4.1 INTRODUCTION

The Laplace Transform is a mathematical tool by which time domain is transformed into frequency domain or frequency domain to time domain.

In order to transform a given function of time f(t) into its corresponding Laplace transform first multiply f(t) by e ^?st, where s is a complex number ( s = ? + J ?). Integrate this product with respect to time with a limit from zero to infinity. This integration results in a Laplace transform of f/(t) and this is denoted F( s) or Lf(t).

s is a complex frequency ( ? + J ?).

? is known as a Neper frequency and ? is known as a real or natural frequency.

The term "Laplace transform of f(t)" is used for the letter Lf(t).

Similarly,

The term "L ^?1" is called the inverse Laplace transformation.

The time function f(t) and its Laplace transform F( s) are a transform pair.

4.2 LAPLACE TRANSFORM OF SOME COMMONLY USED FUNCTIONS

1. The Laplace transform of e ^?at is

   
   
   
   

   The inverse Laplace transform of e ^?at is

   
   

2. Similarly,

   
   

   Put a = -1 into Equation (4.2) and find

   
   

3. f(t) = 1 = e ^{0 t}.

   Put a = 0 into Equation (4.2) and

   
   

4. In the function f(t) = e ^?at put a = J ?.

   Hence, f(t) = e ^?J?t.

   By...