# How to simplify analytic expression without introducing Abs[]?

I am working with analytical expressions that, as you will see from expr below, stem from certain expressions involving Abs[]. However, for further calculus I do not want Abs[] to appear in my expressions, so that I use ComplexExpand[] to get rid of it. Yet, there are terms in the ComplexExpanded expressions that are certainly zero, but Mathematica does not realize this immediately during the evaluation...

ClearAll["Global*"];
\$Assumptions = Element[{f1, f2, th, gb, wb, eb, deg, dwe, dwg}, Reals];

(* shorthand notation *)
sx = PauliMatrix[1];
sy = PauliMatrix[2];
sz = PauliMatrix[3];
id = PauliMatrix[0];

(* define some stuff *)
a1 = 2*Sqrt[wb^2 + gb^2 + dwg^2];
n1 = {gb, dwg, wb}/(a1/2);
u1 = Cos[a1/2]*id - I*n1.{sx, sy, sz}*Sin[a1/2];
unitaryD = {{Exp[-I*f1]*Cos[th], -Exp[I*f2]*Sin[th]}, {Exp[-I*f2]*Sin[th], Exp[I*f1]*Cos[th]}};


Now evaluate expr = ComplexExpand[Abs[Tr[u1.unitaryD]]] and observe that Simplify@expr[[1, 1]] is zero. So expr could actually be written much shorter... However, a naive Simplify@expr returns an expression involving Abs again. How can I avoid this? I am pretty sure the most obvious method is the TransformationFunctions option of Simplify, but I am absolutely clueless about how to use it.

For reference: happens on Mathematica 10.0.1 on Gentoo Linux 64bit

Edit: While replacing greek symbols for readability reasons, I forgot to replace one \[Alpha] which caused the Abs not to appear. Sorry for that.

• no expression with Abs is produced for me if I run Simplify@expr. This is what I get: 2 \[Sqrt](( 1/(\[Alpha]1^2))((\[Alpha]1 Cos[f1] Cos[th] Cos[Sqrt[ dwg^2 + gb^2 + wb^2]] - 2 (wb Cos[th] Sin[f1] + (dwg Cos[f2] + gb Sin[f2]) Sin[th]) Sin[ Sqrt[dwg^2 + gb^2 + wb^2]])^2)) – glS Sep 13 '16 at 14:54
• @glS Hm that's weird. I have already left office but will try on my desktop machine at home. Issue happens on Linux 64bit, MMA 10.0.1 – Lukas Sep 13 '16 at 15:26
• I get the same expression as glS on a Linux 64 bit MMA 10.2.0.0 – andy269 Sep 13 '16 at 15:31
• The closest version I have is v10.0.2.0 on a Mac. I do not see an Abs in expr and Simplify works fine with it. Perhaps you need to add the option TargetFunctions->{Re, Im} to your ComplexExpand – Bob Hanlon Sep 13 '16 at 15:48
• No Abs on MMA 10.4.1, Linux Mint 17.3. – corey979 Sep 13 '16 at 17:21

When I provide the same input as you, my result is the same. Now when we look at expr[[1,1]]:
I notice that this has already been simplified to 0 when we simplify expr. Compare the expressions:
So it seems to me that Simplify@expr can't be written any shorter as you say. I can only surmise that your problem with the output of Simplify@expr then is just that there's an Abs in there. Well that can be removed quite easily by another application of ComplexExpand:
• You're right. That must've been some instance of symbol-blindness or the like... Stared too much at longish expressions and couldn't realize that the disturbing Abs` does not lead to any longer expressions if expanded... Thank you! – Lukas Sep 14 '16 at 12:56