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I ran into the problem that Mathematica's (full-) simplification does not replace 

t Conjugate[t]

by 

Abs[t]^2

In my case, I get pretty ugly expressions which greatly simplify when I replace the expressions manually. As an example: 

FullSimplify[( 2 (-((Abs[a - b + Sqrt[(a - b)^2 + 4 t Conjugate[t]]]^2 + 
  4 t Conjugate[t])/(
 2 (a + b - 2 ω + Sqrt[(a - b)^2 + 4 t Conjugate[t]]))) + (
Abs[-a + b + Sqrt[(a - b)^2 + 4 t Conjugate[t]]]^2 + 
 4 t Conjugate[t])/(
2 (-a - b + 2 ω + 
   Sqrt[(a - b)^2 + 4 t Conjugate[t]]))))/((a - b) (Conjugate[a] -
  Conjugate[b]) + 4 t Conjugate[t] +   Sqrt[(a - b)^2 + 4 t Conjugate[t]]
Conjugate[Sqrt[(a - b)^2 + 4 t Conjugate[t]]]), {{a, 
b, ω} ∈ Reals, t ∈ Complexes}]

Does not yield any simplification. However, if I replace

t Conjugate[t] -> Abs[t]^2

or, if I change $t$ to be real, I find the desired result $\frac{b-\omega}{ |t|^2-(a-\omega)(b-\omega)}$

PS: I just realized this is appears to be the same problem as posted here: FullSimplify on complex numbers seems inconsistent. However I'd much appreciate a less "crazy" solution…

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    $\begingroup$ may be ComplexExpand[t Conjugate[t]]. Lookup up ComplexExpand in the help. $\endgroup$ – Nasser Jul 1 '17 at 9:09
  • $\begingroup$ Thanks, but that does not help. In my case I get terms like Arg[(a - b)^2 + 4 t Conjugate[t]] which are zero as the argument is positive. $\endgroup$ – Leo Stenzel Jul 1 '17 at 10:57
  • $\begingroup$ If split $t$ into real and imaginary part explicitly, and tell Simplify that both parts are real, then I find the final simplified result. But… that's not a great solution? $\endgroup$ – Leo Stenzel Jul 1 '17 at 11:01
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    $\begingroup$ Adding TransformationFunctions -> {Automatic, # /. x_ Conjugate[x_] :> Abs[x]^2 &} does give the result you want, for what it's worth. $\endgroup$ – Mr.Wizard Jul 1 '17 at 11:53
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You can use ComplexExpand with a suitable TargetFunctions option and Simplify. Here is your expression:

expr = (2 (-((Abs[a - b + Sqrt[(a - b)^2 + 4 t Conjugate[t]]]^2 + 
        4 t Conjugate[t])/(2 (a + b - 2 ω + 
          Sqrt[(a - b)^2 + 4 t Conjugate[t]]))) + (Abs[-a + b + 
         Sqrt[(a - b)^2 + 4 t Conjugate[t]]]^2 + 
      4 t Conjugate[t])/(2 (-a - b + 2 ω + 
        Sqrt[(a - b)^2 + 4 t Conjugate[t]]))))/((a - b) (Conjugate[a] - 
    Conjugate[b]) + 4 t Conjugate[t] + 
 Sqrt[(a - b)^2 + 4 t Conjugate[t]] Conjugate[
   Sqrt[(a - b)^2 + 4 t Conjugate[t]]]);

Then, you can use:

Simplify[
    ComplexExpand[expr, t, TargetFunctions->{Abs}],
    (a | b | ω) ∈ Reals
]

(b - ω)/((a - ω) (-b + ω) + Abs[t]^2)

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  • $\begingroup$ Thanks, I had not looked into the documentation of ComplexExpand, properly. It bugs me that Mathematica is so sensitive form of the input, but your answer is a pretty good "workaround". $\endgroup$ – Leo Stenzel Jul 1 '17 at 15:09

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