--- title: "LaplaceTransform" language: "en" type: "Symbol" summary: "LaplaceTransform[f[t], t, s] gives the symbolic Laplace transform of f[t] in the variable t as F[s] in the variable s. LaplaceTransform[f[t], t, OverscriptBox[s, ^]] gives the numeric Laplace transform at the numerical value OverscriptBox[s, ^]. LaplaceTransform[f[t1, ..., tn], {t1, ..., tn}, {s1, ..., sn}] gives the multidimensional Laplace transform of f[t1, ..., tn]." keywords: - transfer function - transient analysis - unit impulse - step response - Dirac delta function - function transform - integral operator - integral transform - Integrate - inverse Laplace transform - Laplace transform - operational calculus - inttrans - laplace canonical_url: "https://reference.wolfram.com/language/ref/LaplaceTransform.html" source: "Wolfram Language Documentation" related_guides: - title: "Fractional Calculus" link: "https://reference.wolfram.com/language/guide/FractionalCalculus.en.md" - title: "Fourier Analysis" link: "https://reference.wolfram.com/language/guide/FourierAnalysis.en.md" - title: "Generalized Functions" link: "https://reference.wolfram.com/language/guide/GeneralizedFunctions.en.md" - title: "Signal Transforms" link: "https://reference.wolfram.com/language/guide/SignalTransforms.en.md" - title: "Calculus" link: "https://reference.wolfram.com/language/guide/Calculus.en.md" - title: "Integral Transforms" link: "https://reference.wolfram.com/language/guide/IntegralTransforms.en.md" - title: "Signal Processing" link: "https://reference.wolfram.com/language/guide/SignalProcessing.en.md" - title: "Summation Transforms" link: "https://reference.wolfram.com/language/guide/SummationTransforms.en.md" related_functions: - title: "InverseLaplaceTransform" link: "https://reference.wolfram.com/language/ref/InverseLaplaceTransform.en.md" - title: "BilateralLaplaceTransform" link: "https://reference.wolfram.com/language/ref/BilateralLaplaceTransform.en.md" - title: "FourierTransform" link: "https://reference.wolfram.com/language/ref/FourierTransform.en.md" - title: "UnilateralConvolve" link: "https://reference.wolfram.com/language/ref/UnilateralConvolve.en.md" - title: "ZTransform" link: "https://reference.wolfram.com/language/ref/ZTransform.en.md" - title: "Integrate" link: "https://reference.wolfram.com/language/ref/Integrate.en.md" - title: "Piecewise" link: "https://reference.wolfram.com/language/ref/Piecewise.en.md" - title: "TransferFunctionModel" link: "https://reference.wolfram.com/language/ref/TransferFunctionModel.en.md" - title: "MellinTransform" link: "https://reference.wolfram.com/language/ref/MellinTransform.en.md" - title: "Asymptotic" link: "https://reference.wolfram.com/language/ref/Asymptotic.en.md" - title: "FractionalD" link: "https://reference.wolfram.com/language/ref/FractionalD.en.md" - title: "CaputoD" link: "https://reference.wolfram.com/language/ref/CaputoD.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" --- # LaplaceTransform LaplaceTransform[f[t], t, s] gives the symbolic Laplace transform of f[t] in the variable t as F[s] in the variable s. LaplaceTransform[f[t], t, Overscript[s, ^ ]] gives the numeric Laplace transform at the numerical value Overscript[s, ^ ]. LaplaceTransform[f[t1, …, tn], {t1, …, tn}, {s1, …, sn}] gives the multidimensional Laplace transform of f[t1, …, tn]. ## 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 Laplace transform of a function $f(t)$ is defined to be $\int _0^{\infty }f(t)e^{-st}dt$. * The multidimensional Laplace transform is given by $\int _0^{\infty }\int _0^{\infty } \cdots f\left(t_1,t_2,\ldots \right)e^{-\left(s_1 t_1+s_2 t_2+\ldots \right)}dt_1dt_2\ldots$. * The integral is computed using numerical methods if the third argument, ``s``, is given a numerical value. * The asymptotic Laplace transform can be computed using ``Asymptotic``. * The Laplace transform of $f(t)$ exists only for complex values of ``s`` in a half-plane $\Re(s)>\alpha$. [image] * The lower limit of the integral is effectively taken to be $0_-$, so that the Laplace transform of the Dirac delta function $\delta (t)$ is equal to 1. » * 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 | | PrincipalValue | False | whether to find Cauchy principal value | | WorkingPrecision | Automatic | the precision used in internal computations | * Use ``GenerateConditions -> "ConvergenceRegion"`` to obtain the region of convergence for the Laplace transform. * In ``TraditionalForm``, ``LaplaceTransform`` is output using $\mathcal{L}$. » --- ## Examples (93) ### Basic Examples (4) Compute the Laplace transform of a function: ```wl In[1]:= LaplaceTransform[ Exp[-t], t, s] Out[1]= (1/1 + s) ``` --- Define a piecewise function: ```wl In[1]:= f[t_] := t UnitStep[t - 1] In[2]:= Plot[f[t], {t, 0, 3}] Out[2]= [image] ``` Compute its Laplace transform: ```wl In[3]:= LaplaceTransform[f[t], t, s] Out[3]= E^-s ((1/s^2) + (1/s)) ``` Compute the transform at a single point: ```wl In[4]:= LaplaceTransform[f[t], t, 0.3] Out[4]= 10.7007 ``` --- Compute the Laplace transform of a multivariate function: ```wl In[1]:= LaplaceTransform[x Exp[-y], {x, y}, {p, q}] Out[1]= (1/p^2 (1 + q)) ``` --- Define a multivariate piecewise function: ```wl In[1]:= f[x_, y_] = UnitStep[x + y - 1] x; In[2]:= Plot3D[f[x, y], {x, 0, 1}, {y, 0, 1}] Out[2]= [image] ``` Compute its Laplace transform: ```wl In[3]:= LaplaceTransform[f[x, y], {x, y}, {p, q}]//FullSimplify Out[3]= ((E^-q/q) + (E^-p (q + p (-2 - p + q))/p^2)/(p - q)^2) ``` ### Scope (67) #### Basic Uses (4) Laplace transform of a function for a symbolic parameter ``s`` : ```wl In[1]:= LaplaceTransform[1, t, s] Out[1]= (1/s) ``` --- Laplace transforms of trigonometric functions: ```wl In[1]:= LaplaceTransform[Sin[ω t], t, s] Out[1]= (ω/s^2 + ω^2) In[2]:= LaplaceTransform[Cos[ω t], t, s] Out[2]= (s/s^2 + ω^2) ``` --- Evaluate the Laplace transform for a numerical value of the parameter ``s`` : ```wl In[1]:= LaplaceTransform[Sin[t], t, 1.2] Out[1]= 0.409836 ``` --- ``TraditionalForm`` formatting: ```wl In[1]:= LaplaceTransform[f[x], x, s]//TraditionalForm Out[1]//TraditionalForm= $$\mathcal{L}_x[f(x)](s)$$ ``` #### Elementary Functions (13) Laplace transform of a power function: ```wl In[1]:= LaplaceTransform[t^n, t, s] Out[1]= s^-1 - n Gamma[1 + n] ``` --- Square root function: ```wl In[1]:= LaplaceTransform[Sqrt[t], t, s] Out[1]= (Sqrt[π]/2 s^3 / 2) ``` --- Laplace transforms of polynomials: ```wl In[1]:= LaplaceTransform[3t ^ 2 + 5t - 7, t, s] Out[1]= (6/s^3) + (5/s^2) - (7/s) In[2]:= LaplaceTransform[Sum[a[i]t^i, {i, 1, n}], t, s] Out[2]= Underoverscript[∑, i = 1, n]s^-1 - i a[i] Gamma[1 + i] ``` --- Exponential function: ```wl In[1]:= LaplaceTransform[E ^ (-a t), t, s] Out[1]= (1/a + s) ``` --- Product of an exponential and a linear function: ```wl In[1]:= LaplaceTransform[(1 + a t)Exp[a t], t, s]//Together Out[1]= (s/(a - s)^2) ``` --- Expressions involving trigonometric functions: ```wl In[1]:= LaplaceTransform[Exp[a t] Sin[ω t], t, s] Out[1]= (ω/(-a + s)^2 + ω^2) In[2]:= LaplaceTransform[(1/2ω)(Sin[ω t] + ω t Cos[ω t]), t, s]//Simplify Out[2]= (s^2/(s^2 + ω^2)^2) ``` --- Expressions involving hyperbolic functions: ```wl In[1]:= LaplaceTransform[(b Sinh[a t] - a Sinh[b t]/a^2 - b^2), t, s]//Simplify Out[1]= -(a b/(a^2 - s^2) (-b^2 + s^2)) In[2]:= LaplaceTransform[t Cosh[a t], t, s] Out[2]= (a^2 + s^2/(-a^2 + s^2)^2) ``` --- Ratio of an exponential and a linear function: ```wl In[1]:= LaplaceTransform[(Exp[t] - 1/t), t, s] Out[1]= Log[(s/-1 + s)] ``` --- Ratio of sine and linear functions: ```wl In[1]:= LaplaceTransform[(Sin[3 t]/t), t, s] Out[1]= ArcTan[(3/s)] ``` --- Composition of elementary functions: ```wl In[1]:= LaplaceTransform[(a Exp[-(a^2/4 t)]/2 Sqrt[π] t^3 / 2), t, s, Assumptions -> a > 0] Out[1]= E^-a Sqrt[s] In[2]:= LaplaceTransform[Sin[t ^ 2], t, s] Out[2]= (1/2) Sqrt[(π/2)] (Cos[(s^2/4)] (1 - 2 FresnelC[(s/Sqrt[2 π])]) + (1 - 2 FresnelS[(s/Sqrt[2 π])]) Sin[(s^2/4)]) ``` --- Logarithmic function: ```wl In[1]:= LaplaceTransform[Log[t], t, s] Out[1]= -(EulerGamma + Log[s]/s) ``` --- Product of logarithmic and power functions: ```wl In[1]:= LaplaceTransform[t^ν - 1Log[t], t, s] Out[1]= s^-ν Gamma[ν] (-Log[s] + PolyGamma[0, ν]) ``` --- Square of a logarithmic function: ```wl In[1]:= LaplaceTransform[Log[t] ^ 2, t, s] Out[1]= (6 EulerGamma^2 + π^2 + 6 Log[s] (2 EulerGamma + Log[s])/6 s) ``` #### Special Functions (10) Laplace transform of error and square root functions composition: ```wl In[1]:= Simplify[LaplaceTransform[(1/Sqrt[a])Erf[Sqrt[a t]], t, s], Assumptions -> {a > 0, s > 0}] Out[1]= (1/s Sqrt[a + s]) ``` --- Bessel functions: ```wl In[1]:= LaplaceTransform[BesselJ[0, a t], t, s] Out[1]= (1/Sqrt[a^2 + s^2]) In[2]:= LaplaceTransform[BesselI[0, a t], t, s] Out[2]= (1/Sqrt[-a^2 + s^2]) ``` Products involving Bessel functions: ```wl In[3]:= Simplify[LaplaceTransform[((t/2a))^ν - 1 / 2BesselJ[ν - 1 / 2, a t], t, s], Assumptions -> a > 0] Out[3]= ((a^2 + s^2)^-ν Gamma[ν]/Sqrt[π]) In[4]:= Simplify[LaplaceTransform[((t/2a))^ν - 1 / 2BesselI[ν - 1 / 2, a t], t, s], Assumptions -> a > 0] Out[4]= ((-a^2 + s^2)^-ν Gamma[ν]/Sqrt[π]) ``` --- Sine integral function: ```wl In[1]:= LaplaceTransform[SinIntegral[t], t, s] Out[1]= (ArcCot[s]/s) ``` --- Laguerre polynomials: ```wl In[1]:= LaplaceTransform[LaguerreL[n, t], t, s] Out[1]= (-1 + s)^n s^-1 - n ``` --- Airy function: ```wl In[1]:= LaplaceTransform[AiryAi[t], t, s] Out[1]= (1/4 3^5 / 6 π)E^-(s^3/3) s (-2 3^2 / 3 s ExpIntegralE[(1/3), -(s^3/3)] Gamma[(4/3)] + 3 ExpIntegralE[(2/3), -(s^3/3)] Gamma[(5/3)]) ``` --- Chebyshev polynomial: ```wl In[1]:= LaplaceTransform[((1 + t)^n/Sqrt[t])ChebyshevT[n, (1 - t/1 + t)], t, s] Out[1]= 2^-2 n Sqrt[π] s^-(1/2) - n HermiteH[2 n, Sqrt[s]] ``` --- Struve function: ```wl In[1]:= LaplaceTransform[(StruveH[1 / 2, a t]/Sqrt[t]), t, s] Out[1]= (Sqrt[(1/a)] Log[1 + (a^2/s^2)]/Sqrt[2 π]) ``` --- Fresnel function: ```wl In[1]:= LaplaceTransform[FresnelC[Sqrt[(2/Pi)]Sqrt[a * t]], t, s] Out[1]= (Sqrt[a] Sqrt[s + Sqrt[a^2 + s^2]]/2 s Sqrt[a^2 + s^2]) ``` --- Gamma function: ```wl In[1]:= LaplaceTransform[Gamma[nu, a / t], t, s] Out[1]= (2 (a s)^nu / 2 BesselK[nu, 2 Sqrt[a s]]/s) ``` --- Hypergeometric function: ```wl In[1]:= LaplaceTransform[t^5 / 3HypergeometricPFQ[{-(7/4), (11/4)}, {1, (8/3)}, t], t, s] Out[1]= (Gamma[(8/3)] LegendreP[(7/4), 1 - (2/s)]/s^8 / 3) ``` #### Piecewise Functions (9) Laplace transform of a piecewise function: ```wl In[1]:= f[t_] = Piecewise[{{t, 0 ≤ t ≤ 1}, {1, t > 1}}]; In[2]:= Plot[f[t], {t, 0, 3}] Out[2]= [image] In[3]:= LaplaceTransform[f[t], t, s] Out[3]= (E^-s (-1 + E^s)/s^2) ``` --- Restriction of a sine function to a half-period: ```wl In[1]:= f[t_] = Piecewise[{{Sin[ω t], 0 < t < (π/ω)}, {0, (π/ω) ≤ t}}]; In[2]:= Plot[f[t] /. ω -> π, {t, 0, 2}] Out[2]= [image] In[3]:= LaplaceTransform[f[t], t, s, Assumptions -> ω > 0] Out[3]= (E^-(π s/ω) (1 + E^(π s/ω)) ω/s^2 + ω^2) ``` --- Exponential function with a left cutoff: ```wl In[1]:= f[t_] = Piecewise[{{2, t ≤ 1}, {Exp[t], t > 1}}]; In[2]:= Plot[f[t], {t, 0, 2}] Out[2]= [image] In[3]:= LaplaceTransform[f[t], t, s]//FullSimplify Out[3]= (2 + (E^-s (2 + (-2 + E) s)/-1 + s)/s) ``` --- Triangular function: ```wl In[1]:= f[t_] = Piecewise[{{t, 0 ≤ t ≤ 1}, {2 - t, 1 < t ≤ 2}, {0, 2 < t }}]; In[2]:= Plot[f[t], {t, 0, 4}] Out[2]= [image] In[3]:= LaplaceTransform[f[t], t, s] Out[3]= (E^-2 s (-1 + E^s)^2/s^2) ``` --- Polynomial function with a left cutoff: ```wl In[1]:= f[t_] = Piecewise[{{0, 0 ≤ t < 1}, {(t - 1)^2, 1 ≤ t}}]; In[2]:= Plot[f[t], {t, 0, 3}] Out[2]= [image] In[3]:= LaplaceTransform[f[t], t, s] Out[3]= (2 E^-s/s^3) ``` --- Ramp: ```wl In[1]:= LaplaceTransform[Ramp[t - a], t, s, Assumptions -> a > 0] Out[1]= (E^-a s/s^2) ``` --- ``UnitStep`` : ```wl In[1]:= LaplaceTransform[UnitStep[t - a], t, s, Assumptions -> a > 0] Out[1]= (E^-a s/s) ``` --- Product of ``UnitStep`` and cosine functions: ```wl In[1]:= LaplaceTransform[UnitStep[t - π]Cos[t - π], t, s] Out[1]= (E^-π s s/1 + s^2) ``` --- Laplace transform of ``Floor`` : ```wl In[1]:= LaplaceTransform[Floor[t], t, s] Out[1]= (1/(-1 + E^s) s) In[2]:= Plot[Floor[t], {t, 0, 10}] Out[2]= [image] ``` #### Periodic Functions (5) Laplace transform of ``SquareWave`` : ```wl In[1]:= LaplaceTransform[SquareWave[t], t, s] Out[1]= (Tanh[(s/4)]/s) In[2]:= Plot[SquareWave[t], {t, 0, 4}] Out[2]= [image] ``` --- ``TriangleWave`` : ```wl In[1]:= LaplaceTransform[TriangleWave[t], t, s] Out[1]= (4 - 4 Sech[(s/4)]/s^2) In[2]:= Plot[TriangleWave[t], {t, 0, 4}] Out[2]= [image] ``` --- ``SawtoothWave`` : ```wl In[1]:= LaplaceTransform[SawtoothWave[t], t, s] Out[1]= (2 + s - s Coth[(s/2)]/2 s^2) In[2]:= Plot[SawtoothWave[t], {t, 0, 6}] Out[2]= [image] ``` --- Full-wave-rectified function with period $\pi$ : ```wl In[1]:= LaplaceTransform[RealAbs[Sin[t]], t, s]//FullSimplify