Laplace Transform

The LaplaceTransform is given by the expression: L{f(t)} = Integral of (f(t).exp(-s.t)) between the limits of 0 and infinity. The LaplaceTransform of f(t) is also denoted by F(s).

t = time, s = a + i.w, i = sqrt(-1), w = 2.PI.f, f = frequency and 'a' is the abscissa of convergence which is 0 for engineering problems.

Neat things about the LaplaceTransform of interest to engineers is that

differentiation in the time domain corresponds to multiplication by s in the S domain.
integration in the time domain corresponds to division by s in the S domain.
translation in the time domain corresponds to multiplication by an exponential in the s domain.

The LaplaceTransform is used heavily in electronics and electrical circuits to simplify analysis. For example, the response of a simple circuit of a resistor, capacitor and inductor in parallel, subjected to a sinusoidal stimulus is described in the time domain by an integrodifferential equation that is tricky to solve. Transforming the problem into the S domain results in a simple algebraic expression that is simple to solve.

More usefully, to evaluate the overall response of a set of cascaded systems requires convolving the time impulse responses of those systems. In the S domain the overall response can be found by multiplying the transforms of each subsystem - much easier.