--- title: "InverseLaplaceTransform" language: "en" type: "Symbol" summary: "InverseLaplaceTransform[F[s], s, t] gives the symbolic inverse Laplace transform of F[s] in the variable s as f[t] in the variable t. InverseLaplaceTransform[F[s], s, OverscriptBox[StyleBox[t, TI], ^]] gives the numeric inverse Laplace transform at the numerical value OverscriptBox[t, ^]. InverseLaplaceTransform[F[s1, ..., sn], {s1, s2, ...}, {t1, t2, ...}] gives the multidimensional inverse Laplace transform of F[s 1, ..., s n]." keywords: - Bromwich integral - step response - transfer function - transfer matrix - transient analysis - function transform - impulse response - integral operator - integral transform - Integrate - inverse Laplace transform - Laplace transform - ilaplace - invlaplace canonical_url: "https://reference.wolfram.com/language/ref/InverseLaplaceTransform.html" source: "Wolfram Language Documentation" related_guides: - title: "Fractional Calculus" link: "https://reference.wolfram.com/language/guide/FractionalCalculus.en.md" - title: "Integral Transforms" link: "https://reference.wolfram.com/language/guide/IntegralTransforms.en.md" - title: "Signal Transforms" link: "https://reference.wolfram.com/language/guide/SignalTransforms.en.md" related_functions: - title: "LaplaceTransform" link: "https://reference.wolfram.com/language/ref/LaplaceTransform.en.md" - title: "InverseBilateralLaplaceTransform" link: "https://reference.wolfram.com/language/ref/InverseBilateralLaplaceTransform.en.md" - title: "InverseFourierTransform" link: "https://reference.wolfram.com/language/ref/InverseFourierTransform.en.md" - title: "InverseZTransform" link: "https://reference.wolfram.com/language/ref/InverseZTransform.en.md" - title: "InverseMellinTransform" link: "https://reference.wolfram.com/language/ref/InverseMellinTransform.en.md" - title: "Integrate" link: "https://reference.wolfram.com/language/ref/Integrate.en.md" - title: "Asymptotic" link: "https://reference.wolfram.com/language/ref/Asymptotic.en.md" related_tutorials: - title: "Integral Transforms and Related Operations" link: "https://reference.wolfram.com/language/tutorial/Calculus.en.md#26017" - title: "Introduction to Fractional Calculus" link: "https://reference.wolfram.com/language/tutorial/FractionalCalculus.en.md" --- # InverseLaplaceTransform InverseLaplaceTransform[F[s], s, t] gives the symbolic inverse Laplace transform of F[s] in the variable s as f[t] in the variable t. InverseLaplaceTransform[F[s], s, Overscript[t, ^ ]] gives the numeric inverse Laplace transform at the numerical value Overscript[t, ^ ]. InverseLaplaceTransform[F[s1, …, sn], {s1, s2, …}, {t1, t2, …}] gives the multidimensional inverse Laplace transform of F[s1, …, sn]. ## Details and Options * Laplace transforms are typically used to transform differential and partial differential equations to algebraic equations, solve and then inverse transform back to a solution. * Laplace transforms are also extensively used in control theory and signal processing as a way to represent and manipulate linear systems in the form of transfer functions and transfer matrices. The Laplace transform and its inverse are then a way to transform between the time domain and frequency domain. * The inverse Laplace transform of a function $F(s)$ is defined to be $\frac{1}{2\pi i}\int _{\gamma -i\infty }^{\gamma +i\infty }F(s)e^{st}ds$, where ``γ`` is an arbitrary positive constant chosen so that the contour of integration lies to the right of all singularities in $F(s)$. [image] * The multidimensional inverse Laplace transform of a function $F$ is given by a contour integral of the form $\int _{\gamma -i\infty }^{\gamma +i\infty }\int _{\delta -i\infty }^{\delta +i\infty } \cdots F\left(s_1,s_2,\ldots \right)e^{\left(s_1 t_1+s_2 t_2+\ldots \right)}ds_1ds_2\ldots$. * The integral