Out[1]= (Coth[(π s/2)]/1 + s^2) In[2]:= Plot[RealAbs[Sin[t]], {t, 0, 12}] Out[2]= [image] ``` --- Rectified wave: ```wl In[1]:= f[t_] = Piecewise[{{Sin[t], 0 ≤ Mod[(t/π), 2] ≤ 1}, {0, 1 < Mod[(t/π), 2] < 2}}]; In[2]:= LaplaceTransform[f[t], t, s] Out[2]= (1/1 + s^2 - E^-π s (1 + s^2)) In[3]:= Plot[f[t], {t, 0, 24}] Out[3]= [image] ``` #### Generalized Functions (5) Laplace transform of ``HeavisideTheta`` : ```wl In[1]:= LaplaceTransform[HeavisideTheta[t], t, s] Out[1]= (1/s) ``` --- ``DiracDelta`` : ```wl In[1]:= LaplaceTransform[DiracDelta[t - 1], t, s] Out[1]= E^-s ``` --- Derivative of ``DiracDelta`` : ```wl In[1]:= LaplaceTransform[DiracDelta'[t - 1], t, s] Out[1]= E^-s s ``` --- ``HeavisideLambda`` : ```wl In[1]:= Plot[HeavisideLambda[t - 1], {t, 0, 4}] Out[1]= [image] In[2]:= LaplaceTransform[HeavisideLambda[t - 1], t, s] Out[2]= (E^-2 s (-1 + E^s)^2/s^2) ``` --- ``HeavisidePi`` : ```wl In[1]:= LaplaceTransform[HeavisidePi[t - (3/2)], t, s] Out[1]= (E^-2 s (-1 + E^s)/s) In[2]:= Plot[HeavisidePi[t - (3/2)], {t, 0, 3}] Out[2]= [image] ``` #### Multivariate Functions (9) Bivariate Laplace transform of a constant: ```wl In[1]:= LaplaceTransform[1, {x, y}, {p, q}] Out[1]= (1/p q) ``` --- Exponential function: ```wl In[1]:= LaplaceTransform[Exp[a x + b y], {x, y}, {p, q}] Out[1]= (1/(-a + p) (-b + q)) ``` --- Power function: ```wl In[1]:= LaplaceTransform[(x^m y^n/Gamma[m + 1]Gamma[n + 1]), {x, y}, {p, q}] Out[1]= p^-1 - m q^-1 - n ``` --- ``BesselJ`` : ```wl In[1]:= LaplaceTransform[BesselJ[0, 2 Sqrt[a x y]], {x, y}, {p, q}]//Simplify Out[1]= (1/a + p q) ``` --- Square root: ```wl In[1]:= LaplaceTransform[(1/Sqrt[π (x + y)]), {x, y}, {p, q}, Assumptions -> {p > 0, q > 0}] Out[1]= (1/Sqrt[p] (q + Sqrt[p q])) ``` --- Composition of cosine and square root: ```wl In[1]:= LaplaceTransform[(Cos[Sqrt[x y ]]/Sqrt[x y ]), {x, y}, {p, q}] Out[1]= (2 π/Sqrt[p] Sqrt[(1/p) + 4 q]) ``` --- Laplace transform of a multivariate power function: ```wl In[1]:= LaplaceTransform[(x^m y^nz^k/Gamma[m + 1]Gamma[n + 1]Gamma[k + 1]), {x, y, z}, {p, q, r}] Out[1]= p^-1 - m q^-1 - n r^-1 - k ``` --- Cosine: ```wl In[1]:= LaplaceTransform[Cos[x + y + z], {x, y, z}, {p, q, r}]//Simplify Out[1]= -(p + q + r - p q r/(1 + p^2) (1 + q^2) (1 + r^2)) ``` --- Logarithm: ```wl In[1]:= Assuming[{p > 0, q > 0, r > 0}, FullSimplify[LaplaceTransform[Log[x y z], {x, y, z}, {p, q, r}]]] Out[1]= -(3 EulerGamma + Log[p q r]/p q r) ``` #### Formal Properties (6) The Laplace transform is a linear operator: ```wl In[1]:= LaplaceTransform[a f[t] + b g[t], t, s] Out[1]= a LaplaceTransform[f[t], t, s] + b LaplaceTransform[g[t], t, s] ``` --- Laplace transform of $f(t) \exp (a t)$ is the Laplace transform of $f(t)$ evaluated at $s-a$ : ```wl In[1]:= LaplaceTransform[Exp[a t]f[t], t, s] Out[1]= LaplaceTransform[f[t], t, -a + s] ``` --- Laplace transform of a first-order derivative: ```wl In[1]:= LaplaceTransform[f'[t], t, s] Out[1]= -f[0] + s LaplaceTransform[f[t], t, s] ``` --- Laplace transform of a second-order derivative: ```wl In[1]:= LaplaceTransform[f''[t], t, s] Out[1]= -s f[0] + s^2 LaplaceTransform[f[t], t, s] - Derivative[1][f][0] ``` --- Laplace transform of a product with monomials: ```wl In[1]:= LaplaceTransform[t f[t], t, s] Out[1]= -Subscript[∂, s]LaplaceTransform[f[t], t, s] In[2]:= LaplaceTransform[t ^ 2 f[t], t, s] Out[2]= Subscript[∂, {s, 2}]LaplaceTransform[f[t], t, s] ``` --- Laplace transform threads itself over equations: ```wl In[1]:= LaplaceTransform[f'[t] == Log[t], t, s] Out[1]= -f[0] + s LaplaceTransform[f[t], t, s] == -(EulerGamma + Log[s]/s) ``` #### Numerical Evaluation (3) Calculate the Laplace transform at a single point: ```wl In[1]:= LaplaceTransform[(Erf[t]/Sqrt[t]), t, 14.2] Out[1]= 0.018575 ``` Alternatively, calculate the Laplace transform symbolically: ```wl In[2]:= LaplaceTransform[(Erf[t]/Sqrt[t]), t, s] Out[2]= (1/3 Sqrt[π])((3 π/Sqrt[s]) - 6 Gamma[(3/4)] HypergeometricPFQ[{(1/4), (3/4)}, {(1/2), (5/4)}, (s^2/4)] + 2 s Gamma[(5/4)] HypergeometricPFQ[{(3/4), (5/4)}, {(3/2), (7/4)}, (s^2/4)]) ``` Then evaluate it for specific value of $s$ : ```wl In[3]:= N[% /. s -> Rationalize[14.2], 20] Out[3]= 0.018574955819252467322 ``` Plot the Laplace transform using numerical values only: ```wl In[4]:= f[s_ ? NumericQ] := LaplaceTransform[(Erf[t]/Sqrt[t]), t, s] In[5]:= Plot[f[s], {s, 0, 5}] Out[5]= [image] ``` --- For some functions, the Laplace transform cannot be evaluated symbolically: ```wl In[1]:= LaplaceTransform[(Erfc[t]/t^2 + 1), t, s] Out[1]= LaplaceTransform[(Erfc[t]/1 + t^2), t, s] ``` Evaluate the Laplace transform numerically and plot it: ```wl In[2]:= f[s_ ? NumericQ] := LaplaceTransform[(Erfc[t]/t^2 + 1), t, s] In[3]:= f[0.4] Out[3]= 0.403521 In[4]:= Plot[f[s], {s, 0, 10}] Out[4]= [image] ``` --- Calculate a multivariate Laplace transform at a single point in the plane: ```wl In[1]:= LaplaceTransform[(1/x + y + 1), {x, y}, {5.4, 4.5}] Out[1]= 0.0303025 ``` #### Fractional Calculus (3) Laplace transform of the ``MittagLefflerE`` functions: ```wl In[1]:= LaplaceTransform[(-MittagLefflerE[(1/2), (1/2), (Sqrt[t]/2)] + MittagLefflerE[(1/2), (1/2), (2 Sqrt[t]/3)]/Sqrt[t]), t, s] Out[1]= -(2/-1 + 2 Sqrt[s]) + (3/-2 + 3 Sqrt[s]) ``` ``ComplexPlot`` in the $s$-domain: ```wl In[2]:= ComplexPlot[%, {s, -1 - I, 1 + I}] Out[2]= [image] ``` Inverse Laplace transform to the time domain: ```wl In[3]:= InverseLaplaceTransform[-(2/-1 + 2 Sqrt[s]) + (3/-2 + 3 Sqrt[s]), s, t] Out[3]= (1/2) (-E^t / 4 - (2/Sqrt[π] Sqrt[t]) - E^t / 4 Erf[(Sqrt[t]/2)]) + (1/3) (2 E^4 t / 9 + (3/Sqrt[π] Sqrt[t]) + 2 E^4 t / 9 Erf[(2 Sqrt[t]/3)]) ``` --- Laplace transform of the ``MittagLefflerE`` functions involving parameters: ```wl In[1]:= LaplaceTransform[MittagLefflerE[α, a t^α], t, s] Out[1]= (s^-1 + α/-a + s^α) ``` Inverse Laplace transform to the time domain: ```wl In[2]:= InverseLaplaceTransform[%, s, t, Assumptions -> α > 0] Out[2]= MittagLefflerE[α, a t^α] ``` --- Laplace transform of the ``CaputoD`` fractional derivative: ```wl In[1]:= LaplaceTransform[CaputoD[f[t], {t, 3 / 2}], t, s] Out[1]= -Sqrt[s] f[0] + s^3 / 2 LaplaceTransform[f[t], t, s] - (Derivative[1][f][0]/Sqrt[s]) ``` Apply to sine function: ```wl In[2]:= % /. {f[t_] -> Sin[t], f'[t_] -> Cos[t]}//Simplify Out[2]= -(1/Sqrt[s] (1 + s^2)) ``` Compare this with the ``LaplaceTransform`` of the ``CaputoD`` derivative of the sine function: ```wl In[3]:= CaputoD[Sin[t], {t, 3 / 2}] Out[3]= -(1/Sqrt[π] Sqrt[t]) + (Sqrt[2 π] HypergeometricPFQ[{1}, {(1/4), (3/4)}, -(t^2/4)]/Sqrt[t] Gamma[(1/4)] Gamma[(3/4)]) In[4]:= LaplaceTransform[%, t, s]//FullSimplify Out[4]= -(1/Sqrt[s] (1 + s^2)) ``` ### Options (4) #### Assumptions (1) Specify the range for a parameter using ``Assumptions`` : ```wl In[1]:= LaplaceTransform[ArcTan[t / a], t, s, Assumptions -> a > 0] Out[1]= (2 CosIntegral[a s] Sin[a s] + Cos[a s] (π - 2 SinIntegral[a s])/2 s) ``` #### GenerateConditions (1) Use ``GenerateConditions -> True`` to get parameter conditions for when a result is valid: ```wl In[1]:= LaplaceTransform[E ^ (-a t ^ 2), t, s, GenerateConditions -> True] Out[1]= ConditionalExpression[(E^(s^2/(4*a))*Sqrt[Pi]*Erfc[s/(2*Sqrt[a])])/(2*Sqrt[a]), (Re[a] == 0 && Re[s] > 0) || Re[a] > 0] ``` #### Principal Value (1) The Laplace transform of the following function is not defined due to the singularity at $t=1$ : ```wl In[1]:= LaplaceTransform[(Sin[t]/t - 1), t, s] Out[1]= LaplaceTransform[(Sin[t]/-1 + t), t, s] ``` Use ``PrincipalValue`` to obtain the Cauchy principal value for the integral: ```wl In[2]:= LaplaceTransform[(Sin[t]/t - 1), t, s, PrincipalValue -> True]//FullSimplify Out[2]= (1/2) I E^-I - s (E^2 I ExpIntegralEi[-I + s] - ExpIntegralEi[I + s]) ``` #### Working Precision (1) Use ``WorkingPrecision`` to obtain a result with arbitrary precision: ```wl In[1]:= LaplaceTransform[(Exp[-t^2]/t^2 + 5t + 2), t, 15.7] Out[1]= 0.0276643 In[2]:= LaplaceTransform[(Exp[-t^2]/t^2 + 5t + 2), t, 15.7, WorkingPrecision -> 10] Out[2]= 0.02766429794 In[3]:= LaplaceTransform[(Exp[-t^2]/t^2 + 5t + 2), t, 15.7, WorkingPrecision -> 20] Out[3]= 0.027664297937238794917 ``` ### Applications (12) #### Ordinary Differential Equations (5) Solve a differential equation using Laplace transforms: ```wl In[1]:= LaplaceTransform[y''[t] + y[t] == 1, t, s] /. {y[0] -> 0, y'[0] -> 0} Out[1]= LaplaceTransform[y[t], t, s] + s^2 LaplaceTransform[y[t], t, s] == (1/s) ``` Solve for the Laplace transform: ```wl In[2]:= Solve[%, LaplaceTransform[y[t], t, s]] Out[2]= {{LaplaceTransform[y[t], t, s] -> (1/s (1 + s^2))}} ``` Find