Inverse Laplace Transform: As mentioned earlier, inverse Laplace transform is calculated by partial fraction method rather than complex integration evaluation. Let F(s) is the Laplace transform of f(t) then the inverse Laplace transform is denoted as, The F(s), in partial fraction method, is written in the form as, where N(s) = Numerator polynomial in s […]

Convolution Theorem: The convolution theorem of Laplace transform states that, let f1 (t) and f2 (t) are the Laplace transformable functions and F1 (s), F2 (s) are the Laplace transforms of f1 (t) and f2 (t) respectively. Then the product of F1 (s) and F2 (s) is the Laplace transform of f(t) which is obtained […]

Laplace Transform of Periodic Function: Consider a Laplace Transform of Periodic Function of time period T satisfying the condition, where n is positive or negative integer. The Laplace transform of such periodic function is given by , where F1 (s) is the Laplace transform of the first cycle of the periodic function. Proof: Consider the […]

Laplace Transform of Standard Functions: The Laplace Transform of Standard Functions is given by Step Function: The unit step function is, If there is step of amplitude A i.e. f(t) = A u(t) then its Laplace is given by A/s. If the unit step is delayed by T instants then, From the shifting theorem of […]

Relationships Between Standard Time Function: The Relationships between Standard Time Function contains two parts, namely Relation Between Unit Step and Unit Ramp Relation Between Unit Step and Unit Impulse 1. Relation Between Unit Step and Unit Ramp: The unit step is given by, while the unit ramp is given by, Differentiating ramp function with respect […]

Laplace Transform of Impulse Function: The Laplace Transform of Impulse Function is a function which exists only at t = 0 and is zero, elsewhere. The impulse function is also called delta function. The unit impulse function is denoted as δ(t). The amplitude of impulse function is infinitely large at t = 0, but for […]