is computed using numerical methods if the third argument $t$ is given a numerical value. The available method settings include ``"Crump"``, ``"Durbin"``, ``"Papoulis"``, ``"Piessens"``, ``"Stehfest"``, ``"Talbot"`` and ``"Weeks"``. * The asymptotic inverse Laplace transform can be computed using ``Asymptotic``. * The following options can be given: | | | | | ------------------- | ----------------- | ------------------------------------------------------------------ | | AccuracyGoal | Automatic | digits of absolute accuracy sought | | Assumptions | \$Assumptions | assumptions to make about parameters | | GenerateConditions | False | whether to generate answers that involve conditions on parameters | | Method | Automatic | method to use | | PerformanceGoal | \$PerformanceGoal | aspects of performance to optimize | | PrecisionGoal | Automatic | digits of precision sought | | WorkingPrecision | Automatic | the precision used in internal computations | * In ``TraditionalForm``, ``InverseLaplaceTransform`` is output using ``ℒ^-1``. » ## Examples (72) ### Basic Examples (4) Compute the inverse Laplace transform of a function: ```wl In[1]:= InverseLaplaceTransform[(1/s + 1), s, t] Out[1]= E^-t ``` --- Inverse Laplace transform of a function with parameters: ```wl In[1]:= InverseLaplaceTransform[Log[(s + a/s + b)], s, t] Out[1]= (-E^-a t + E^-b t/t) ``` Plot the result: ```wl In[2]:= Plot[Evaluate[Table[% /. a -> 1, {b, 2, 5}]], {t, 0, 4}] Out[2]= [image] ``` --- Compute a numerical inverse Laplace transform: ```wl In[1]:= InverseLaplaceTransform[(1/s ^ (3 / 2) + s ^ 2 + 1), s, 0.3] Out[1]= 0.207358 ``` --- Inverse Laplace transform of a bivariate function: ```wl In[1]:= InverseLaplaceTransform[(1/Sqrt[1 + p q]), {p, q}, {x, y}] Out[1]= (Cosh[2 Sqrt[-x y]]/π Sqrt[x] Sqrt[y]) ``` ### Scope (56) #### Basic Uses (3) Compute the inverse Laplace transform of a function for a symbolic parameter ``t`` : ```wl In[1]:= InverseLaplaceTransform[(1/Sqrt[s + 1]), s, t] Out[1]= (E^-t/Sqrt[π] Sqrt[t]) ``` --- Use a numerical value for the parameter: ```wl In[1]:= InverseLaplaceTransform[(1/Sqrt[s + 1]), s, 1.2] Out[1]= 0.155125 ``` --- ``TraditionalForm`` formatting: ```wl In[1]:= TraditionalForm[InverseLaplaceTransform[f[t], t, s]] Out[1]//TraditionalForm= $$\mathcal{L}_t^{-1}[f(t)](s)$$ ``` #### Rational Functions (5) Reciprocal of a linear function: ```wl In[1]:= InverseLaplaceTransform[(1/a s + b), s, t] Out[1]= (E^-(b t/a)/a) ``` --- Reciprocal of a quadratic function: ```wl In[1]:= InverseLaplaceTransform[(1/a s^2 + b s + c), s, t] Out[1]= -(E^(-(b/2 a) - (Sqrt[b^2 - 4 a c]/2 a)) t - E^(-(b/2 a) + (Sqrt[b^2 - 4 a c]/2 a)) t/Sqrt[b^2 - 4 a c]) ``` --- Functions whose inverses are trigonometric functions: ```wl In[1]:= InverseLaplaceTransform[(a/s^2 + a^2), s, t] Out[1]= Sin[a t] In[2]:= InverseLaplaceTransform[(s/s^2 + a^2), s, t] Out[2]= Cos[a t] ``` --- Functions whose inverses are hyperbolic functions: ```wl In[1]:= InverseLaplaceTransform[(a/s^2 - a^2), s, t]//FullSimplify Out[1]= Sinh[a t] In[2]:= InverseLaplaceTransform[(s/s^2 - a^2), s, t]//FullSimplify Out[2]= Cosh[a t] ``` --- More examples of rational functions: ```wl In[1]:= InverseLaplaceTransform[(s/(s^2 + a^2)^2), s, t] Out[1]= (t Sin[a t]/2 a) In[2]:= InverseLaplaceTransform[(s/(s^2 + a^2)(s^2 + b^2)), s, t] Out[2]= (-Cos[a t] + Cos[b t]/a^2 - b^2) In[3]:= InverseLaplaceTransform[(s^2/(s^2 - a^2)(s^2 - b^2)), s, t]//FullSimplify Out[3]= (a Sinh[a t] - b Sinh[b t]/a^2 - b^2) ``` #### Elementary Functions (5) Function involving a square root: ```wl In[1]:= InverseLaplaceTransform[(1/s Sqrt[s + 1]), s, t] Out[1]= Erf[Sqrt[t]] ``` --- Function involving exponential and square root functions: ```wl In[1]:= InverseLaplaceTransform[Exp[-5s](1/Sqrt[s](s - 2)), s, t] Out[1]= (E^2 (-5 + t) GammaRegularized[(1/2), 0, 2 (-5 + t)] HeavisideTheta[-5 + t]/Sqrt[2]) ``` --- Other algebraic functions: ```wl In[1]:= InverseLaplaceTransform[(1/ Sqrt[s + a] + b), s, t] Out[1]= E^-a t ((1/Sqrt[π] Sqrt[t]) - b E^b^2 t Erfc[b Sqrt[t]]) In[2]:= InverseLaplaceTransform[(Sqrt[s + b]/s + a), s, t] Out[2]= (E^-b t/Sqrt[π] Sqrt[t]) + Sqrt[-a + b] E^-a t Erf[Sqrt[(-a + b) t]] ``` --- Inverse Laplace transform leading to a Bessel function: ```wl In[1]:= InverseLaplaceTransform[(1/Sqrt[s^2 + b s + a^2]), s, t] Out[1]= E^-(b t/2) BesselJ[0, (1/2) Sqrt[4 a^2 - b^2] t] ``` --- Inverse of an arctangent function: ```wl In[1]:= InverseLaplaceTransform[ArcTan[(1/2s)], s, t] Out[1]= (Sin[(t/2)]/t) ``` #### Logarithmic Functions (4) Inverse Laplace transform of a logarithmic function: ```wl In[1]:= InverseLaplaceTransform[(Log[s]/s - 1), s, t] Out[1]= -E^t ExpIntegralEi[-t] ``` --- Logarithm of a rational function: ```wl In[1]:= InverseLaplaceTransform[Log[(s^2 + 1/s^2 + 2)], s, t] Out[1]= (2 (-Cos[t] + Cos[Sqrt[2] t])/t) ``` Plot the result: ```wl In[2]:= Plot[%, {t, 0, 15}] Out[2]= [image] ``` --- Product of a logarithm and a power function: ```wl In[1]:= InverseLaplaceTransform[(Log[4 s] /s^1 / 3), s, t] Out[1]= (Log[4] - Log[t] + PolyGamma[0, (1/3)]/t^2 / 3 Gamma[(1/3)]) ``` --- Function involving the square of a logarithm: ```wl In[1]:= InverseLaplaceTransform[(Log[s]^2/s), s, t] Out[1]= -(π^2/6) + (EulerGamma + Log[t])^2 ``` #### Special Functions (12) Function involving ``BesselK`` : ```wl In[1]:= InverseLaplaceTransform[(BesselK[4, Sqrt[s]]/s^2), s, t] Out[1]= 8 E^-(1/4) / t t^3 ``` --- Ratio of an incomplete gamma function and a power function: ```wl In[1]:= InverseLaplaceTransform[(Gamma[(1/3), s]/s^(1/3)), s, t] Out[1]= (HeavisideTheta[-1 + t]/t^2 / 3) ``` --- Inverse of a polygamma function: ```wl In[1]:= InverseLaplaceTransform[PolyGamma[7, s], s, t] Out[1]= (E^t t^7/-1 + E^t) ``` --- Inverse of a function involving an error function: ```wl In[1]:= InverseLaplaceTransform[Exp[9s^2]Erfc[3s], s, t] Out[1]= (E^-(t^2/36)/3 Sqrt[π]) ``` --- Error function composed with a square root: ```wl In[1]:= InverseLaplaceTransform[Erfc[ Sqrt[3s]], s, t] Out[1]= (Sqrt[3] HeavisideTheta[-3 + t]/π Sqrt[-3 + t] t) In[2]:= InverseLaplaceTransform[(Erf[ Sqrt[2s]]/Sqrt[s]), s, t] Out[2]= (HeavisideTheta[2 - t]/Sqrt[π] Sqrt[t]) ``` --- Inverse transforms of the ``LegendreP`` and ``LegendreQ`` functions: ```wl In[1]:= InverseLaplaceTransform[LegendreP[-3 / 5, s], s, t] Out[1]= (Sqrt[5 + Sqrt[5]] BesselK[-(1/10), t]/2 π^3 / 2 Sqrt[t]) In[2]:= InverseLaplaceTransform[LegendreQ[3 / 5, s], s, t] Out[2]= Sqrt[(π/2)] Sqrt[(1/t)] BesselI[(11/10), t] ``` Complex plots of the Legendre functions: ```wl In[3]:= {ComplexPlot[LegendreP[-3 / 5, s], {s, -3 - 3I, 3 + 3I}], ComplexPlot[LegendreQ[+3 / 5, s], {s, -3 - 3I, 3 + 3I}]} Out[3]= {[image], [image]} ``` --- Product of ``BesselI`` and ``Exp`` : ```wl In[1]:= InverseLaplaceTransform[Exp[-2 s]BesselI[0, 2 s], s, t] Out[1]= (UnitStep[4 - t]/π Sqrt[-((-4 + t) t)]) ``` ``ComplexPlot`` of the frequency-domain function: ```wl In[2]:= ComplexPlot[Exp[-2 s]BesselI[0, 2 s], {s, -3 - 3I, 3 + 3I}] Out[2]= [image] ``` --- Product of ``BesselJ`` and ``Gamma`` functions: ```wl In[1]:= InverseLaplaceTransform[(Gamma[s]/(5 / 2)^s)BesselJ[1 / 2 + s, 5], s, t] Out[1]= Sqrt[(2/5 π)] Sin[5 Sqrt[1 - E^-t]] ``` ``ComplexPlot`` of the frequency-domain function: ```wl In[2]:= ComplexPlot[(Gamma[s]/(5 / 2)^s)BesselJ[1 / 2 + s, 5], {s, -3 - 2I, 3 + 2I}] Out[2]= [image] ``` --- Composition of exponential integral and square root functions: ```wl In[1]:= InverseLaplaceTransform[(ExpIntegralEi[-5 Sqrt[s]]/Sqrt[s]), s, t] Out[1]= (ExpIntegralEi[-(25/4 t)]/2 Sqrt[π] Sqrt[t]) ``` --- Product of two ``ParabolicCylinderD`` functions: ```wl In[1]:= InverseLaplaceTransform[ParabolicCylinderD[-1 / 3, Sqrt[s + Sqrt[s^2 - 