Mathquery

Unit Step Function :         A unit step function is a function which has a zero value for t≤t₀ and then rises instantaneously to a sustained unity value.  This is shown in the figure below. In electric circuit , and in mechanical system it could represent a force suddenly impressed on the system .            This unit step function starting at zero time will be designated by u(t) , and that starting at time t₀ is u(t-t₀) . Thus                           { 0 for 0<t≤t₀            u(t-t₀) ={                           { 1 for t>t₀               .....(1)    The Laplace transform of unit step function is                         ∞    L{u(t-t₀)} = ∫ e⁻ᵖᵗ u(t-t₀) dt                         0                        ∞                           ∞                     = ∫ e⁻ᵖᵗ u(t-t₀) dt + ∫ e⁻ᵖᵗ u(t-t₀) dt                        0                           t₀                       ∞                               ∞                    = ∫ e⁻ᵖᵗ dt =