$$\require{cancel}$$

# 14.2: Table of Laplace Transforms

It is easy, by using Equation 14.1.2, to derive all of the transforms shown in the following table, in which t > 0. (Do it!)

\begin{array}{c c c c}
y(t) &&& \bar{y}(s) \\
1 &&& 1/s \\
t &&& 1/s^2 \\
\frac{t^{n-1}}{(n-1)!} &&& 1/s^n \\
\sin at &&& \frac{a}{s^2+a^2}\\
\cos at &&& \frac{s}{s^2+a^2}\\
\sinh at &&& \frac{a}{s^2-a^2}\\
\cosh at &&& \frac{s}{s^2-a^2}\\
e^{at} &&& \frac{1}{s-a}\\
\end{array}

This table can, of course, be used to find inverse Laplace transforms as well as direct transforms. Thus, for example, $$\textbf{L}^{-1} \frac{1}{s-1}=e^t$$. In practice, you may find that you are using it more often to find inverse transforms than direct transforms.

These are really all the transforms that it is necessary to know – and they need not be committed to memory if this table is handy. For more complicated functions, there are rules for finding the transforms, as we shall see in the following sections, which introduce a number of theorems. Although I shall derive some of these theorems, I shall merely state others, though perhaps with an example. Many (not all) of them are straightforward to prove, but in any case I am more anxious to introduce their applications to circuit theory than to write a formal course on the mathematics of Laplace transforms.

After you have understood some of these theorems, you may well want to apply them to a number of functions and hence greatly expand your table of Laplace transforms with results that you will discover on application of the theorems.