s)\). Here \(F(t)\) is an input function and \(f(s)\) is an output function.
When trying to analyze a given complex function or while performing the functional analysis (maximum value, minimum value, critical value etc) sometimes it becomes very difficult or time-consuming or sometimes it is very difficult to represent the values in the present in the graph.
In these cases, we need to transform the function so that it becomes easier to analyze the functional properties. There are various types of transformation e.g. Fourier transforms, Laplace transforms, Z-transforms.
Laplace Transform is very important because when we transform a function using Laplace transform we get an algebraic function. It is also very useful to solve ODE, PDEs as this method is easier, simpler and less time-consuming compared to the conventional methods.
Definition: If
\(F(t), ~ t > 0\)
then the Laplace transform of
\(F(t)\)
is given by,
$$ \mathcal{L} \left\{ F(t) \right\}= \int^\infty_0 e^{- st} ~ F(t) ~dt = f(s )$$
This is an improper integral. Here, \(F(t)\) is the given function where the variable is \(t\) and \(f(s)\) is the transformed function the variable in this transformed function is \(s\).
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