stylistic:style_transfer
Evaluating improper integrals

Let L { f (t) } = F (s) {\displaystyle {\mathcal {L}}\left\{f(t)\right\}=F(s)}. Then (see the table above)

∂ s L { f (t) t } = ∂ s ∫ 0 ∞ f (t) t e − s t d t = − ∫ 0 ∞ f (t) e − s t d t = − F (s) {\displaystyle \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)}

From which one gets:

L { f (t) t } = ∫ s ∞ F (p) d p. {\displaystyle {\mathcal {L}}\left\{{\frac {f(t)}{t}}\right\}=\int _{s}^{\infty }F(p)\,dp.}

In the limit s → 0 {\displaystyle s\rightarrow 0}, one gets ∫ 0 ∞ f (t) t d t = ∫ 0 ∞ F (p) d p, {\displaystyle \int _{0}^{\infty }{\frac {f(t)}{t}}\,dt=\int _{0}^{\infty }F(p)\,dp,} provided that the interchange of limits can be justified. This is often possible as a consequence of the final value theorem. Even when the interchange cannot be justified the calculation can be suggestive. For example, with a ≠ 0 ≠ b, proceeding formally one has ∫ 0 ∞ cos (a t) − cos (b t) t d t = ∫ 0 ∞ (p p 2 + a 2 − p p 2 + b 2) d p = 0 ∞ = 1 2 ln b 2 a 2 = ln | b a |. {\displaystyle {\begin{aligned}\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\\&=\left_{0}^{\infty }={\frac {1}{2}}\ln {\frac {b^{2}}{a^{2}}}=\ln \left|{\frac {b}{a}}\right|.\end{aligned}}}

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

Another example is Dirichlet integral.