# inverse Laplace transform of derivatives

It may be shown that the Laplace transform   $F(s)=\int_{0}^{\infty}e^{-st}f(t)\,dt$  is always differentiable   and that its derivative can be formed by differentiating under the integral sign (http://planetmath.org/DifferentiationUnderIntegralSign), i.e. one has

 $F^{\prime}(s)=\int_{0}^{\infty}\frac{\partial(e^{-st}f(t))}{\partial s}\,dt=% \int_{0}^{\infty}e^{-st}(-t)f(t)\,dt.$

This gives the rule

 $\displaystyle\mathcal{L}^{-1}\{F^{\prime}(s)\}=-tf(t).$ (1)

Applying (1) to $F^{\prime}(s)$ instead of $F(s)$ gives

 $\mathcal{L}^{-1}\{F^{\prime\prime}(s)\}=t^{2}f(t).$

Continuing this way we can obtain the general rule

 $\displaystyle\mathcal{L}^{-1}\{F^{(n)}(s)\}=(-1)^{n}t^{n}f(t),$ (2)

or equivalently

 $\displaystyle\mathcal{L}\{t^{n}f(t)\}=(-1)^{n}\cdot\frac{d^{n}\mathcal{L}\{f(t% )\}}{ds^{n}},$ (3)

for any  $n=1,\,2,\,3,\,\ldots$ (and of course for  $n=0$).

 $J_{0}(t):=\sum_{m=0}^{\infty}\frac{(-1)^{m}}{(m!)^{2}}\left(\frac{t}{2}\right)% ^{2m},$

which is the solution $y(t)$ of the Bessel’s equation

 $\displaystyle ty^{\prime\prime}(t)+y^{\prime}(t)+ty(t)=0$ (4)

satisfying the initial condition  $y(0)=1$.  The equation implies that  $y^{\prime}(0)=0$.

 $-\frac{d\mathcal{L}\{y^{\prime\prime}(t)\}}{ds}+\mathcal{L}\{y^{\prime}(t)\}-% \frac{d\mathcal{L}\{y(t)\}}{ds}=0.$

Using here twice the rule 5 in the parent (http://planetmath.org/LaplaceTransform) entry gives us

 $-\frac{d(s^{2}Y(s)-s)}{ds}+sY(s)-1-\frac{dY(s)}{ds}=0,$

which is simplified to

 $(s^{2}+1)\frac{dY}{ds}+sY=0,$

i.e. to

 $\frac{dY}{Y}=-\frac{s\,ds}{s^{2}+1}.$

Integrating this gives

 $\ln Y=-\frac{1}{2}\ln(s^{2}+1)+\ln C=\ln\frac{C}{\sqrt{s^{2}+1}},$

i.e.

 $Y(s)=\frac{C}{\sqrt{s^{2}+1}}.$

The initial condition enables to justify that the integration constant $C$ must be 1.  Thus we have the result

 $\mathcal{L}\{J_{0}(t)\}=\frac{1}{\sqrt{s^{2}+1}}.$

## References

• 1 K. Väisälä: Laplace-muunnos.  Handout Nr. 163. Teknillisen korkeakoulun ylioppilaskunta, Otaniemi, Finland (1968).
Title inverse Laplace transform of derivatives InverseLaplaceTransformOfDerivatives 2013-03-22 16:46:27 2013-03-22 16:46:27 pahio (2872) pahio (2872) 11 pahio (2872) Derivation msc 44A10 differentiation of Laplace transform MellinsInverseFormula SeparationOfVariables KalleVaisala TableOfLaplaceTransforms