LAPLACE TRANSFORM | MATHEMATICS | LECTURE 01 | Important Formulae | All University

Introduction

This tutorial provides a comprehensive guide to understanding the Laplace Transform, a fundamental concept in mathematics, particularly in engineering and physics. The Laplace Transform is used to transform differential equations into algebraic equations, making it easier to analyze linear time-invariant systems. This guide will cover important formulae and key applications of the Laplace Transform.

Step 1: Understanding the Basics of Laplace Transform

• The Laplace Transform converts a time-domain function ( f(t) ) into a complex frequency-domain function ( F(s) ).

• The formula for the Laplace Transform is given by:

  [ F(s) = \int_0^{\infty} e^{-st} f(t) dt ]

• Here, ( s ) is a complex number, usually expressed as ( s = \sigma + j\omega ), where ( j ) is the imaginary unit.

Step 2: Important Properties of Laplace Transform

Familiarize yourself with these key properties which simplify calculations:

1. Linearity:

   • If ( a ) and ( b ) are constants, then:

     [ L{af(t) + bg(t)} = aL{f(t)} + bL{g(t)} ]

2. Time Shifting:

   • If ( f(t) ) is shifted by ( t_0 ):

     [ L{f(t - t_0)u(t - t_0)} = e^{-st_0}F(s) ]

3. Frequency Shifting:

   • If you multiply by an exponential:

     [ L{e^{at}f(t)} = F(s - a) ]

4. Differentiation:

   • For the first derivative:

     [ L{f'(t)} = sF(s) - f(0) ]

5. Integration:

   • For the integral of a function:

     [ L\left{\int_0^t f(\tau) d\tau\right} = \frac{F(s)}{s} ]

Step 3: Common Laplace Transforms

Memorize these common transforms to facilitate your calculations:

• ( L{1} = \frac{1}{s} )
• ( L{t^n} = \frac{n!}{s^{n+1}} )
• ( L{e^{at}} = \frac{1}{s - a} )
• ( L{\sin(at)} = \frac{a}{s^2 + a^2} )
• ( L{\cos(at)} = \frac{s}{s^2 + a^2} )

Step 4: Applications of Laplace Transform

• Solving Ordinary Differential Equations: Use Laplace Transform to convert ODEs into algebraic equations, solve for ( F(s) ), and then apply the inverse transform to find ( f(t) ).
• Control Systems: Analyze system stability and response by working in the frequency domain.
• Electrical Engineering: Analyze circuits with time-varying components using Laplace Transforms.

Conclusion

The Laplace Transform is a powerful mathematical tool that simplifies the process of solving differential equations and analyzing dynamic systems. By understanding its properties, common transforms, and applications, you can effectively use it in engineering and mathematical contexts. To further your learning, practice solving different types of differential equations using the Laplace Transform and explore its applications in real-world engineering problems.