Is it possible to calculate inverse Laplace Transform of exponent?

[Paul Royik]

I want to calculate inverse laplace of e^(-2s) which is dirac(t-2), but sympy gives unevaluated.

[Kalevi Suominen]

On Friday, July 3, 2015 at 1:58:06 PM UTC+3, Paul Royik wrote:

I want to calculate inverse laplace of e^(-2s) which is dirac(t-2), but sympy gives unevaluated.

It is currently not possible to compute in SymPy. The inverse transform involves integration along a vertical line in the complex plane and the absolute value of e^(--2s) is constant along any such line. Hence the function is not integrable on a vertical line in the usual sense, and its integration is currently not supported by SymPy.

[Paul Royik]

OK.

Thanks.

[Aaron Meurer]

More generally, SymPy's integrator doesn't know how to return delta
functions. It only knows how to deal with delta functions as arguments
to integrals.

Aaron Meurer

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[Kalevi Suominen]

On Tuesday, July 7, 2015 at 9:31:41 PM UTC+3, Aaron Meurer wrote:

More generally, SymPy's integrator doesn't know how to return delta
functions. It only knows how to deal with delta functions as arguments
to integrals.

 
It would not be too hard to extend SymPy's integrator to return delta functions,

their derivatives and even other tempered distributions. They can be represented
as boundary values of analytic functions provided a suitable interface is defined.
I have discussed this in a slightly different connection in #9483.

There are more details in the attachment.

[Andrés Prieto]

Some weeks ago, we were working on a (dirty) patch to be able to compute inverse Laplace transforms of exponentials in Octsympy (https://github.com/cbm755/octsympy/pull/261#issuecomment-122077921). The trick is based on the linearity of the inverse Laplace transform and on the expansion of all terms to be transformed in terms of exponentials. In what follows, a code snippet is attached to illustrate its feasibility:

import sympy as sp

from sympy.abc import t, s, c

F=5*sp.exp(-3*s)+2*sp.exp(c*s)-2*sp.exp(-2*s)/s

f=0; a_ = sp.Wild("a_"); b_ = sp.Wild("b_")

Fr=F.rewrite(sp.exp)

if type(Fr)==sp.Add:

    terms=Fr.expand().args

else:

    terms=(Fr,)

for term in terms:

    #compute the Laplace transform for each term

    r=sp.simplify(term).match(a_*sp.exp(b_))

    if r!=None and sp.diff(term,s)!=0:

        rlist=list(r.values())

        modulus=rlist[0]

        phase=rlist[1]/s

        # if a is constant and b/s is constant

        if sp.diff(modulus,s)==0 and sp.diff(phase,s)==0:

            f = f + modulus*sp.DiracDelta(t+phase)

        else:

            f = f + sp.Subs(sp.inverse_laplace_transform(term, s, t),sp.Heaviside(t),1).doit()

    elif sp.diff(term,s)==0:

        f = f + term*sp.DiracDelta(t)

    else:

        f = f + sp.Subs(sp.inverse_laplace_transform(term, s, t),sp.Heaviside(t),1).doit()

     I hope that this approach could be helpful to handle Dirac deltas in the context of (i)Laplace transforms.

[Aaron Meurer]

Perhaps we just need to add some cases to manualintegrate to handle this.

Aaron Meurer

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