Assume that the value of an integral is real

Question

I would like to give a condition that the integral I am handling are not complexes.

Consider

$Assumptions=Element[a,Reals] && Element[b,Reals] && Element[t,Reals] && Element[f[t],Reals] && Element[R,Reals] && Element[Integrate[b f[t],{t,-R,R}],Reals]
ep = a + I b;
B=Integrate[ComplexExpand[f[t] ep],{t,-R,R}] // Distribute
Re[B] //Distribute

The output is:

Re[Integrate[a f[t], {t, -R, R}]] + Re[Integrate[I b f[t], {t, -R, R}]]

I think that is can't simplify because it may happen that the value of the integral (even if the integrand is real) is complex, how can I tell mathematica to give the result : Integrate[a f[t], {t, -R, R}]

score 1 · Accepted Answer · answered Aug 01 '20 at 09:22

First, let's look at what happens if we factor the ep term out of the integral, like this

ClearAll[a, b, f, R, t]
ep = a + I b;
B = Integrate[f[t] , {t, -R, R}] ep;
With[{$Assumptions = Element[Integrate[ f[t], {t, -R, R}], Reals]},
 Re[B] // ComplexExpand // Simplify
 ]
(*   aIntegrate[f[t], {t, -R, R}]   )

So, it looks like MMA is able to apply the assumption that the integral is real. Note that we used With to make a temporary change to the global $Assumptions and we applied the assumptions using Simplify. (The Simplify takes about 12 seconds on my desktop. I wonder why.)

Next, we start with the ep term inside the integral. This time we will use With to set the $Assumptions and to factor ep, or any other contants, out of the integral, like this

ClearAll[a, b, f, R, t]
ep = a + I b;
B = Integrate[f[t] ep, {t, -R, R}]
With[{$Assumptions = Im@Integrate[ f[t], {t, -R, R}] == 0,
  B = B //.
    Integrate[q1___ r__ q2___, {v_, s___}] /; FreeQ[{r}, v] :>
     r Integrate[q1 q2, {v, s}]},
(Re[B] // ComplexExpand // Simplify) /.
  Times[r_, Integrate[q_, {v_, s___}]] /; FreeQ[{r}, v] :> 
   Integrate[r q, {v, s}]
 ]
(*   Integrate[af[t], {t, -R, R}]   )

Note the different, but equivalent, assumptions in the two With statements. Also note that factoring the constants out of the integral is done with code provided by @dr-belisarius in his answer to How to do algebra on unevaluated integrals?. Another reference that may be useful is How to simplify symbolic integration.

Assume that the value of an integral is real

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