InverseLaplaceTransform

The inverse Laplace transform, as a small dictionary of common images ( $s^{-n}$ , $1/(s-a)$ , $s/(s^2 \pm a^2)$ , $a/(s^2 \pm a^2)$ ) plus linearity — not a general residue calculus.

✓ checked against that system · ! checked, with disagreements (hover a chip)Wolfram LanguageInverseLaplaceTransform✓

InverseLaplaceTransform(F, s, t)

f(t)

such that

\mathcal{L}\{f\}(s) = F(s)

, read off a fixed table of images.

Typed signatures

Type	Package	Applies
`(expression, expression, expression) -> expression`	`analytic`	always

Domain: Transforms

Details

Covered: $s^{-n}$ ( $n$ a positive integer) $\to t^{n-1}/(n-1)!$ ; $1/(s-a) \to e^{at}$ ; $s/(s^2+a^2) \to \cos(at)$ ; $a/(s^2+a^2) \to \sin(at)$ ; $s/(s^2-a^2) \to \cosh(at)$ ; $a/(s^2-a^2) \to \sinh(at)$ ; sums and a constant factor.
Also, from the usual tables (each confirmed against wolframscript): $s^{-\nu}\to t^{\nu-1}/\Gamma(\nu)$ for a non-whole $\nu>0$ and its first two $\nu$ -derivatives (a factor $\ln(cs)$ or $\ln^2(cs)$ , $c>0$ ); the shifting theorems ( $e^{-cs}F\to f(t-c)\,\theta(t-c)$ , and $F(s-a)\to e^{at}f(t)$ when a $\sqrt{s-a}$ appears, once); $1/((s-a)\sqrt s)\to e^{at}\operatorname{erf}(\sqrt{at})/\sqrt a$ and $\sqrt s/(s-a)$ ; $\ln s/(s-a)\to e^{at}(\ln a-\mathrm{Ei}(-at))$ for a concrete $a>0$ ; $\Gamma(\nu,s)/s^\nu\to t^{\nu-1}\theta(t-1)$ ; $\operatorname{erf}(\sqrt{as})/\sqrt s\to\theta(a-t)/\sqrt{\pi t}$ for $a>0$ ; and the elliptic pairs $K(a^2/s^2)-\pi/2$ and $s(\pi/2-E(a^2/s^2))$ , both Bessel $I_0I_1$ products.
$s^{-\nu/2}K_\nu(b\sqrt s)\to 2^{\nu-1}b^{-\nu}t^{\nu-1}e^{-b^2/4t}$ for $b>0$ and the order matching the power of $s$ (from $\mathcal{L}\{t^{\nu-1}e^{-a/t}\}=2(a/s)^{\nu/2}K_\nu(2\sqrt{as})$ ).
Declined: anything outside those — a general rational function of s (partial-fraction decomposition is not attempted), a non-positive power of s, a bare logarithm, an image that doesn't depend on s at all.

Examples

Basic #

$\mathcal{L}^{-1}\{1/s^2\} = t$

In

\mathrm{InverseLaplaceTransform}(\frac{1}{s^2}, s, t)

Out

t

$\mathcal{L}^{-1}\{1/(s-a)\} = e^{at}$

In

\mathrm{InverseLaplaceTransform}(\frac{1}{s-a}, s, t)

Out

\exp(at)

$\mathcal{L}^{-1}\{s/(s^2+a^2)\} = \cos(at)$

In

\mathrm{InverseLaplaceTransform}(\frac{s}{s^2+a^2}, s, t)

Out

\cos(at)

In

\mathrm{InverseLaplaceTransform}(\frac{1}{s+1}, s, t)

Out

\exp(-t)

In

\mathrm{InverseLaplaceTransform}(\ln(\frac{a+s}{b+s}), s, t)

Out

\frac{1}{t}(\exp(-(bt))-\exp(-(at)))

Scope #

In

\mathrm{InverseLaplaceTransform}(\frac{a}{a^2+s^2}, s, t)

Out

\sin(at)

In

\mathrm{InverseLaplaceTransform}(\frac{s}{a^2+s^2}, s, t)

Out

\cos(at)

In

\mathrm{FullSimplify}(\mathrm{InverseLaplaceTransform}(\frac{a}{s^2-a^2}, s, t))

Out

\sinh(at)

