Editing Laplace transform (section)

=== Evaluating improper integrals ===
Let <math>\mathcal{L}\left\{f(t)\right\} = F(s)</math>. Then (see the table above)

<math display="block">\partial_s\mathcal{L} \left\{\frac{f(t)} t \right\} = \partial_s\int_0^\infty \frac{f(t)}{t}e^{-st}\, dt = -\int_0^\infty f(t)e^{-st}dt = - F(s) </math>

From which one gets:

<math display=block>
 \mathcal{L} \left\{\frac{f(t)} t \right\} = \int_s^\infty F(p)\, dp.</math>

In the limit <math>s \rightarrow 0</math>, one gets
<math display=block>\int_0^\infty \frac{f(t)} t \, dt = \int_0^\infty F(p)\, dp,</math>
provided that the interchange of limits can be justified. This is often possible as a consequence of the [[Final value theorem#Final Value Theorem for improperly integrable functions (Abel's theorem for integrals)|final value theorem]]. Even when the interchange cannot be justified the calculation can be suggestive. For example, with {{math|''a'' ≠ 0 ≠ ''b''}}, proceeding formally one has
<math display=block>
\begin{align}
\int_0^\infty \frac{ \cos(at) - \cos(bt) }{t} \, dt 
&=\int_0^\infty \left(\frac p {p^2 + a^2} - \frac{p}{p^2 + b^2}\right)\, dp \\[6pt]
&=\left[ \frac{1}{2}  \ln\frac{p^2 + a^2}{p^2 + b^2} \right]_0^\infty = \frac{1}{2} \ln \frac{b^2}{a^2} = \ln \left| \frac {b}{a} \right|.
\end{align}
</math>

The validity of this identity can be proved by other means. It is an example of a [[Frullani integral]].

Another example is [[Dirichlet integral]].