Unilateral Laplace Transform

The unilateral Laplace transform is a special case of the bilateral Laplace transform in which all signals are restricted to being causal, such that:

$$X(s)={\int }_{0^{-}}^{\infty }x(t)e^{-st}dt$$

Using $0^{-}$ as the lower limit of integration allows the inclusion of an impulse function at $t=0$ but also allows us to use initial conditions at $0^{-}$.

The Z-transform version of this is:

$$X[z]=\sum _{n=0^{-}}^{\infty }x[n]z^{-n}$$