4]]]ParabolicCylinderD[-1 / 3, Sqrt[s - Sqrt[s^2 - 4]]], s, t] Out[1]= (E^-2 Sqrt[-(1/4) + t^2] ((1/2) + t)^1 / 6 UnitStep[-(1/2) + t]/2 (-(1/4) + t^2)^5 / 6 Gamma[(1/3)]) ``` --- Inverse transforms involving elliptic integral functions: ```wl In[1]:= InverseLaplaceTransform[EllipticK[a^2 / s^2] - π / 2, s, t] Out[1]= (1/2) a π BesselI[0, (a t/2)] BesselI[1, (a t/2)] In[2]:= InverseLaplaceTransform[s(π / 2 - EllipticE[a^2 / s^2]), s, t] Out[2]= (a π BesselI[0, (a t/2)] BesselI[1, (a t/2)]/2 t) ``` --- Inverse transform of ``EllipticTheta`` : ```wl In[1]:= InverseLaplaceTransform[(EllipticTheta[2, 0, Exp[-Pi^2s] ]/s), s, t] Out[1]= 2 (1 + Floor[-(1/2) + (Sqrt[t]/π)]) ``` #### Piecewise Functions (5) The inverse of the following function is a piecewise function: ```wl In[1]:= InverseLaplaceTransform[(Exp[-3s]/s + 1), s, t] Out[1]= E^3 - t HeavisideTheta[-3 + t] ``` Plot the inverse: ```wl In[2]:= Plot[%, {t, 0, 8}, PlotRange -> All] Out[2]= [image] ``` --- Inverse of an exponential function leading to a box function: ```wl In[1]:= InverseLaplaceTransform[(Exp[-2s] - Exp[-5s]/s), s, t] Out[1]= -HeavisideTheta[-5 + t] + HeavisideTheta[-2 + t] In[2]:= Plot[%, {t, 0, 8}] Out[2]= [image] ``` --- Inverse of an exponential function leading to a piecewise trigonometric function: ```wl In[1]:= InverseLaplaceTransform[((1 - Exp[-6s])/s^2 + 1), s, t] Out[1]= HeavisideTheta[-6 + t] Sin[6 - t] + Sin[t] In[2]:= Plot[%, {t, 0, 27}] Out[2]= [image] ``` --- Inverse leading to a staircase function: ```wl In[1]:= InverseLaplaceTransform[(1/s Sinh[3s]), s, t] Out[1]= (t/3) + Inactive[Sum][Piecewise[{{-((I*E^((1/3)*I*\[FormalN]*Pi*(3 + t)))/(\[FormalN]*Pi)), \[FormalN] != 0}}, 0], {\[FormalN], -Infinity, Infinity}] In[2]:= Activate[%] Out[2]= (t/3) - (I (-Log[1 + E^(I π t/3)] + Log[E^-(1/3) I π t (1 + E^(I π t/3))])/π) In[3]:= Plot[%, {t, 0, 40}, Exclusions -> None] Out[3]= [image] ``` --- More involved staircase function: ```wl In[1]:= InverseLaplaceTransform[(1/s Sinh[3s]Sinh[5s]), s, t] Out[1]= (1/90) (-34 + 3 t^2) + Inactive[Sum][Piecewise[{{(E^(I*\[FormalN]*Pi*t)*(1 - I*\[FormalN]*Pi*t))/(15*\[FormalN]^2*Pi^2), \[FormalN] != 0}}, 0] + Piecewise[{{-((E^((1/5)*I*\[FormalN]*Pi*(5 + t))*Csc[(3*\[FormalN]*Pi)/5])/(\[FormalN]*Pi)), Mod[\[FormalN], 5] != 0}}, 0] + Piecewise[{{-((E^((1/3)*I*\[FormalN]*Pi*(3 + t))*Csc[(5*\[FormalN]*Pi)/3])/(\[FormalN]*Pi)), Mod[\[FormalN], 3] != 0}}, 0], {\[FormalN], -Infinity, Infinity}] In[2]:= TruncateSum[%, 45]; In[3]:= Plot[%, {t, 0, 60}] Out[3]= [image] ``` #### Periodic Functions (4) The inverse of the following function is periodic with respect to ``t`` : ```wl In[1]:= InverseLaplaceTransform[(1/s(Exp[s] + 1)), s, t] Out[1]= (1/2) + Inactive[Sum][(I*E^(I*(1 + 2*\[FormalN])*Pi*t))/(Pi + 2*\[FormalN]*Pi), {\[FormalN], -Infinity, Infinity}] ``` Activate the sum: ```wl In[2]:= Assuming[t > 0, FullSimplify[Activate[%]]] Out[2]= (1/2) + (I (-ArcCoth[E^I π t] + ArcTanh[E^I π t])/π) ``` Plot to verify that the result is a square wave: ```wl In[3]:= Plot[%, {t, 0, 7}] Out[3]= [image] ``` --- Function whose inverse is the absolute value of the sine function: ```wl In[1]:= InverseLaplaceTransform[(1/s^2 + 1)Coth[π (s/2)], s, t] Out[1]= Inactive[Sum][(2*E^(2*I*\[FormalN]*t))/(Pi - 4*\[FormalN]^2*Pi), {\[FormalN], -Infinity, Infinity}] ``` Activate and plot the result: ```wl In[2]:= Plot[Evaluate[Activate[%]//FullSimplify], {t, 0, 10}] Out[2]= [image] ``` --- Function whose inverse is a trapezoidal wave: ```wl In[1]:= InverseLaplaceTransform[((1 - E^-3 s) (1 - E^-s)/2 (1 - E^-4 s) s^2), s, t] Out[1]= (1/2) ((1/3) + (1/8) t (4 + t) + HeavisideTheta[-4 + t] ((1/3) + (1/8) (-4 + t) t + Inactive[Sum][Piecewise[{{-(1/(E^((1/2)*I*\[FormalN]*Pi*(-4 + t))*(\[FormalN]^2*Pi^2))), \[FormalN] != 0}}, 0], {\[FormalN], -Infinity, Infinity}]) - HeavisideTheta[-3 + t] ((1/3) + (1/8) (-3 + t) ( ... (1/8) (-1 + t) (3 + t) + Inactive[Sum][Piecewise[{{-(1/(E^((1/2)*I*\[FormalN]*Pi*(-1 + t))*(\[FormalN]^2*Pi^2))), \[FormalN] != 0}}, 0], {\[FormalN], -Infinity, Infinity}]) + Inactive[Sum][Piecewise[{{-(1/(E^((1/2)*I*\[FormalN]*Pi*t)*(\[FormalN]^2*Pi^2))), \[FormalN] != 0}}, 0], {\[FormalN], -Infinity, Infinity}]) In[2]:= Plot[Evaluate[Activate[%]], {t, 0, 14}] Out[2]= [image] ``` --- A more involved piecewise example: ```wl In[1]:= InverseLaplaceTransform[((Exp[s] - 1) Sin[π / 5]/s(Exp[2s] - 2 Exp[s]Cos[π / 5] + 1)), s, t] Out[1]= Sqrt[(5/8) - (Sqrt[5]/8)] ((1/2) (-3 - Sqrt[5]) + (1/2) (3 + Sqrt[5]) + Inactive[Sum][(Sqrt[(5/2)*(5 + Sqrt[5])]*E^((1/5)*I*(-1 + 10*\[FormalN])*Pi*t)* (1 + E^((2*I*Pi*t)/5) - 10*\[FormalN]*(-1 + E^((2*I*Pi*t)/5))))/((-1 + 100*\[FormalN]^2)*Pi), {\[FormalN], -Infinity, Infinity}] - Inactive[Sum][(E^((1/5)*I*(1 + 10*\[FormalN])*Pi*t)*Root[15625 + 3125*#1^2 + 250*#1^4 + #1^8 & , 2, 0])/ (Pi + 10*\[FormalN]*Pi) + (E^((1/5)*I*(-1 + 10*\[FormalN])*Pi*t)*Root[15625 + 3125*#1^2 + 250*#1^4 + #1^8 & , 8, 0])/((-1 + 10*\[FormalN])*Pi), {\[FormalN], -Infinity, Infinity}]) In[2]:= Plot[{Evaluate[TruncateSum[%, 6]], Sin[π Floor[t] / 5]}, {t, 0, 15}] Out[2]= [image] ``` #### Generalized Functions (3) Inverse of an exponential function is the Dirac function: ```wl In[1]:= InverseLaplaceTransform[Exp[-s], s, t] Out[1]= DiracDelta[-1 + t] ``` --- Inverse of the product of an exponential function and a power of ``s`` is a derivative of the Dirac function: ```wl In[1]:= InverseLaplaceTransform[Exp[-s]s ^ 2, s, t] Out[1]= HeavisideTheta[-1 + t] Derivative[2][DiracDelta][-1 + t] ``` --- The inverse of the hyperbolic secant is a series of Dirac functions: ```wl In[1]:= InverseLaplaceTransform[Sech[s], s, t] Out[1]= 2 Underoverscript[∑, K[1] = 0, ∞](-1)^K[1] DiracDelta[-1 + t - 2 K[1]] ``` Hyperbolic cotangent: ```wl In[2]:= InverseLaplaceTransform[Coth[3s], s, t] Out[2]= DiracDelta[t] + 2 Underoverscript[∑, K[1] = 1, ∞]DiracDelta[t - 6 K[1]] ``` #### Multivariate Functions (8) Inverse Laplace transform of a bivariate rational function: ```wl In[1]:= InverseLaplaceTransform[(1/p + q), {p, q}, {x, y}] Out[1]= DiracDelta[-x + y] ``` Verify the result: ```wl In[2]:= LaplaceTransform[%, {x, y}, {p, q}] Out[2]= (1/p + q) ``` --- Rational function that is separable with respect to ``p`` and ``q`` : ```wl In[1]:= InverseLaplaceTransform[(1/(-a + p) (-b + q)), {p, q}, {x, y}] Out[1]= E^a x + b y ``` --- Rational function that is not separable with respect to ``p`` and ``q`` : ```wl In[1]:= InverseLaplaceTransform[(1/(q - 1)(p + q)), {p, q}, {x, y}] Out[1]= E^-x + y HeavisideTheta[-x + y] ``` --- Rational function whose inverse is related to ``BesselJ`` : ```wl In[1]:= InverseLaplaceTransform[(1/1 + p q), {p, q}, {x, y}] Out[1]= BesselJ[0, 2 Sqrt[x] Sqrt[y]] ``` --- Inverse Laplace transform of a bivariate function involving a radical: ```wl In[1]:= InverseLaplaceTransform[(1/Sqrt[p + q]), {p, q}, {x, y}] Out[1]= (DiracDelta[-x + y]/Sqrt[π] Sqrt[x]) ``` --- Inverse Laplace transform of a multivariate function in three variables: ```wl In[1]:= InverseLaplaceTransform[(1/p^mq^nr^k), {p, q, r}, {x, y, z}] Out[1]= (x^-1 + m y^-1 + n z^-1 + k/Gamma[k] Gamma[m] Gamma[n]) ``` --- Rational function: ```wl In[1]:= InverseLaplaceTransform[-(p + q + r - p q r/(1 + p^2) (1 + q^2) (1 + r^2)), {p, q, r}, {x, y, z}]//Simplify Out[1]= Cos[x + y + z] ``` --- Function involving a logarithm: ```wl In[1]:= InverseLaplaceTransform[-(3 EulerGamma + Log[p q r]/p q r), {p, q, r}, {x, y, z}] Out[1]= Log[x] + Log[y] + Log[z] ``` #### Numerical Inversion (4) Calculate the inverse Laplace transform at a single point: ```wl In[1]:= InverseLaplaceTransform[(s/(s + 2)^2), s, 2.3] Out[1]= -0.0361866 ``` Alternatively, calculate the inverse Laplace transform symbolically: ```wl