--- title: "InverseBilateralLaplaceTransform" language: "en" type: "Symbol" summary: "InverseBilateralLaplaceTransform[expr, s, t] gives the inverse bilateral Laplace transform of expr. InverseBilateralLaplaceTransform[expr, {s1, s2, ..., sn}, {t1, t2, ..., tn}] gives the multidimensional inverse bilateral Laplace transform of expr." keywords: - function transform - bilateral transform - two-sided Laplace transform - impulse response - integral operator - integral transform - Integrate - inverse Laplace transform - Laplace transform - step response - transfer function - transient analysis - ilaplace - invlaplace canonical_url: "https://reference.wolfram.com/language/ref/InverseBilateralLaplaceTransform.html" source: "Wolfram Language Documentation" related_guides: - title: "Integral Transforms" link: "https://reference.wolfram.com/language/guide/IntegralTransforms.en.md" related_functions: - title: "BilateralLaplaceTransform" link: "https://reference.wolfram.com/language/ref/BilateralLaplaceTransform.en.md" - title: "InverseLaplaceTransform" link: "https://reference.wolfram.com/language/ref/InverseLaplaceTransform.en.md" - title: "LaplaceTransform" link: "https://reference.wolfram.com/language/ref/LaplaceTransform.en.md" - title: "InverseFourierTransform" link: "https://reference.wolfram.com/language/ref/InverseFourierTransform.en.md" - title: "InverseMellinTransform" link: "https://reference.wolfram.com/language/ref/InverseMellinTransform.en.md" - title: "InverseZTransform" link: "https://reference.wolfram.com/language/ref/InverseZTransform.en.md" - title: "InverseBilateralZTransform" link: "https://reference.wolfram.com/language/ref/InverseBilateralZTransform.en.md" --- # InverseBilateralLaplaceTransform InverseBilateralLaplaceTransform[expr, s, t] gives the inverse bilateral Laplace transform of expr. InverseBilateralLaplaceTransform[expr, {s1, s2, …, sn}, {t1, t2, …, tn}] gives the multidimensional inverse bilateral Laplace transform of expr. ## Details and Options * The inverse bilateral Laplace transform of a function $F(s)$ is defined to be $(1/2πi)Subsuperscript[∫, γ - i∞, γ + i∞]F(s)e^stds$, where the integration is along a vertical line $Re(s)=γ$, lying in a strip $α < Re(s) < β$ in which the function $F(s)$ is holomorphic. In some cases, the strip of analyticity may extend to a half-plane. [image] * The multidimensional inverse bilateral Laplace transform of a function $f$ is given by a contour integral of the form $Subsuperscript[∫, γ - i∞, γ + i∞]Subsuperscript[∫, δ - i∞, δ + i∞] ⋯ f(Subscript[s, 1], Subscript[s, 2], …)E^(Subscript[s, 1] Subscript[t, 1] + Subscript[s, 2] Subscript[t, 2] + …)\[DifferentialD]Subscript[s, 1]\[DifferentialD]Subscript[s, 2]…$. * The integral is computed using numerical methods if the third argument, $t$, is given a numerical value. * 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 | --- ## Examples (21) ### Basic Examples (2) Inverse bilateral Laplace transform of a function: ```wl In[1]:= InverseBilateralLaplaceTransform[ConditionalExpression[(1/(1 + s)(1 - s)), -1 < Re[s] < 1], s, t] Out[1]= Piecewise[{{1/(E^t*2), t >= 0}}, E^t/2] In[2]:= Plot[%, {t, -2, 2}, PlotRange -> All] Out[2]= [image] ``` --- Function with a parameter: ```wl In[1]:= InverseBilateralLaplaceTransform[ConditionalExpression[(Exp[a s]/(s^2 + 1)), Re[s] < 0], s, t] Out[1]= Piecewise[{{0, a + t >= 0}}, ((1/2)*I*(-1 + E^(2*I*(a + t))))/E^(I*(a + t))] ``` Plot the result: ```wl In[2]:= Plot[Evaluate[Table[%, {a, -2, 2}]], {t, -3, 3}, PlotRange -> All] Out[2]= [image] ``` ### Scope (13) Inverse bilateral Laplace transform of rational function with two real poles: ```wl In[1]:= InverseBilateralLaplaceTransform[ConditionalExpression[(1/s(s - 1)), 0 < Re[s] < 1], s, t] Out[1]= Piecewise[{{-1, t >= 0}}, -E^t] ``` --- Rational function with two real and two complex poles: ```wl In[1]:= InverseBilateralLaplaceTransform[ConditionalExpression[(s/(3 + s) (1 + s^2)(s - 5)), 0 < Re[s] < 1], s, t] Out[1]= Piecewise[{{-((5*E^(5*t))/208), t < 0}}, (E^((-3 - I)*t)*(39*E^(I*t) - (32 + 4*I)*E^(3*t) - (32 - 4*I)*E^((3 + 2*I)*t)))/1040] ``` --- The following function has two real and four complex poles: ```wl In[1]:= F[s_] := (1/(s^2 - 1)(s^2 + 1)((s + 1)^2 + 4)) In[2]:= ComplexPlot[F[s], {s, -3 - 3I, 3 + 3I}] Out[2]= [image] In[3]:= InverseBilateralLaplaceTransform[ConditionalExpression[F[s], -1 < Re[s] < 0], s, t] Out[3]= Piecewise[{{(1/320)*E^((-1 - 2*I)*t)*((-3 - I) - 20*E^(2*I*t) - (3 - I)*E^(4*I*t)), t >= 0}}, ((-(1/160))*((4 - 8*I) + (4 + 8*I)*E^(2*I*t) + 5*E^((1 + I)*t)))/E^(I*t)] In[4]:= Plot[%, {t, -4, 4}] Out[4]= [image] ``` --- Inverse bilateral Laplace transform of a product of rational and exponential functions: ```wl In[1]:= InverseBilateralLaplaceTransform[ConditionalExpression[(s Exp[4s]/(3 + s) (1 + s^2)), -1 < Re[s] < 0], s, t] Out[1]= Piecewise[{{-(3/10)/E^(3*(4 + t)), t >= -4}}, ((1/20)*((-3 - I) - (3 - I)*E^(2*I*(4 + t))))/ E^(I*(4 + t))] ``` --- A rational function with different strips of convergences has different inverse bilateral Laplace transforms: ```wl In[1]:= InverseBilateralLaplaceTransform[ConditionalExpression[(1/s(s - 4)(s - 8)), 0 < Re[s] < 4], s, t] Out[1]= Piecewise[{{1/32, t >= 0}}, (-(1/32))*E^(4*t)*(-2 + E^(4*t))] In[2]:= Plot[%, {t, -1, 1}] Out[2]= [image] In[3]:= InverseBilateralLaplaceTransform[ConditionalExpression[(1/s(s - 4)(s - 8)), 4 < Re[s] < 8], s, t] Out[3]= Piecewise[{{-(E^(8*t)/32), t < 0}}, (1/32)*(1 - 2*E^(4*t))] In[4]:= Plot[%, {t, -1, 1}] Out[4]= [image] ``` --- Rational function whose region of convergence is in the left half-plane: ```wl In[1]:= InverseBilateralLaplaceTransform[ConditionalExpression[(s/(s ^ 2 - 9)(s - 1)), Re[s] < -3], s, t] Out[1]= Piecewise[{{((1/8)*(1 + E^(4*t) - 2*E^(6*t)))/E^(3*t), t < 0}}, 0] ``` --- Function with region of convergence in the right half-plane: ```wl In[1]:= InverseBilateralLaplaceTransform[ConditionalExpression[(Exp[7s]/s^3(s - I / 3)), Re[s] > 0], s, t] Out[1]= Piecewise[{{-27*I + 27*I*E^((1/3)*I*(7 + t)) + 9*(7 + t) + (3/2)*I*(7 + t)^2, 7 + t >= 0}}, 0] In[2]:= ReImPlot[%, {t, -4, 4}] Out[2]= [image] ``` --- The inverse bilateral Laplace transform of the following rational function is a decaying sinusoidal wave: ```wl In[1]:= InverseBilateralLaplaceTransform[ConditionalExpression[-(2 (-101 + s^2)/(101 - 2 s + s^2) (101 + 2 s + s^2)), -1 < Re[s] < 1], s, t]//FullSimplify Out[1]= Piecewise[{{Cos[10*t]/E^t, t >= 0}}, E^t*Cos[10*t]] In[2]:= Plot[%, {t, -3, 3}] Out[2]= [image] ``` --- Inverse bilateral Laplace transform of a function that is analytic in the whole complex plane: ```wl In[1]:= InverseBilateralLaplaceTransform[(Exp[s] - 1/s), s, t] Out[1]= (1/2) (-Sign[t] + Sign[1 + t]) In[2]:= Plot[%, {t, -2, 2}, Exclusions -> None] Out[2]= [image] ``` --- Inverse bilateral Laplace transform leading to a Gaussian function: ```wl In[1]:= InverseBilateralLaplaceTransform[Exp[s^2 / 2], s, t] Out[1]= (E^-(t^2/2)/Sqrt[2 π]) ``` --- Inverse bilateral Laplace transform of a constant is a Dirac delta function: ```wl In[1]:= InverseBilateralLaplaceTransform[1, s, t] Out[1]= DiracDelta[t] ``` --- Evaluate the