Laplace Transform 3 different types of transforms 1

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Laplace Transform

Laplace Transform

3 different types of transforms 1. Laplace-transforms 2. Fourier-transforms 3. Z-transforms

3 different types of transforms 1. Laplace-transforms 2. Fourier-transforms 3. Z-transforms

The “French Newton” Pierre-Simon Laplace • Developed mathematics in astronomy, physics, and statistics •

The “French Newton” Pierre-Simon Laplace • Developed mathematics in astronomy, physics, and statistics • Began work in calculus which led to the Laplace Transform • Focused later on celestial mechanics • One of the first scientists to suggest the existence of black holes

 • The Laplace transform was widely adapted to engineering problems in the last

• The Laplace transform was widely adapted to engineering problems in the last century. * Its utility lies in the ability to convert differential equations to algebraic forms that are more easily solved. • The notation has become very common in certain areas as a form of engineering “language” for dealing with systems.

Steps involved in using the Laplace transform.

Steps involved in using the Laplace transform.

Notation of Laplace Transformation The Laplace Transform of piecewise continuous function, f(t) is denoted

Notation of Laplace Transformation The Laplace Transform of piecewise continuous function, f(t) is denoted as:

Definition of Laplace Transformation

Definition of Laplace Transformation

Another Notation of Laplace Transform

Another Notation of Laplace Transform

Definition and Notation of Laplace Transformation

Definition and Notation of Laplace Transformation

Example-1: • Derive the Laplace transform for unit step function, i. e, f(t) =

Example-1: • Derive the Laplace transform for unit step function, i. e, f(t) = 1. • In short, compute

Example -2. Derive the Laplace transform of the exponential function

Example -2. Derive the Laplace transform of the exponential function

Home work: Derive the Laplace transform of the function

Home work: Derive the Laplace transform of the function

Hint: • Integration by parts to be used twice • Answer:

Hint: • Integration by parts to be used twice • Answer:

Basic Theorems of Linearity

Basic Theorems of Linearity

 • The Laplace transform of a product is not the product of the

• The Laplace transform of a product is not the product of the transforms.

In short,

In short,

Common transform pairs.

Common transform pairs.

The Laplace Transform The above is a statement that f(t) and F(s) are transform

The Laplace Transform The above is a statement that f(t) and F(s) are transform pairs. What this means is that for each f(t) there is a unique F(s) and for each F(s) there is a unique f(t). If we can remember the Pair relationships between approximately 10 of the Laplace transform pairs we can go a long way.

Class work • Find the Laplace transform for the following functions: • 1.

Class work • Find the Laplace transform for the following functions: • 1.

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2.

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