In[2]:= InverseLaplaceTransform[(s/(s + 2)^2), s, t] Out[2]= E^-2 t (1 - 2 t) ``` Then evaluate it for a specific value of ``t`` : ```wl In[3]:= % /. t -> 2.3 Out[3]= -0.0361866 ``` --- Plot the inverse Laplace transform numerically and compare it with the exact result: ```wl In[1]:= f[t_ ? NumericQ] := InverseLaplaceTransform[(s/(s + 2)^2), s, t] In[2]:= g[t_] = InverseLaplaceTransform[(s/(s + 2)^2), s, t] Out[2]= E^-2 t (1 - 2 t) In[3]:= Show[ DiscretePlot[f[t], {t, 0, 5, 0.1}, PlotStyle -> Red, PlotRange -> All], Plot[g[t], {t, 0, 5}, PlotRange -> All] ] Out[3]= [image] ``` --- Function whose inverse is a piecewise function with respect to ``t`` : ```wl In[1]:= f[t_ ? NumericQ] := InverseLaplaceTransform[((1 - Exp[-s])^2/s), s, t, Method -> "Durbin"] In[2]:= f[0.7] Out[2]= 0.984805 In[3]:= DiscretePlot[f[t], {t, 0.1, 6, 0.1}] Out[3]= [image] ``` --- Function whose inverse is a periodic function with respect to ``t`` : ```wl In[1]:= f[t_ ? NumericQ] := InverseLaplaceTransform[((1 - Exp[-3s]) (1 - Exp[-s])/2 (1 - Exp[-4s]) s^2), s, t, Method -> "Durbin"] In[2]:= f[0.7] Out[2]= 0.350077 In[3]:= DiscretePlot[f[t], {t, 0.1, 12, 0.2}] Out[3]= [image] ``` #### Fractional Calculus (3) ``ComplexPlot`` of an algebraic function in the $s$-domain: ```wl In[1]:= ComplexPlot[(s^2 / 3/(s^2 / 3 - 1) (2s^2 / 3 + 5)), {s, -2 - 2I, 2 + 2I}] Out[1]= [image] ``` Inverse Laplace transform of this algebraic function: ```wl In[2]:= InverseLaplaceTransform[(s^2 / 3/(s^2 / 3 - 1) (2s^2 / 3 + 5)), s, t] Out[2]= (1/28 t^2 / 3)(-2 MittagLefflerE[(1/3), (1/3), -t^1 / 3] + 2 MittagLefflerE[(1/3), (1/3), t^1 / 3] + I Sqrt[10] (MittagLefflerE[(1/3), (1/3), -(I Sqrt[5] t^1 / 3/Sqrt[2])] - MittagLefflerE[(1/3), (1/3), (I Sqrt[5] t^1 / 3/Sqrt[2])])) ``` Laplace transform to the $s$-domain: ```wl In[3]:= LaplaceTransform[%, t, s]//Simplify Out[3]= (s^2 / 3/-5 + 3 s^2 / 3 + 2 s^4 / 3) ``` --- Inverse Laplace transform of the algebraic function: ```wl In[1]:= f[t_] = InverseLaplaceTransform[(7s^1 / 5 - 6/-1 + 3 s^1 / 5 - 4 s^2 / 5 + 12 s^3 / 5), s, t] Out[1]= (1/52 t^4 / 5)7 ((-2 + 3 I) MittagLefflerE[(1/5), (1/5), -(1/2) I t^1 / 5] - (2 + 3 I) MittagLefflerE[(1/5), (1/5), (1/2) I t^1 / 5] + 4 MittagLefflerE[(1/5), (1/5), (t^1 / 5/3)]) - (1/13 t^4 / 5)3 ((-3 - 2 I) MittagLefflerE[(1/5), (1/5), -(1/2) I t^1 / 5] - (3 - 2 I) MittagLefflerE[(1/5), (1/5), (1/2) I t^1 / 5] + 6 MittagLefflerE[(1/5), (1/5), (t^1 / 5/3)]) ``` Plot in time domain: ```wl In[2]:= Plot[f[t], {t, 0, 20}] Out[2]= [image] ``` Laplace transform to the $s$-domain: ```wl In[3]:= LaplaceTransform[f[t], t, s]//Simplify Out[3]= (6 - 7 s^1 / 5/1 - 3 s^1 / 5 + 4 s^2 / 5 - 12 s^3 / 5) ``` --- Inverse Laplace transform of the algebraic function involving parameters: ```wl In[1]:= InverseLaplaceTransform[(s^-β/1 - a s^-α), s, t, Assumptions -> α > 0 && β > 0] Out[1]= t^-1 + β MittagLefflerE[α, β, a t^α] ``` Laplace transform to the $s$-domain: ```wl In[2]:= LaplaceTransform[%, t, s] Out[2]= (s^α - β/-a + s^α) ``` ### Options (3) #### GenerateConditions (1) By default, ``InverseLaplaceTransform`` assumes that the result is defined for non-negative ``t`` : ```wl In[1]:= InverseLaplaceTransform[1 / s ^ 2, s, t] Out[1]= t ``` Use ``GenerateConditions`` to obtain the range of validity for the result: ```wl In[2]:= InverseLaplaceTransform[1 / s ^ 2, s, t, GenerateConditions -> True] Out[2]= t HeavisideTheta[t] In[3]:= LaplaceTransform[%, t, s] Out[3]= (1/s^2) ``` #### Method (1) Use the default method for numerical evaluation: ```wl In[1]:= InverseLaplaceTransform[(1/Sqrt[s + 1] + s), s, 5.3] Out[1]= 0.02096 ``` Use ``Method`` to obtain the result from different methods: ```wl In[2]:= InverseLaplaceTransform[(1/Sqrt[s + 1] + s), s, 5.3, Method -> #]& /@ {"Stehfest", "Piessens", "Talbot", "Durbin", "Crump", "Papoulis", "Weeks"}//Chop Out[2]= {0.02096, 