In

\mathrm{FullSimplify}(\mathrm{InverseLaplaceTransform}(\frac{s}{s^2-a^2}, s, t))

Out

\cosh(at)

In

\mathrm{InverseLaplaceTransform}(\frac{s}{(a^2+s^2)^2}, s, t)

Out

\frac{t\sin(at)}{2a}

In

\mathrm{InverseLaplaceTransform}(\frac{s}{(a^2+s^2)(b^2+s^2)}, s, t)

Out

\frac{\cos(at)}{b^2-a^2}+\frac{\cos(bt)}{a^2-b^2}

In

\mathrm{FullSimplify}(\mathrm{InverseLaplaceTransform}(\frac{s^2}{(s^2-a^2)(s^2-b^2)}, s, t))

Out

\frac{a\sinh(at)}{a^2-b^2}+\frac{b\sinh(bt)}{b^2-a^2}

In

\mathrm{InverseLaplaceTransform}(\frac{\exp(-5s)}{(s-2)\sqrt{s}}, s, t)

Out

\frac{1}{2}(\sqrt{2}\operatorname{erf}(\sqrt{2(t-5)})\mathrm{HeavisideTheta}(t-5)\exp(2(t-5)))

In

\mathrm{InverseLaplaceTransform}(\frac{\sqrt{b+s}}{a+s}, s, t)

Out

(\operatorname{erf}(\sqrt{t×(b-a)})\exp(t×(b-a))\sqrt{b-a}+\frac{1}{\sqrt{π}\sqrt{t}})\exp(-(bt))

In

\mathrm{InverseLaplaceTransform}(\frac{\ln(s)}{s-1}, s, t)

Out

-(\mathrm{ExpIntegralEi}(-t)e^{t})

In

\mathrm{InverseLaplaceTransform}(\ln(\frac{s^2+1}{s^2+2}), s, t)

Out

-\frac{1}{t}(2(\cos(t)-\cos(\sqrt{2}t)))

In

\mathrm{InverseLaplaceTransform}(\frac{\ln(4s)}{\sqrt[3]{s}}, s, t)

Out

\frac{-\ln(t)+2\ln(2)-\frac{3\ln(3)}{2}-\frac{\sqrt{3}π}{6}-γ}{Γ(\frac{1}{3})t^{\frac{2}{3}}}

In

\mathrm{InverseLaplaceTransform}(\frac{\ln(s)^2}{s}, s, t)

Out

(-\ln(t)-γ)^2-\frac{π^2}{6}

In

\mathrm{InverseLaplaceTransform}(\frac{Γ(\frac{1}{3}, s)}{\sqrt[3]{s}}, s, t)

Out

\frac{\mathrm{HeavisideTheta}(t-1)}{t^{\frac{2}{3}}}

In

\mathrm{InverseLaplaceTransform}(\frac{\operatorname{erf}(\sqrt{2s})}{\sqrt{s}}, s, t)

Out

\frac{\mathrm{HeavisideTheta}(2-t)}{\sqrt{π}\sqrt{t}}

In

\mathrm{InverseLaplaceTransform}(\mathrm{EllipticK}(\frac{a^2}{s^2})-\frac{π}{2}, s, t)

Out

\frac{1}{2}(πa\operatorname{I}_{0}(\frac{at}{2})\operatorname{I}_{1}(\frac{at}{2}))

In

\mathrm{InverseLaplaceTransform}(s×(\frac{π}{2}-\mathrm{EllipticE}(\frac{a^2}{s^2})), s, t)

Out

\frac{πa\operatorname{I}_{0}(\frac{at}{2})\operatorname{I}_{1}(\frac{at}{2})}{2t}

In

\mathrm{InverseLaplaceTransform}(\frac{1}{\sqrt{s+1}}, s, t)

Out

\frac{\exp(-t)}{\sqrt{π}\sqrt{t}}

In

\mathrm{InverseLaplaceTransform}(\frac{\operatorname{K}_{4}(\sqrt{s})}{s^2}, s, t)

Out

8t^3\exp(-\frac{1}{4t})

About

Examples	24
Checked against	Wolfram Language 24 agree
Crosswalk	Wolfram Language

Implementation

primitive · kernela kernel — an algorithm over mutable state that would not be readable as a tree

nativeenginetypescriptpackages/symbols/analysis/analytic/src/transforms.ts

See also: LaplaceTransform, InverseFourierTransform