inverse bilateral Laplace transform at a single point: ```wl In[1]:= InverseBilateralLaplaceTransform[ConditionalExpression[(s^3 + 1/s(s^2 - 100)(s - 64)), 0 < Re[s] < 10], s, 6.7] Out[1]= 0.00015625 ``` --- Inverse bilateral Laplace transform at a single point for an analytic function: ```wl In[1]:= InverseBilateralLaplaceTransform[(11(Exp[s^2] - 1)/s), s, -1.5] Out[1]= 1.58864  + 0. I ``` ### Options (4) #### Assumptions (4) Specify the range for a parameter using ``Assumptions`` : ```wl In[1]:= InverseBilateralLaplaceTransform[ConditionalExpression[(1/s(s - a)), 0 < Re[s] < 1], s, t, Assumptions -> a > 1] Out[1]= Piecewise[{{-(1/a), t >= 0}}, -(E^(a*t)/a)] ``` --- Use ``Assumptions`` to place a pole outside the strip of convergence: ```wl In[1]:= InverseBilateralLaplaceTransform[ConditionalExpression[(1/s(s - a)(s - b)), 0 < Re[s] < a], s, t, Assumptions -> b > a] Out[1]= Piecewise[{{1/(a*b), t >= 0}}, ((-b)*E^(a*t) + a*E^(b*t))/(a*(a - b)*b)] ``` --- Use ``Assumptions`` to restrict the right end of the convergence strip in the left half-plane: ```wl In[1]:= InverseBilateralLaplaceTransform[ConditionalExpression[(11/(s + 1 / 2)(s + a)s), -1 / 2 < Re[s] < a], s, t, Assumptions -> a < 0] Out[1]= Piecewise[{{44/(E^(t/2)*(1 - 2*a)), t >= 0}}, (22*(-1 + 1/(E^(a*t)*(1 - 2*a))))/a] ``` --- Specify the range for a parameter using ``Assumptions`` for a multivariate case: ```wl In[1]:= InverseBilateralLaplaceTransform[ConditionalExpression[(1/p(s - a)), 0 < Re[s] < 1], {s, p}, {x, y}, Assumptions -> a > 1] Out[1]= (Piecewise[{{-E^(a*x), x < 0}}, 0]) UnitStep[y] ``` ### Properties & Relations (1) ``InverseBilateralLaplaceTransform`` and ``BilateralLaplaceTransform`` are mutual inverses: ```wl In[1]:= InverseBilateralLaplaceTransform[BilateralLaplaceTransform[f[t], t, s], s, t] Out[1]= f[t] In[2]:= BilateralLaplaceTransform[InverseBilateralLaplaceTransform[F[s], s, t], t, s] Out[2]= F[s] ``` ### Neat Examples (1) Create a table of basic inverse bilateral Laplace transforms: ```wl In[1]:= Flist = {...}; In[2]:= Grid[Prepend[{#, FullSimplify[InverseBilateralLaplaceTransform[#1, s, t]]}& /@ Flist, {F[s], InverseBilateralLaplaceTransform[F[s], s, t]}], IconizedObject[«Grid options»]]//TraditionalForm Out[2]//TraditionalForm= | | | | -------------------------------------------------------- | --------------------------------------------- | | F(s) ... *t))*(cos*t), t < 0}}, 0] | | ConditionalExpression[a/((s - a)^2 + 1), Re*s > a] | Piecewise[{{a*E^(a*t)*(sin*t), t >= 0}}, 0] | | ConditionalExpression[a/((s - a)^2 + 1), Re*s < a] | Piecewise[{{(-a)*E^(a*t)*(sin*t), t < 0}}, 0] | ``` ## See Also * [`BilateralLaplaceTransform`](https://reference.wolfram.com/language/ref/BilateralLaplaceTransform.en.md) * [`InverseLaplaceTransform`](https://reference.wolfram.com/language/ref/InverseLaplaceTransform.en.md) * [`LaplaceTransform`](https://reference.wolfram.com/language/ref/LaplaceTransform.en.md) * [`InverseFourierTransform`](https://reference.wolfram.com/language/ref/InverseFourierTransform.en.md) * [`InverseMellinTransform`](https://reference.wolfram.com/language/ref/InverseMellinTransform.en.md) * [`InverseZTransform`](https://reference.wolfram.com/language/ref/InverseZTransform.en.md) * [`InverseBilateralZTransform`](https://reference.wolfram.com/language/ref/InverseBilateralZTransform.en.md) ## Related Guides * [Integral Transforms](https://reference.wolfram.com/language/guide/IntegralTransforms.en.md) ## History * [Introduced in 2021 (12.3)](https://reference.wolfram.com/language/guide/SummaryOfNewFeaturesIn123.en.md)