Tool:       The following list of identities allows one to generate complicated Laplace transforms from basic Laplace transforms. Proofs follow from the definition of the Laplace transform:

Other terms used are , which is the impulse (or delta) function, and u(t), which is the unit step function.

    with   ,              

List:                   Name                           t-domain                      s-domain          Condition                   

        definition                                                                                 

        linearity                                                

        delay                                                                a ≥ 0

        multiply by t                                                                    

        multiply by e−at                                                           a ≥ 0

        divide by t                                                                     

        derivative                                                          

        nth derivative                                    

        integral                                                                         

        time scaling                                                                       a ≥ 0

Ref:          James A. Nilsson, Susan A. Riedel, Electric Circuits, 8th Ed., Upper Saddle River, NJ: Prentice Hall, 2007.