Properties of Laplace and Z-Transform

1) Linearity

For Laplace

If   with ROC R1 and   with ROC R2,

then

  with ROC containing .

The ROC of X(s) is at least the intersection of  R1 and R2,which could be empty,in which case x(t) has no Laplace transform.

For z-transform :

If  with ROC = R1 and with ROC = R2

then

  with ROC containing .