Electrical Engineering ⇒ Topic : Equation of Laplace Transform

LAPLACE TRANSFORM

Though the Fourier transform can be applied to a large variety of functions, due to the restriction imposed by Eq. (1)

                                          ............ (1)

it is impossible to apply this transform to many functions, e.g., ramp, parabolic, etc. because the integral is not converging. So these kind of functions are not Fourier transformable. It is possible to handle such kind of functions by Laplace transform. The Laplace transform of any function in time domain t is given by

                  ........... (2)

                                                                    ............ (3)

The notation LT indicates "the Laplace transform of f(t)" In Eq. (2) the lower limit is taken as 0 instead of -∞ because the convergence factor of e-σt will diverge for t →-∞. All information contained in f(t) before t = 0 is being ignored by the transformation in Eq. (2). Therefore, Eq. (2) represents one side or unilateral Laplace transform because the limits are 0 and ∞. A function is said to be Laplace transformable when

                                                       .............. (4)

is satisfied for some limit σ, where ,σ is the real part of s. Here, f(t) is defined in the interval O ≤ t ≤ ∞.