[f(t)] F(s) f(t)e-sdt 0 (6-27) = = +

The integral of Eq. (6-27) cannot be evaluated for all f(t)'s but when it can, it establishes a unique pair of functions, f(t) in the time domain and F(s) in the s domain. It is conventional to use capital letters for s functions and lowercase for t functions. Since comprehensive tables of such Laplace transforms have been published, it is rarely necessary for a transform user to actually work out Eq. (6-27). Table 6-l is a brief table adequate for the purposes of this book and most practical applications. When we use Laplace transforms to solve differential equations, we must transform entire equations, not just isolatedf(t) functions, so several theorems necessary for this will now be stated, without proof.