0.02096, 0.02096, 0.0205214, 0.0233592, 0.0208867, 0.0209602} ``` #### Working Precision (1) Use ``WorkingPrecision`` to obtain a result with arbitrary precision: ```wl In[1]:= InverseLaplaceTransform[(15Exp[-1 / Sqrt[s]]) / (s^5 / 3 + 1), s, 0.3] Out[1]= 4.40864 In[2]:= InverseLaplaceTransform[(15Exp[-1 / Sqrt[s]]) / (s^5 / 3 + 1), s, 0.3, WorkingPrecision -> 10] Out[2]= 4.408640327 In[3]:= InverseLaplaceTransform[(15Exp[-1 / Sqrt[s]]) / (s^5 / 3 + 1), s, 0.3, WorkingPrecision -> 20] Out[3]= 4.4086403270659850060 ``` ### Applications (5) Compute the step response to the linear system with transfer function $\frac{1}{s^2+s+1}$ : ```wl In[1]:= InverseLaplaceTransform[1 / (s ^ 2 + s + 1) 1 / s, s, t] Out[1]= 1 - (E^-t / 2 (Sqrt[3] Cos[(Sqrt[3] t/2)] + Sin[(Sqrt[3] t/2)])/Sqrt[3]) In[2]:= Plot[%, {t, 0, 10}] Out[2]= [image] ``` --- Solve a differential equation using Laplace transforms: ```wl In[1]:= LaplaceTransform[f''[t] + f[t] == E ^ (-3t), t, s] Out[1]= -s f[0] + LaplaceTransform[f[t], t, s] + s^2 LaplaceTransform[f[t], t, s] - Derivative[1][f][0] == (1/3 + s) ``` Solve for the Laplace transform: ```wl In[2]:= Solve[%, LaplaceTransform[f[t], t, s]] Out[2]= {{LaplaceTransform[f[t], t, s] -> ((1/3 + s) + s f[0] + Derivative[1][f][0]/1 + s^2)}} ``` Find the inverse transform: ```wl In[3]:= InverseLaplaceTransform[%, s, t] Out[3]= {{f[t] -> (1/10) (E^-3 t + Cos[t] (-1 + 10 f[0]) + Sin[t] (3 + 10 Derivative[1][f][0]))}} ``` Apply initial conditions: ```wl In[4]:= sol1 = f[t] /. %[[1]] /. {f[0] -> 1, f'[0] -> 3} Out[4]= (1/10) (E^-3 t + 9 Cos[t] + 33 Sin[t]) ``` Find the solution directly using ``DSolve`` : ```wl In[5]:= sol2 = f[t] /. DSolve[{f''[t] + f[t] == Exp[-3t], f[0] == 1, f'[0] == 3}, f[t], t][[1]] Out[5]= (1/10) E^-3 t (9 E^3 t Cos[t] + Cos[t]^2 + 33 E^3 t Sin[t] + Sin[t]^2) In[6]:= Simplify[sol1 - sol2] Out[6]= 0 ``` --- Solve a fractional-order differential equation of order ``3 / 2`` : ```wl In[1]:= eqn = CaputoD[y[t], {t, 3 / 2}] - D[y[t], {t, 1}] == t; ``` Solve for the Laplace transform: ```wl In[2]:= Solve[LaplaceTransform[eqn, t, s] /. {y[0] -> 0, y'[0] -> 1}, LaplaceTransform[y[t], t, s]] Out[2]= {{LaplaceTransform[y[t], t, s] -> (1 + s^3 / 2/(-1 + Sqrt[s]) s^3)}} ``` Find the inverse transform: ```wl In[3]:= InverseLaplaceTransform[%, s, t] Out[3]= {{y[t] -> t (MittagLefflerE[(1/2), 2, Sqrt[t]] + t^3 / 2 MittagLefflerE[(1/2), (7/2), Sqrt[t]])}} ``` Plot the solution: ```wl In[4]:= Plot[y[t] /. %, {t, 0, 4}] Out[4]= [image] ``` Find the solution directly using ``DSolve`` : ```wl In[5]:= DSolve[{CaputoD[y[t], {t, 3 / 2}] - D[y[t], {t, 1}] == t, y[0] == 0, y'[0] == 1}, y[t], t] Out[5]= {{y[t] -> t (MittagLefflerE[(1/2), 2, Sqrt[t]] + t^3 / 2 MittagLefflerE[(1/2), (7/2), Sqrt[t]])}} ``` --- Solve a fractional-order differential equation of order ``21 / 10`` : ```wl In[1]:= LaplaceTransform[CaputoD[y[t], {t, 21 / 10}] + 10y[t] == 0, t, s] /. {y[0] -> 1, y'[0] -> 0, y''[0] -> 0} Out[1]= -s^11 / 10 + 10 LaplaceTransform[y[t], t, s] + s^21 / 10 LaplaceTransform[y[t], t, s] == 0 ``` Solve for the Laplace transform: ```wl In[2]:= Solve[%, LaplaceTransform[y[t], t, s]] Out[2]= {{LaplaceTransform[y[t], t, s] -> (s^11 / 10/10 + s^21 / 10)}} ``` Find the inverse transform: ```wl In[3]:= InverseLaplaceTransform[%, s, t] Out[3]= {{y[t] -> MittagLefflerE[(21/10), -10 t^21 / 10]}} ``` Plot the solution: ```wl In[4]:= Plot[y[t] /. %, {t, 0, 10}] Out[4]= [image] ``` Find the solution directly using ``DSolve`` : ```wl In[5]:= DSolve[{CaputoD[y[t], {t, 21 / 10}] + 10y[t] == 0, y[0] == 1, y'[0] == 0, y''[0] == 0}, y[t], t] Out[5]= {{y[t] -> MittagLefflerE[(21/10), -10 t^21 / 10]}} ``` --- Solve a system of fractional DEs using ``LaplaceTransform`` : ```wl In[1]:= Solve[LaplaceTransform[{CaputoD[y[t], {t, 1 / 3}] == -z[t], CaputoD[z[t], {t, 1 / 3}] == y[t]}, t, s] /. {y[0] -> 