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• the direct integration approach of section 1.1

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Unformatted text preview: • 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. • Try to verify a few of them yourself (we will also try some in the Tutorials)! More: • Maths formula Tables (page 13), • HELM 20.2 (page 5) 1.2 Laplace Transforms by Table 1.2 Laplace Transforms by Table 1.2 Laplace Transforms by Table 1.2 Laplace Transforms by Table 1.2 Laplace Transforms by Table 1.2 Laplace Transforms by Table Example 1.2.1 • Use the tables to find • i) • ii) 4 t 1.2 Laplace Transforms by Table [ ] sin 5 t • Note that the Laplace Transform is a linear process i.e. • and [ ] [ ] [ ] ( ) ( ) ( ) ( ) f t g t f t g t + = + 1.2 Laplace Transforms by Table [ ] [ ] ( ) ( ) k f t k f t = Do we always have to calculate the Laplace transform of a function by integration? • More generally • These linearity properties can be used in the same way as for derivatives and integrals. • They help us to calculate Laplace transforms for some of the more complicated expressions by just using the elementary components which we find in the standard Laplace transform tables and combining them linearly. 1.2 Laplace Transforms by Table [ ] [ ] [ ] 1 2 1 2 ( ) ( ) ( ) ( ) k f t k g t k f t k g t + = + Example 1.2.2 • Use the tables and linearity properties to find 1.2 Laplace Transforms by Table 5 2 4 sin 3 7 t t e t e- - + Basic Trig/Hyp Polytrig Heavy Exptrig Next Monday: • Inverse Laplace Transform • The Laplace Transform of Derivatives and Integrals • Convolutions and the Laplace Transform...
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• The direct integration approach of section 1.1 will...

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