Transforms - Definition

 
   
     
       

Laplace Transform

       

Definition of Laplace Transform

       

          The Laplace Transform is an integral transform used to convert functions from the time domain into the complex frequency domain. It is mainly used for solving linear differential equations.        

        Laplace Transform Definition Formula.      
     
       
         

Definition:

         
           

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

         
         

Key Properties:

         
               
  • Linearity: \( \mathcal{L}\{af + bg\} = aF + bG \)
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  • Transforms derivatives into algebraic terms.
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  • Defined for functions where exponential order convergence exists.
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Applications:

         
               
  • Control Systems (transfer functions)
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  • Solving ODEs with initial conditions
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  • Circuit analysis (voltage/current equations)
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