Context: when dealing with control systems, taking derivatives and integration is common, but is too error-prone, computation-intensive, and complicated. The Laplace transform of a function will allow us to deal with the system in frequency domain. As a result, derivation and integration turns into multiplication and division of , the complex frequency.
The definition of full Laplace transform is given as:
The function is integrated from time being to . However, we don’t care about what happens the time far far before. We care about the system after time 0. Thus we have the half Laplace transform:
Where is a “infinite spike” at , and is the unit step function.