Extract real part of a complex expression better than Re does

Question

I have a complex expression with real positive variables only.

Mathematica Input Style:

PP = -((α (γ Cosh[y1 α] + α Sinh[y1 α])(-γ Cosh[(-L +  y2) α] + α Sinh[(-L + y2) α]))
    /(s (2 α γ Cosh[L α] + (α^2 + γ^2) Sinh[L α]))) /. α -> Sqrt[s/d] /. 
    γ -> (kd + s)/ka /. s -> -ω*I

which leads to

$ PP=-\frac{i \sqrt{-\frac{i \omega }{d}} \left(\frac{(\text{kd}-i \omega ) \text{Cosh}\left[\text{y1} \sqrt{-\frac{i \omega }{d}}\right]}{\text{ka}}+\sqrt{-\frac{i \omega }{d}} \text{Sinh}\left[\text{y1} \sqrt{-\frac{i \omega }{d}}\right]\right) \left(-\frac{(\text{kd}-i \omega ) \text{Cosh}\left[(-L+\text{y2}) \sqrt{-\frac{i \omega }{d}}\right]}{\text{ka}}+\sqrt{-\frac{i \omega }{d}} \text{Sinh}\left[(-L+\text{y2}) \sqrt{-\frac{i \omega }{d}}\right]\right)}{\omega \left(\frac{2 (\text{kd}-i \omega ) \sqrt{-\frac{i \omega }{d}} \text{Cosh}\left[L \sqrt{-\frac{i \omega }{d}}\right]}{\text{ka}}+\left(\frac{(\text{kd}-i \omega )^2}{\text{ka}^2}-\frac{i \omega }{d}\right) \text{Sinh}\left[L \sqrt{-\frac{i \omega }{d}}\right]\right)} $

Now I want to extract the real part of this by assuming real only variables and then Re[PP]. Mathematica 8 can't extract it.

I saw on different forums that many people have had difficulties with extracting real or imaginary parts of expressions with built-in Re.

Some propose to replace Re by ComplexExpand[ ( PP + Conjugate[PP] )/2 ], which seems to work well in some cases. Do you have other suggestions?

All the variables are real and positive, so that I define at the beginning of my Mathematica notebook:

$Assumptions = ω > 0 && d > 0 && L > 0 && y1 > 0 && y2 > 0 && 
    y1 < y2 && ka > 0 && kd > 0 && s > 0 && y1 < L && y2 < L

My version of Mathematica is $Version = "8.0 for Linux x86 (64-bit) (October 10, 2011)"

Answer 1

Removing the imaginary portion of an expression is done by doing

ComplexExpand[Re[expression]].

Using just Re alone will not work as Re does no evaluation on symbols with unknown complex parts.

Now as stated in the problem and the comments above this particular problem requires a fair amount of assumptions. The simplest way to add local assumptions is to use Assuming. But this will not work, so we must instead make use of Simplify. For example:

Simplify[ComplexExpand[Re[expression]], a > 0]

where a is a symbol used in expression and is a real number greater than zero.