# Properties of Laplace Transform

## Abstract

Similar to the Fourier transform, the Laplace transform has many useful and illustrative properties. Almost all of the Laplace transform properties follow the Fourier counterparts, but there are some subtle differences in some cases mostly relating to initial conditions and to the absence of delta functions when doing time integration. Also beware of negative signs since in the Fourier world we deal with *jω* while in the Laplace world we deal with *s*; for example when inverting *jω* we get \(-j\frac {1}{\omega }\) while when inverting *s* we don’t pick a negative sign; also when squaring *jω* we get − *ω*^{2} while when squaring *s* we simply get *s*^{2}! There are also a couple of new theorems which are the initial and final value theorems. For completeness we cover all properties and thoroughly illustrate them with relevant examples to drive the point home and to get valuable experience in solving similar problems.