Unit Impulse Function

Unit Impulse Function 

              The above figure shows a function which has a zero value when t is negative , rises instantaneously to a value 1/t₀ when 0<t<t₀ , and is zero thereafter.

                 If we let t₀--> 0 , this function  tends towards infinity as t₀--> 0, but width decreases inversely with the magnitude ; hence area under the curve remains finite .

The unit impulse function is represented by δ(t) .

 Among Physicists, the unit impulse function is referred to as the 'Dirac δ-function ' after the name of great Physicist Dirac , who is the first to use this function in systematic manner .  The Dirac δ - function at the point t=a as in above figure is represented by δ(t-a) . Thus 

                     {0 for t < a

        δ(t-a) ={ 1/t₀ for a<t<a+t₀ 

                     {0 for t> a+t₀              .......(1)

Where t₀ --> 0 in the limit . 

     The Laplace transform of δ(t-a) is 

                            ∞

        L{δ(t-a)} = ∫ e⁻ᵖᵗ δ(t-a) dt 

                           0

         a                                   a+t₀

      = ∫ e⁻ᵖᵗ δ(t-a) dt + lim   ∫ e⁻ᵖᵗ δ(t-a) dt

         0                          t₀-->0 a

              ∞

          +  ∫ e⁻ᵖᵗ δ(t-a) dt 

            a+t₀

                             a+t₀

            = lim 1/t₀  ∫  e⁻ᵖᵗ dt 

                              a

                                             a+t₀

            = lim 1/t₀ [-1/p e⁻ᵖᵗ ]

                                              a

          = e⁻ᵖᵃ/p  lim   1-e⁻ᵖᵗ / t₀ = e⁻ᵖᵃ  .....(2)

                        t₀-->0

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