I am trying to simplify the (exponential) result of an inverse Laplace Transform. The expression could be simpler than shown and I don't understand why the simplification will not work. Usually when this happens to me it's because of some assumptions I have implicitly made about the constants, but this has simply to do with multiplication/cancellation of exponentials which doesn't seem to need assumptions. Why is this happening and how can I get the desired result?
Here are my steps:
F[s_, a_, b_, c_] := (a b)/((a + s) (b + s) (c + s))
That defines the function to subject to the inverse Laplace Transform. Then I do the Laplace transform:
f[t_, a_, b_, c_] := FullSimplify[InverseLaplaceTransform[F[s, a, b, c], s, t]]
f[t, a, b, c]
That seems to work:
(a b ((b - c) E^(-a t) + (-a + c) E^(-b t) + (a - b) E^(-c t)))/((a - b) (a - c) (b - c))
The problem comes when I divide that result by another function and try to simplify that.
g[t_, a_, b_, c_] := f[t, a, b, c]/Exp[-a t]
g[t, a, b, c]
(a b E^(a t) ((b - c) E^(-a t) + (-a + c) E^(-b t) + (a - b) E^(-c t)))/((a - b) (a - c) (b - c))
But then:
FullSimplify[g[t, a, b, c]]
(a b E^(a t) ((b - c) E^(-a t) + (-a + c) E^(-b t) + (a - b) E^(-c t)))/((a - b) (a - c) (b - c))
The FullSimplify
result of g
is the same. It clearly seems like (independent of assumptions about a,b,c) it should be able to multiply through by Exp[a t]
and remove some complexity.