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Basic Theorems of Laplace Transform

  1. Linearity Theorem If Laplace transform of the functions f1(t) and f2(t) are F1(s) and F2(s) respectively, then Laplace transform of the functions [K1 f1(t) + K2 f2(t)] will be [K1 F1(s) + K2 F2(s)].
     
    [K1 f1(t) + K2 f2(t)] = [K1 F1(s) + K2 F2(s)]
     
    where, K1 and K2 are constants.
  2. Scaling Theorem If Laplace transform of f (t) is F(s), then
     
    [f(Kt)] = Description: 4782.png, where K is a constant and K > 0.
  3. Time Differentiation Theorem If Laplace transform of f (t) is F(s), then,
     
    Description: 4789.png
  4. Frequency Differentiation Theorem If Laplace transform of f (t) is F(s), then,
     
    Description: 4795.png
  5. Time Integration Theorem If Laplace transform of f (t) is F(s), then,
     
    Description: 4801.png
     
    In general, for nth order integration,
     
    Description: 4807.png
  6. Shifting Theorem The shifting may be done with respect to time or frequency.
    1. Time Shifting Theorem
       
      If Laplace transform of f (t) is F (s), then
       
      Description: 4813.png
    2. Frequency Shifting Theorem
       
      If Laplace transform of f(t) is F(s), then
       
      Description: 4819.png
  7. Initial Value Theorem If the Laplace Transform of f (t) is F (s) and the first derivative of f (t) is Laplace transformable, then, the initial value of f (t) is,
     
    Description: 4831.png
  8. Final Value Theorem If a function f (t) and its derivatives are Laplace transformable, then the final value of f (t) is,
     
    Description: 4885.png




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