Laplace Transform Of Periodic Functions

Laplace Transform Of Periodic Functions :

Definition :

           A function f(t) is said to be periodic , with a period 'l' , if it satisfies the functional equation 

                 f(t±l) = f(t) 

   So, if t>0 , a periodic function f(t) can be written as 

          f(t) = f(t + nl) , n= 0,1,2.. ......(1)

     For example sin t = sin(t+2πn) , n=0,1,2.... is a periodic function with period 2π . 

     In case a function f(t) is periodic , the Laplace transform can be expressed as an integral over one cycle of the function instead of an integral over an infinite range.

           The transform of equation(1) is 

                     ∞

          f̅(p) = ∫ e⁻ᵖᵗ f(t) dt 

                    a

                       l     2l   3l

                 = [ ∫ + ∫ + ∫  .....]e⁻ᵖᵗ f(t) dt

                      0     l    2l

                  ∞ (n+1)l

                = Σ    ∫  e⁻ᵖᵗ f(t) dt   ............(2)  

                  n=0 nl

Now  let us substitute 

               t = τ + nl ,

Where τ is a new variable , so that 

       f(t) = f(τ+nl) = f(τ) .

   We have dt = dτ. In the integration , when t= nl , we have τ = 0 and when t= (n+1)l ,     τ= l.

  Hence from (2) we get 

                ∞     l

      f̅(p) = Σ    ∫ f(τ+nl) e^ -p(τ+nl) dτ

               n=0 0

                ∞             l

             = Σ   e⁻ᵖⁿᵗ ∫ f(τ) e^-pτ dτ .......(3)

               n=0        0

Now 

           ∞

           Σ e⁻ᵖⁿᵗ = 1+e⁻ᵖᵗ + e⁻²ᵖᵗ+ e⁻³ᵖᵗ + ...

         n=0

                       = 1/ 1-e⁻ᵖᵗ , since e⁻ᵖᵗ < 1 .......(4)

   From the relations (3) and (4) , we have 

                                     l

        f̅(p) = 1/ 1-e⁻ᵖᵗ   ∫ e⁻ᵖᵗ f(t) dt .

                                   0

   We have thus established the theorem i.e.

                                     l

      L{f(t)} = 1/ 1-e⁻ᵖᵗ ∫ e⁻ᵖᵗ f(t) dt .

                                    0 

     Which is known as Laplace Transform Of Periodic Function Theorem .

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