the inverse transform: ```wl In[3]:= InverseLaplaceTransform[%, s, t] Out[3]= {{y[t] -> 1 - Cos[t]}} ``` Plot the solution: ```wl In[4]:= Plot[y[t] /. %, {t, 0, 5Pi}] Out[4]= [image] ``` Find the solution directly using ``DSolve`` : ```wl In[5]:= DSolve[{y''[t] + y[t] == 1, y[0] == 0, y'[0] == 0}, y[t], t] Out[5]= {{y[t] -> 1 - Cos[t]}} ``` --- Solve the following differential equation: ```wl In[1]:= eqn = 4y''[t] - y[t] == UnitStep[t - 1]; ``` Solve for the Laplace transform: ```wl In[2]:= Solve[LaplaceTransform[eqn, t, s] /. {y[0] -> 0, y'[0] -> 1 / 25}, LaplaceTransform[y[t], t, s]] Out[2]= {{LaplaceTransform[y[t], t, s] -> (E^-s (25 + 4 E^s s)/25 s (-1 + 4 s^2))}} ``` Find the inverse transform: ```wl In[3]:= InverseLaplaceTransform[%, s, t]//FullSimplify Out[3]= {{y[t] -> (-1 + Cosh[(1 - t/2)]) HeavisideTheta[-1 + t] + (2/25) Sinh[(t/2)]}} ``` Plot the solution: ```wl In[4]:= Plot[y[t] /. %, {t, 0, 3}] Out[4]= [image] ``` --- Solve an RL circuit to find the current $i(t)$ : ```wl In[1]:= Solve[LaplaceTransform[L i'[t] + R i[t] == E0 Sin[ω t], t, s] /. i[0] -> 0, LaplaceTransform[i[t], t, s]] Out[1]= {{LaplaceTransform[i[t], t, s] -> (E0 ω/(R + L s) (s^2 + ω^2))}} In[2]:= InverseLaplaceTransform[%, s, t]//Expand Out[2]= {{i[t] -> (E^-(R t/L) E0 L ω/R^2 + L^2 ω^2) - (E0 L ω Cos[t ω]/R^2 + L^2 ω^2) + (E0 R Sin[t ω]/R^2 + L^2 ω^2)}} ``` Verify with ``DSolveValue`` : ```wl In[3]:= DSolveValue[{L i'[t] + R i[t] == E0 Sin[ω t], i[0] == 0}, i[t], t]//Expand Out[3]= (E^-(R t/L) E0 L ω/R^2 + L^2 ω^2) - (E0 L ω Cos[t ω]/R^2 + L^2 ω^2) + (E0 R Sin[t ω]/R^2 + L^2 ω^2) ``` --- Green's function for an RL circuit: ```wl In[1]:= Solve[LaplaceTransform[L i'[t] + R i[t] == DiracDelta[t - τ], t, s] /. i[0] -> 0, LaplaceTransform[i[t], t, s]] Out[1]= {{LaplaceTransform[i[t], t, s] -> (E^-s τ HeavisideTheta[τ]/R + L s)}} In[2]:= GreenF[t_, τ_] = i[t] /. Assuming[τ > 0, Refine[InverseLaplaceTransform[%, s, t]]][[1]] Out[2]= (E^(R (-t + τ)/L) HeavisideTheta[t - τ]/L) ``` Use the Green's function to solve the RL circuit: ```wl In[3]:= Integrate[GreenF[t, τ] * E0 Sin[ω τ], {τ, 0, ∞}, Assumptions -> {t > 0, R > 0, L > 0}]//Expand Out[3]= (E^-(R t/L) E0 L ω/R^2 + L^2 ω^2) - (E0 L ω Cos[t ω]/R^2 + L^2 ω^2) + (E0 R Sin[t ω]/R^2 + L^2 ω^2) ``` --- Solve a system of ODEs: ```wl In[1]:= Solve[LaplaceTransform[{y'[t] == -z[t], z'[t] == y[t]}, t, s] /. {y[0] -> 1, z[0] -> 0}, LaplaceTransform[{y[t], z[t]}, t, s]] Out[1]= {{LaplaceTransform[y[t], t, s] -> (s/1 + s^2), LaplaceTransform[z[t], t, s] -> (1/1 + s^2)}} In[2]:= InverseLaplaceTransform[%, s, t] Out[2]= {{y[t] -> Cos[t], z[t] -> Sin[t]}} ``` #### Fractional Differential Equations (3) Solve a fractional-order differential equation using Laplace transforms: ```wl In[1]:= LaplaceTransform[CaputoD[y[t], {t, 5 / 3}] + y[t] == 1, t, s] /. {y[0] -> 0, y'[0] -> -1} Out[1]= (1/s^1 / 3) + LaplaceTransform[y[t], t, s] + s^5 / 3 LaplaceTransform[y[t], t, s] == (1/s) ``` Solve for the Laplace transform: ```wl In[2]:= Solve[%, LaplaceTransform[y[t], t, s]] Out[2]= {{LaplaceTransform[y[t], t, s] -> (1 - s^2 / 3/s (1 + s^5 / 3))}} ``` Find the inverse transform: ```wl In[3]:= InverseLaplaceTransform[%, s, t] Out[3]= {{y[t] -> 1 - MittagLefflerE[(5/3), -t^5 / 3] - t MittagLefflerE[(5/3), 2, -t^5 / 3]}} ``` Plot the solution: ```wl In[4]:= Plot[y[t] /. %, {t, 0, 6}] Out[4]= [image] ``` Find the solution directly using ``DSolve`` : ```wl In[5]:= DSolve[{CaputoD[y[t], {t, 5 / 3}] + y[t] == 1, y[0] == 0, y'[0] == -1}, y[t], t] Out[5]= {{y[t] -> 1 - MittagLefflerE[(5/3), -t^5 / 3] - t MittagLefflerE[(5/3), 2, -t^5 / 3]}} ``` --- Solve the following fractional integro-differential equation: ```wl In[1]:= eqn = CaputoD[y[t], {t, 1 / 2}] - CaputoD[y[t], {t, -1 / 2}] == 1; ``` Solve for the Laplace transform: ```wl In[2]:= Solve[LaplaceTransform[eqn, t, s] /. y[0] -> 0, LaplaceTransform[y[t], t, s]] Out[2]= {{LaplaceTransform[y[t], t, s] -> (1/(-1 + s) Sqrt[s])}} ``` Find the inverse transform: ```wl In[3]:= InverseLaplaceTransform[%, s, t] Out[3]= {{y[t] -> E^t GammaRegularized[(1/2), 0, t]}} ``` Find the solution directly using ``DSolve`` : ```wl In[4]:= DSolve[{eqn, y[0] == 0}, y[t], t] Out[4]= {{y[t] -> E^t GammaRegularized[(1/2), 0, t]}} ``` --- The following equation describes a fractional harmonic oscillator of order 1.9: ```wl In[1]:= eqn = CaputoD[x[t], {t, 1.9}] + 5x[t] == 0; ``` Solve for the Laplace transform: ```wl In[2]:= Solve[LaplaceTransform[eqn, t, s] /. {x[0] -> 0, x'[0] -> 1}, LaplaceTransform[x[t], t, s]] Out[2]= {{LaplaceTransform[x[t], t, s] -> (1./s^0.1 (5.  + s^1.9))}} ``` Find the inverse transform: ```wl In[3]:= InverseLaplaceTransform[%, s, t] Out[3]= {{x[t] -> 1. t^1. MittagLefflerE[1.9, 2., -5. t^1.9]}} ``` Plot the solution: ```wl In[4]:= Plot[x[t] /. %, {t, 0, 10}] Out[4]= [image] ``` Find the solution directly using ``DSolve`` : ```wl In[5]:= DSolve[{eqn, x[0] == 0, x'[0] == 1}, x[t], t] Out[5]= {{x[t] -> 1. t^1. MittagLefflerE[1.9, 2., -5. t^1.9]}} ``` #### Evaluation of Integrals (2) Calculate the following integral: ```wl In[1]:= Inactivate[Integrate[(u Sin[t u]/1 + u^2), {u, 0, ∞}], Integrate] Out[1]= Inactive[Integrate][(u*Sin[t*u])/(1 + u^2), {u, 0, Infinity}] ``` Compute the Laplace transform and interchange the order of Laplace transform and integration: ```wl In[2]:= Inactivate[Integrate[LaplaceTransform[(u Sin[t u]/1 + u^2), t, s], {u, 0, ∞}], Integrate] Out[2]= Inactive[Integrate][u^2/((1 + u^2)*(s^2 + u^2)), {u, 0, Infinity}] ``` Perform the integration over $u$ : ```wl In[3]:= Assuming[s > 0, Activate[%]] Out[3]= (π/2 + 2 s) ``` Use ``InverseLaplaceTransform`` to obtain the original integral: ```wl In[4]:= InverseLaplaceTransform[%, s, t] Out[4]= (E^-t π/2) ``` Verify the result: ```wl In[5]:= Integrate[(u Sin[t u]/1 + u^2), {u, 0, ∞}, Assumptions -> t > 0] Out[5]= (E^-t π/2) ``` --- Integral involving the Bessel function: ```wl In[1]:= Inactivate[Integrate[(BesselJ[k, u]/u^k - n + 1), {u, 0, ∞}], Integrate] Out[1]= Inactive[Integrate][u^(-1 - k + n)*BesselJ[k, u], {u, 0, Infinity}] ``` Perform a change of variables $u=2 \sqrt{v}$ and introduce an auxiliary variable $t$ : ```wl In[2]:= f[t_] = Inactivate[(1/2^k - n + 1)Integrate[v^(n/2) - 1((t/v))^(k/2)BesselJ[k, 2 Sqrt[v t]], {v, 0, ∞}], Integrate] Out[2]= 2^-1 - k + n Inactive[Integrate][(t/v)^(k/2)*v^(-1 + n/2)*BesselJ[k, 2*Sqrt[t*v]], {v, 0, Infinity}] ``` Apply the Laplace transform and interchange the order of Laplace transform and integration: ```wl In[3]:= Inactivate[(1/2^k - n + 1)Integrate[v^(n/2) - 1LaplaceTransform[((t/v))^(k/2)BesselJ[k, 2 Sqrt[v t]], t, s], {v, 0, ∞}], Integrate] Out[3]= 2^-1 - k + n Inactive[Integrate][(s^(-1 - k)*(1/v)^(k/2)*v^(-1 + k/2 + n/2))/E^(v/s), {v, 0, Infinity}] ``` Perform the integration over $v$ : ```wl In[4]:= Assuming[{s > 0, n > 0}, Activate[%]] Out[4]= 2^-1 - k + n s^-1 - k + (n/2) Gamma[(n/2)] ``` Use ``InverseLaplaceTransform`` to obtain $f(t)$ : ```wl In[5]:= f[t_] = InverseLaplaceTransform[%, s, t] Out[5]= (2^-1 - k + n t^k - (n/2) Gamma[(n/2)]/Gamma[1 + k - (n/2)]) ``` The original integral equals $f(1)$ : ```wl In[6]:= f[t] /. t -> 1 Out[6]= (2^-1 - k + n Gamma[(n/2)]/Gamma[1 + k - (n/2)]) ``` Verify the result: ```wl In[7]:= Integrate[(BesselJ[k, u]/u^k - n + 1), {u, 0, ∞} , GenerateConditions -> False] Out[7]= (2^-1 - k + n Gamma[(n/2)]/Gamma[1 + k - (n/2)]) ``` #### Other Applications (2) Compute a Laplace transform using a series expansion: ```wl In[1]:= f[t_] = Exp[-t^2]; ``` The odd coefficients vanish: ```wl In[2]:= coeff = SeriesCoefficient[f[t], {t, 0, n}] Out[2]= Piecewise[{{I^n/(n/2)!, Mod[n, 2] == 0 && n >= 0}}, 0] ``` The transformed series can be summed using ``Regularization`` : ```wl In[3]:= Assuming[s > 0, Sum[LaplaceTransform[(coeff /. n -> 2n )t^2n, t, s], {n, 0, ∞}, Regularization -> "Borel"]] Out[3]= (1/2) E^(s^2/4) Sqrt[π] Erfc[(s/2)] ``` Verify the result directly using ``LaplaceTransform`` : ```wl In[4]:= LaplaceTransform[Exp[-t^2], t, s] Out[4]= (1/2) E^(s^2/4) Sqrt[π] Erfc[(s/2)] ``` --- Laplace transform of ``Sinc`` using series expansions: ```wl In[1]:= f[t_] = Sinc[t]; In[2]:= Plot[f[t], {t, 0, 30}, PlotRange -> All] Out[2]= [image] In[3]:= coeff = SeriesCoefficient[f[t], {t, 0, n}] Out[3]= Piecewise[{{(I^n*(1 + (-1)^n))/(2*(1 + n)!), n >= 0}}, 0] ``` Odd coefficients vanish: ```wl In[4]:= Assuming[s > 0, Sum[LaplaceTransform[(coeff /. n -> 2n ) t^2n, t, s], {n, 0, ∞}]] Out[4]= ArcTan[(1/s)] ``` Verify the result: ```wl In[5]:= LaplaceTransform[(Sin[t]/t), t, s] Out[5]= ArcTan[(1/s)] ``` ### Properties & Relations (3) Use ``Asymptotic`` to compute an asymptotic approximation: ```wl In[1]:= Asymptotic[Inactive[LaplaceTransform][1 / (E ^ (t ^ 2) + t ^ 17), t, s], s -> Infinity] Out[1]= (1/s) ``` --- ``LaplaceTransform`` and ``InverseLaplaceTransform`` are mutual inverses: ```wl In[1]:= InverseLaplaceTransform[LaplaceTransform[f[t], t, s], s, t] Out[1]= f[t] In[2]:= LaplaceTransform[InverseLaplaceTransform[g[s], s, t], t, s] Out[2]= g[s] In[3]:= LaplaceTransform[E ^ (-t ^ 2), t, s] Out[3]= (1/2) E^(s^2/4) Sqrt[π] Erfc[(s/2)] In[4]:= InverseLaplaceTransform[%, s, t] Out[4]= E^-t^2 ``` --- Use ``NIntegrate`` for numerical approximation: ```wl In[1]:= LaplaceTransform[1 / (1 + t ^ 2), t, s] Out[1]= CosIntegral[s] Sin[s] + (1/2) Cos[s] (π - 2 SinIntegral[s]) In[2]:= % /. {s -> 0.4} Out[2]= 0.934119 ``` ``NIntegrate`` computes the transform for numeric values of the Laplace parameter ``s`` : ```wl In[3]:= NIntegrate[Exp[-0.4 t] / (1 + t ^ 2), {t, 0, Infinity}] Out[3]= 0.934119 ``` ### Possible Issues (1) Simplification can be required to get back the original form: ```wl In[1]:= LaplaceTransform[Sin[t]Exp[t], t, s] Out[1]= (1/1 + (-1 + s)^2) In[2]:= InverseLaplaceTransform[%, s, t] Out[2]= -(1/2) I E^(1 - I) t (-1 + E^2 I t) In[3]:= FullSimplify[%] Out[3]= E^t Sin[t] ``` ### Neat Examples (2) ``LaplaceTransform`` done in terms of ``MeijerG`` : ```wl In[1]:= LaplaceTransform[1 / (1 + t ^ 8), t, s] Out[1]= (MeijerG[{{(7/8)}, {}}, {{0, (1/8), (1/4), (3/8), (1/2), (5/8), (3/4), (7/8), (7/8)}, {}}, (s^8/16777216)]/32 π^7 / 2) ``` --- Create a table of basic Laplace transforms: ```wl In[1]:= flist = {t^n, E ^ (a t), Sin[a t], t Sin[a t], Sinh[a t], DiracDelta[t - a], Log[t], Floor[t], (1/Sqrt[a])Erf[Sqrt[a t]], BesselJ[n, t], BesselI[n, t], (2/Sqrt[π])Exp[-t^2]}; In[2]:= Grid[Prepend[{#, Assuming[{a > 0, s > 0}, Simplify[LaplaceTransform[#1, t, s]]]}& /@ flist, {f[t], LaplaceTransform[f[t], t, s]}], IconizedObject[«Grid options»]]//TraditionalForm Out[2]//TraditionalForm= $$\begin{array}{cc} f(t) & \mathcal{L}_t[f(t)](s) \\ t^n & s^{-n-1} \Gamma (n+1) \\ e^{a t} & \frac{1}{s-a} \\ \sin (a t) & \frac{a}{a^2+s^2} \\ t \sin (a t) & \frac{2 a s}{\left(a^2+s^2\right)^2} \\ \sinh (a t) & -\frac{a}{a^2-s^2} \\ \delt ... a+s}} \\ J_n(t) & \frac{\left(\sqrt{s^2+1}+s\right)^{-n}}{\sqrt{s^2+1}} \\ I_n(t) & \frac{\left(\sqrt{s^2-1}+s\right)^{-n}}{\sqrt{s^2-1}} \\ \frac{2 e^{-t^2}}{\sqrt{\pi }} & e^{\frac{s^2}{4}} \text{erfc}\left(\frac{s}{2}\right) \\ \end{array}$$ ``` ## See Also * [`InverseLaplaceTransform`](https://reference.wolfram.com/language/ref/InverseLaplaceTransform.en.md) * [`BilateralLaplaceTransform`](https://reference.wolfram.com/language/ref/BilateralLaplaceTransform.en.md) * [`FourierTransform`](https://reference.wolfram.com/language/ref/FourierTransform.en.md) * [`UnilateralConvolve`](https://reference.wolfram.com/language/ref/UnilateralConvolve.en.md) * [`ZTransform`](https://reference.wolfram.com/language/ref/ZTransform.en.md) * [`Integrate`](https://reference.wolfram.com/language/ref/Integrate.en.md) * [`Piecewise`](https://reference.wolfram.com/language/ref/Piecewise.en.md) * [`TransferFunctionModel`](https://reference.wolfram.com/language/ref/TransferFunctionModel.en.md) * [`MellinTransform`](https://reference.wolfram.com/language/ref/MellinTransform.en.md) * [`Asymptotic`](https://reference.wolfram.com/language/ref/Asymptotic.en.md) * [`FractionalD`](https://reference.wolfram.com/language/ref/FractionalD.en.md) * [`CaputoD`](https://reference.wolfram.com/language/ref/CaputoD.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) * [Fourier Analysis](https://reference.wolfram.com/language/guide/FourierAnalysis.en.md) * [Generalized Functions](https://reference.wolfram.com/language/guide/GeneralizedFunctions.en.md) * [Signal Transforms](https://reference.wolfram.com/language/guide/SignalTransforms.en.md) * [`Calculus`](https://reference.wolfram.com/language/guide/Calculus.en.md) * [Integral Transforms](https://reference.wolfram.com/language/guide/IntegralTransforms.en.md) * [Signal Processing](https://reference.wolfram.com/language/guide/SignalProcessing.en.md) * [Summation Transforms](https://reference.wolfram.com/language/guide/SummationTransforms.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)