Laplace Transforms of Unit Step Function:

Unit Step Function ( Heaviside Function ):  The unit step function or Heaviside Function u (t - a) is defined as follows:

Where a ≥ 0; The graph of this function is shown below and where a is the positive constant.

Properties associated with the Unit Step Function:

Proof: (1) Using the definition of Laplace Transform, we have;

     

(2) Heaviside Shifting Theorem:

Where u (t - a) is the unit step function.

Proof: By definition we have;

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