The Inverse Laplace Transform

Given $ F(s) $, how do we transform it back to the time domain and obtain the corresponding $ f(t) $ ? By matching entries in Table 15.2, we avoid using Eq. (15.5) to find $ f(t) $. Suppose $ F(s) $ has the general form of $$F(s)=\frac{N(s)}{D(s)} \tag{1}$$ where $ N(s) $ is the numerator polynomial and $ D(s) $ is the denominator polynomial. The roots of $ N(s)=0 $ are called the zeros of $ F(s) $, while the roots of $ D(s)=0 $ are the poles of $ F(s) $. Although Eq. (1) is similar in form to Transfer Function, here $ F(s) $ is the Laplace transform of a function, which is not necessarily a transfer function. We use partial fraction expansion to break $ F(s) $ down into simple terms whose inverse transform we obtain from Table 2. Thus, finding the inverse Laplace transform of $ F(s) $ involves two steps.
Steps to Find the Laplace Transform:
1. Decompose $ F(s) $ into simple terms using partial fraction expansion.
2. Find the inverse of each term by matching entries in Table 2.
Let us consider the three possible forms F(s) may take and how to apply the two steps to each form.

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