# Laplace Transform (LT)

Laplace transform of x(t) function is

Properties of Laplace Transform:

1. Linearity

1. Time-Reversal : x(tâ†” X(s) then x(â€“tâ†” X(â€“s), ROC = â€“R
2. Shift in S-domain : x(tâ†” X(s) with ROC = R then
3. Time-shifting: It x(t) is input-signal whose ROC is R.

then  then  with ROC = R
4. Differentiation in S-domain: If x(tâ†” X(s) ROC = R, then
5. Differentiation in Time : If  then
6. Integration in Time :  for unilateral LT
7. Frequency Integration :
8. Convolution in Time :  and  with  then
9. Initial and Final Value Theorem : If x(tX(s)

then initial value theorem is x(0) =  s X(s)

Final value theorem is x(âˆž) = s X(s)