# They have the general form they transform functions

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Mathematics. They have the general form: They transform functions of a specific type into functions of another type. Many Engineering problems can be solved easier by first applying suitable integral transforms, then solving the problem in the transformed form, and then ‘transform the result back’ to the original problem.

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Original problem Problem in transformed form Solution of original problem Solution of transformed problem Integral transform Difficult to solve ‘Backward’ integral transform Easier to solve 1.0. Background: Integral transforms
π 2 / ) , ( ist e s t K - = st e s t K - = ) , ( Examples for important integral transforms are: Fourier Transform Laplace Transform Hilbert Transform (and many more) t s s t K - = 1 1 ) , ( π 1.0. Background: Integral transforms

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We will discuss in this module only the Laplace Transform , which has important applications in Electrical Engineering (the others have as well, but you will discuss them, if necessary, when their applications arise). We will in particular focus on its application to solving Ordinary Differential Equations (ODEs) . 1.0. Background: Integral transforms
Example: In the design of RLC circuits , certain ODE’s appear which can be solved by the Laplace Transform technique. We will discuss some examples later. 1.0. Background: Integral transforms ?

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Original problem: solve ODE ODE in transformed form = ALGEBRAIC PROBLEM Solution of ODE Solution of ALGEBRAIC PROBLEM Laplace transform Solve ODE directly: often difficult ! ‘Inverse’ Laplace transform Solve this easier algebraic problem 1.0. Background: Integral transforms So, the goal is to establish the following diagram:
1.1 Introduction to Laplace Transforms Pierre-Simon Laplace 1749 – 1827 French mathematician.

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• Winter '12
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