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 • 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 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.
Notation of Laplace Transformation The Laplace Transform of piecewise continuous function, f(t) is denoted as:
Definition of Laplace Transformation
Another Notation of Laplace Transform
Definition and Notation of Laplace Transformation
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
Home work: Derive the Laplace transform of the function
Hint: • Integration by parts to be used twice • Answer:
Basic Theorems of Linearity
• The Laplace transform of a product is not the product of the transforms.
In short,
Common transform pairs.
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.