# For any function ft the laplace transform is defined

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For any function f(t), the Laplace Transform is defined as 1.1 Introduction to Laplace Transforms ( 29 0 ( ) st f t e f t dt - = So, how exactly is the Laplace Transform defined?

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As this is a definite integral between limits on t, the final result will NOT depend on t. It will, however depend on s. For this reason, [f(t)] is often written as 1.1 Introduction to Laplace Transforms ( 29 0 ( ) st f t e f t dt - = ( ) f s
Example 1.1.1 If f(t) = t, find the Laplace Transform ( 29 [ ] ( ) f t t f s = = 1.1 Introduction to Laplace Transforms Well, in this case, this appears to be a simple integration. Do you remember how to do it ? - = 0 ] [ dt te t st

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( 29 [ ] at f t e = Example 1.1.2 If , find the Laplace Transform Hence, for example with , find 1.1 Introduction to Laplace Transforms ( ) at f t e = - = 0 4 4 ] [ dt e e e st t t 4 = a
Laplace Transforms : An Illustration Consider the function f(t) = t e -st for different values of s (s = 1, 2, 5). 1.1 Introduction to Laplace Transforms

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is given by the area under the curve for the appropriate value of s. 1.1 Introduction to Laplace Transforms [ ] 0 st t t e dt - =
s = 1 : Largish area under curve 1.1 Introduction to Laplace Transforms

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s = 2 : Smaller area under curve 1.1 Introduction to Laplace Transforms
s = 5 : Even smaller area under curve 1.1 Introduction to Laplace Transforms

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The Laplace Transform (area under curve) depends on s. In fact, it is equal to 1/s 2 1.1 Introduction to Laplace Transforms
1.2 Laplace Transforms by Table

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The direct integration approach of section 1.1 will give the Laplace Transform of many functions. Once carried out, these Laplace Transforms can be written in a table.
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