inverse Laplace transform of derivatives

It may be shown that the Laplace transform  $F(s)={\int }_0^{\mathrm{\infty }}{e}^{-st}f(t)𝑑t$  is always differentiable and that its derivative can be formed by differentiating under the integral sign (http://planetmath.org/DifferentiationUnderIntegralSign), i.e. one has

$$F^{\prime }(s)={\int }_0^{\mathrm{\infty }}\frac{\partial (e^{-st}f(t))}{\partial s}𝑑t={\int }_0^{\mathrm{\infty }}{e}^{-st}(-t)f(t)𝑑t.$$

This gives the rule

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

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

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

Continuing this way we can obtain the general rule

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

or equivalently

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

for any  $n=1,\mathrm{\hspace{0.17em}2},\mathrm{\hspace{0.17em}3},\mathrm{\dots }$ (and of course for  $n=0$).

Example.  Let’s find the Laplace transform of the first kind and 0th Bessel function

$$J_0(t):=\sum _{m=0}^{\mathrm{\infty }}\frac{{(-1)}^m}{{(m!)}^2}{\left(\frac{t}{2}\right)}^{2m},$$

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

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

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

By (3), the Laplace transform of the differential equation (4) is

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

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

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

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

$$ℒ\{J_0(t)\}=\frac{1}{\sqrt{s^2+1}}.$$

Title	inverse Laplace transform of derivatives
Canonical name	InverseLaplaceTransformOfDerivatives
Date of creation	2013-03-22 16:46:27
Last modified on	2013-03-22 16:46:27
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	11
Author	pahio (2872)
Entry type	Derivation
Classification	msc 44A10
Synonym	differentiation of Laplace transform
Related topic	MellinsInverseFormula
Related topic	SeparationOfVariables
Related topic	KalleVaisala
Related topic	TableOfLaplaceTransforms