Compute inverse Laplace transform with Integrate

Question

Since inverse Laplace transform is just an integral, while we already have InverseLaplaceTransform built in, can we compute it with Integrate? The direct implementation of the definition of the inverse Laplace transform seems not to work:

In the following piece of code, il is the direct implementation of the Bromwich integral, elements of ex are all taken from the document of InverseLaplaceTransform.

il[f_, s_, t_] := Module[{r}, 1/(2 π I) Integrate[f Exp[s t], {s, r - I ∞, r + I ∞}]]

ex = {1/(1 + s), Log[s]^2/s, Erf[5 Sqrt[s]], s/(s^2 + 1), 1/(s^2 - 4), 
      1/(1 + s + 5 s^2), 1/(1 + Sqrt[s]), E^(-s)/s, Log[s]/s^4,
      ArcTan[2/s], Gamma[s, 0, 1], PolyGamma[2, s], 
      Exp[s + 1] ExpIntegralEi[-s], 1/(s^2 + s + 1) 1/s, 1/(s^2 + 1), 
      MeijerG[{{1/3}, {}}, {{0, 1/3, 1/3, 2/3}, {}}, s^3/27]/(2 Sqrt[3] π)};

Grid[ParallelMap[{InverseLaplaceTransform[#, s, t], il[#, s, t]}&, ex], Frame -> True]

As shown above, il fails in all cases, but is there really nothing more that Integrate can do?

Answer 1

Direct Integration

Possible issues in performing the integrations include choice of Assumptions, branch cuts in the integrands, and how limits are taken. Addressing the first of these gives five solutions.

il[f_, s_, t_] := Module[{r}, 1/(2 π I) Integrate[f Exp[s t], {s, r - I ∞, r + I ∞}, 
    Assumptions -> r > 2 && t > 0]]

Grid[ParallelMap[{InverseLaplaceTransform[#, s, t], il[#, s, t]} &, 
    Extract[ex, {{1}, {2}, {5}, {6}, {10}}], Method -> "FinestGrained"], Frame -> True]

The last result does not agree with the corresponding result from InverseLaplaceTransform, perhaps due to the location of the branch cuts of ArcTan[2/s].

Indefinite Integrals

Although Integrate returns unevaluated for all definite integrals, il, except those in the preceding table, Integrate returns indefinite integrals for many of the expressions in ex. For instance,

int = Grid[ParallelMap[{InverseLaplaceTransform[#, s, t], 
    Integrate[# Exp[s t]/(2 π I), s, Assumptions -> t > 0]} &, 
    Extract[ex, {{1}, {4}, {5}, {6}, {14}, {15}}], Frame -> True];

returns results in which s appears only in terms of the form, ExpIntegralEi[s t + c[t]], where c[t] is a polynomial in t. For instance, ex[[4]] integrates to

(* -((I E^(I t) ExpIntegralEi[(-I + s) t])/(4 π)) 
   - (I E^(-I t) ExpIntegralEi[(I + s) t])/(4 π) *)

For st a large positive or negative imaginary number, c[t] can be ignored, and

ExpIntegralEi[s t] - ExpIntegralEi[- s t] == 2 Pi I

in this limit, as can be seen from Eq (7) the Wolfram MathWorld description of this function or simply by plotting the left side of this last expression.

Plot[Evaluate[ReIm[ExpIntegralEi[I/x] - ExpIntegralEi[-I/x]]], {x, 0, 0.2}]

Hence, int can be simplified in this limit to

(int /. ExpIntegralEi[_] -> 2 Pi I) // Expand // ExpToTrig // FullSimplify

Inverse Laplace Transforms obtained by integration have, of course, been obtained in the first table for elements {1, 5, 6} of ex, and those transforms agree with those in the second table. Incidentally, it may not be obvious that the two results for ex[[15]] in the second table are equal, They are.

FullSimplify[1 - (E^(-t/2) (Sqrt[3] Cos[(Sqrt[3] t)/2] + Sin[(Sqrt[3] t)/2]))/Sqrt[3]]
    - (E^(-(-1)^(1/3) t) (-1 - I Sqrt[3] + 2 (1 + (-1)^(1/3)) E^((-1)^(1/3) t) 
        - 2 E^(I Sqrt[3] t)))/(2 (1 + (-1)^(1/3)))
(* 0 *)

Thus, we now have results for {1, 2, 4, 5, 6, 10, 14, 15} of ex, although the result for ex[[10]] does not agree with the InverseLaplaceTransform result, as explained above.

At first glance, the indefinite integral of ex[[13]] should be evaluated in a similar fashion.

(* -((I E^(1 + s (1 + t)) ExpIntegralEi[-s])/(2 π (1 + t))) 
   + (I E ExpIntegralEi[s t])/(2 π (1 + t)) *)

The second term in this expression reduces in the limit of s t a large imaginary number to -(E/(1 + t)), the result returned InverseLaplaceTransform. However, the first term does not vanish in this limit but instead oscillates at an every faster rate. Probably, this is so, because ex[[13]] has a branch cut along the entire positive real axis, so the path integral in the question cannot be carried out legitimately.

Indefinite integrals also can be computed for ex[[3]] in terms of Erf and Erfi functions, and for ex[[9]] (and also ex[[2]]) in terms of HypergeometricPFQ functions. I have not attempted to obtain limits of these expressions for s t a large imaginary number. Integrate again returns unevaluated for elements {7, 11, 12, 16} of ex.