# How to help Simplify with conjugates without using ComplexExpand

I have a seemingly simple simplification involving conjugates:

$Assumptions = Element[{a}, Reals]; FullSimplify[ x E^(I a) + Conjugate[x] E^(-I a)]  which merely outputs this, E^(-I a) (E^(2 I a) x + Conjugate[x])  I would have expected the expression above to simplify to 2 Re[x E^(I a)]  In order to use ComplexExpand, one needs to explicitly convert every cplx variable into either Cartesian or polar form, which causes quite some extra labour: expr = x E^(I a) + Conjugate[x] E^(-I a); repToCart = {x -> xRE + I xIM}; repToCplx = {xRE -> Re[x], xIM -> Im[x]};$Assumptions = Element[{a, xRE, xIM}, Reals];
Output = FullSimplify[expr /. repToCart] /. repToCplx


outputs this:

2 Cos[a] Re[x] - 2 Im[x] Sin[a]


This is indeed the same as this:

2 Re[x E^(I a)]


But it is

1. More complicated
2. Required quite some extra steps

Any tips on how to speed up the workflow here would be greatly appreciated :)

• You can get the second expression with ComplexExpand[x E^(I a) + Conjugate[x] E^(-I a), {x}, TargetFunctions -> {Re, Im}]. Commented Nov 28, 2020 at 9:09
• Thanks a lot, I was not aware of this wonderful shortcut ! This helps a lot. I would even have marked this comment as an accepted answer if it were an answer. Commented Nov 28, 2020 at 12:50
• Go ahead and post an answer yourself then. Commented Nov 28, 2020 at 13:32

I cannot understand, why not to use ComplexExpand]. However, if not, try this:

expr = x E^(I a) + Conjugate[x] E^(-I a) /. x -> z + I*t;

Simplify[expr // ExpToTrig, {a, z, t} \[Element] Reals] /. {z ->
Re[x], t -> Im[x]}

(*   2 Cos[a] Re[x] - 2 Im[x] Sin[a] *)


Have fun!

Following up to the comment of b.gates.you.know.what, I think that is the most useful answer.

ComplexExpand[x E^(I a) + Conjugate[x] E^(-I a), {x},   TargetFunctions -> {Re, Im}]


This approach spares the user to write replacement rules back and forth. The answer from Alexei Boulbitch is also useful, but not much different than the replacement rules I mentioned in my post.

Thanks to all. Massimo