Simplify complex conjugation and force joining of exponentials

Question

I define the following piecewise functions.

Element[{p, pt, q, qt, rp, rq, tp, tq}, Reals]

\[Chi]Rp = Piecewise[{{Exp[I p z] + rp Exp[-I p z], z < 0}, {tp Exp[I pt z], z > 0}}];
\[Chi]Lp = Piecewise[{{pt/p tp Exp[I p z], z < 0}, {-rp Exp[I p z] + Exp[-I p z], z > 0}}];
\[Chi]Rq = Piecewise[{{Exp[I q z] + rq Exp[-I q z], z < 0}, {tq Exp[I qt z],   z > 0}}];
\[Chi]Lq = Piecewise[{{qt/q tq Exp[I q z], z < 0}, {-rq Exp[I q z] + Exp[-I q z], z > 0}}];

FullSimplify[PiecewiseExpand[\[Chi]Rp SuperStar[\[Chi]Lq]], Assumptions -> {Element[{p, pt, q, qt, rp, rq, tp, tq}, Reals]}]

Mathematica returns the following

Is there a way I can instruct it to apply the complex conjugation only to the exponential ($i\rightarrow -i $) and also join the exponentials into one?

This is a is a simple example. I need automation to deal with much more complicated examples.

Answer 1

You used SuperStar, which has no inherent meaning in Mathematica and stays unevaluated. Use Conjugate instead.

FullSimplify[PiecewiseExpand[χRp Conjugate[χLq]], 
 Assumptions -> {Element[{p, pt, q, qt, rp, rq, tp, tq}, Reals]}]

If you want to have conjugation displayed as an asterisk after the symbol, you have to use \[Conjugate] (which will be displayed as a smaller, superscript asterisk), not the normal *.

χRp Conjugate[χLq]    (* This is correct. *)
χRp χLq\[Conjugate]   (* This is correct. *)
χRp χLq*              (* This is not correct. *)