This document provides an overview of Laplace transforms, a key mathematical concept used in engineering and differential equations. It includes definitions, properties, theorems such as the first and second shifting theorems, and techniques for evaluating integrals and functions using Laplace transforms. The content also addresses applications related to periodic functions and various elementary functions.

1. GUJARAT
TECHNOLOGICAL
UNIVERSITY
Subject : Advanced
Engineering Mathematics
Sem-3 (Chemical)
Topic : Laplace Transforms

2. Topics
 Definition of Laplace Transform
 Linearity of the Laplace Transform
 Laplace Transform of some Elementary Functions
 First Shifting Theorem
 Inverse Laplace Transform
 Differentiation & Integration of Laplace Transform
 Evaluation of Integrals By Laplace Transform
 Convolution Theorem
 Application to Differential Equations
 Laplace Transform of Periodic Functions
 Unit Step Function
 Second Shifting Theorem
 Dirac Delta Function

3. Definition of Laplace Transform
 Let f(t) be a given function of t defined for all
then the Laplace Transform of f(t) denoted by L{f(t)}
or or F(s) or is defined as
provided the integral exists, where s is a parameter real
or complex.
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0
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4. Linearity of the Laplace Transform
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constants a and b
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5. Laplace Transform of some Elementary
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11. Inverse Laplace Transform
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12. Differentiation & Integration of Laplace
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15. Evaluation of Integrals By Laplace
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16. Convolution Theorem
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18. Laplace Transform of Periodic Functions
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21. Second Shifting Theorem
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