1, z[0] -> 0}, LaplaceTransform[{y[t], z[t]}, t, s]] Out[1]= {{LaplaceTransform[y[t], t, s] -> (1/(1 + s^2 / 3) s^1 / 3), LaplaceTransform[z[t], t, s] -> (1/(1 + s^2 / 3) s^2 / 3)}} ``` Find the inverse transform: ```wl In[2]:= InverseLaplaceTransform[%, s, t] Out[2]= {{y[t] -> MittagLefflerE[(2/3), -t^2 / 3], z[t] -> t^1 / 3 MittagLefflerE[(2/3), (4/3), -t^2 / 3]}} ``` ### Properties & Relations (2) Use ``Asymptotic`` to compute an asymptotic approximation: ```wl In[1]:= Asymptotic[Inactive[InverseLaplaceTransform][1 / Sqrt[s ^ 5 + 1], s, t], t -> 0] Out[1]= (4 t^3 / 2/3 Sqrt[π]) ``` --- ``InverseLaplaceTransform`` and ``LaplaceTransform`` are mutual inverses: ```wl In[1]:= LaplaceTransform[InverseLaplaceTransform[f[s], s, t], t, s] Out[1]= f[s] In[2]:= InverseLaplaceTransform[LaplaceTransform[g[t], t, s], s, t] Out[2]= g[t] In[3]:= InverseLaplaceTransform[1 / (s ^ 2 + 1), s, t] Out[3]= Sin[t] In[4]:= LaplaceTransform[%, t, s] Out[4]= (1/1 + s^2) ``` ### Neat Examples (2) ``InverseLaplaceTransform`` of a ``MeijerG`` function: ```wl In[1]:= InverseLaplaceTransform[MeijerG[{{1 / 3}, {}}, {{0, 1 / 3, 1 / 3, 2 / 3}, {}}, s ^ 3 / 27] / (2 * Sqrt[3] * Pi), s, t] Out[1]= (t/1 + t^3) ``` --- Create a table of basic inverse Laplace transforms: ```wl In[1]:= Flist = {1, 1 / (s - a), 1 / s ^ 2, a / (a ^ 2 + s ^ 2), s / (a ^ 2 + s ^ 2), s / (s ^ 2 - a ^ 2) ^ 2, 1 / Sqrt[s + a], 1 / (s * Sqrt[s]), 1 / (s * Sqrt[s + a]), 1 / Sqrt[s ^ 2 + a ^ 2], Exp[-a / s] / Sqrt[s], Exp[(-a) * Sqrt[s]] / Sqrt[s], Exp[(-a) * Sqrt[s]] / s, Log[s] / s, Log[(s + a) / (s + b)]}; In[2]:= Grid[Prepend[{#, Assuming[{a > 0}, Simplify[InverseLaplaceTransform[#1, s, t]]]}& /@ Flist, {F[s], InverseLaplaceTransform[F[s], s, t]}], IconizedObject[«Grid options»]]//TraditionalForm Out[2]//TraditionalForm= $$\begin{array}{cc} F(s) & \mathcal{L}_s^{-1}[F(s)](t) \\ 1 & \delta (t) \\ \frac{1}{s-a} & e^{a t} \\ \frac{1}{s^2} & t \\ \frac{a}{a^2+s^2} & \sin (a t) \\ \frac{s}{a^2+s^2} & \cos (a t) \\ \frac{s}{\left(s^2-a^2\right)^2} & \frac{t e^{-a ... ^{-\frac{a^2}{4 t}}}{\sqrt{\pi } \sqrt{t}} \\ \frac{e^{-a \sqrt{s}}}{s} & \text{erfc}\left(\frac{a}{2 \sqrt{t}}\right) \\ \frac{\log (s)}{s} & -\log (t)-\gamma \\ \log \left(\frac{a+s}{b+s}\right) & \frac{e^{-b t}-e^{-a t}}{t} \\ \end{array}$$ ``` ## See Also * [`LaplaceTransform`](https://reference.wolfram.com/language/ref/LaplaceTransform.en.md) * [`InverseBilateralLaplaceTransform`](https://reference.wolfram.com/language/ref/InverseBilateralLaplaceTransform.en.md) * [`InverseFourierTransform`](https://reference.wolfram.com/language/ref/InverseFourierTransform.en.md) * [`InverseZTransform`](https://reference.wolfram.com/language/ref/InverseZTransform.en.md) * [`InverseMellinTransform`](https://reference.wolfram.com/language/ref/InverseMellinTransform.en.md) * [`Integrate`](https://reference.wolfram.com/language/ref/Integrate.en.md) * [`Asymptotic`](https://reference.wolfram.com/language/ref/Asymptotic.en.md) ## Tech Notes * [Integral Transforms and Related Operations](https://reference.wolfram.com/language/tutorial/Calculus.en.md#26017) * [Introduction to Fractional Calculus](https://reference.wolfram.com/language/tutorial/FractionalCalculus.en.md) ## Related Guides * [Fractional Calculus](https://reference.wolfram.com/language/guide/FractionalCalculus.en.md) * [Integral Transforms](https://reference.wolfram.com/language/guide/IntegralTransforms.en.md) * [Signal Transforms](https://reference.wolfram.com/language/guide/SignalTransforms.en.md) ## Related Links * [MathWorld](https://mathworld.wolfram.com/LaplaceTransform.html) ## History * Introduced in 1999 (4.0) \| [Updated in 2020 (12.2)](https://reference.wolfram.com/language/guide/SummaryOfNewFeaturesIn122.en.md) ▪ [2023 (13.3)](https://reference.wolfram.com/language/guide/SummaryOfNewFeaturesIn133.en.md)