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I'm trying to simplify the expression

ωplus := 1/2 c (I f V + Sqrt[4 k^2 - f (f + 8 I k) V^2])

so that I can get it in the form $a+i b$. I am using the global assumptions at the top of my notebook

$Assumptions = {Element[k, Reals], 
Element[f, Reals], f > 0, Element[c, Reals], c > 0, 
Element[V, Reals], V > 0 }

I can almost get it using ComplexExpand:

In[12]:= Simplify[ComplexExpand[ωplus]]

Out[12]= 1/2 c (64 f^2 k^2 V^4 + (-4 k^2 + f^2 V^2)^2)^(1/4)
         Cos[1/2 Arg[4 k^2 - f (f + 8 I k) V^2]] + 
         1/2 I c (f V + (16 k^4 + f^4 V^4 + 8 f^2 k^2 V^2 (-1 + 8 V^2))^(1/4)
         Sin[1/2 Arg[4 k^2 - f (f + 8 I k) V^2]])

For some reason, Mathematica will not evaluate Arg[4 k^2 - f (f + 8 I k) V^2]. I thought I had made all necessary assumptions. This is particularly puzzling because it was able to ComplexExpand everything else in my expression! If I plug in random numbers for f, k and V then it evaluates just fine. This makes me think that Mathematica isn't listening to my assumptions.

Interestingly, ComplexExpand[Abs[4 k^2 - f^2 V^2 - 8 I f k V^2]] works. However, ComplexExpand[Arg[4 k^2 - f^2 V^2 - 8 I f k V^2]] does not.

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    $\begingroup$ I suspect changing the TargetFunctions might help. $\endgroup$ – chuy Jun 10 '16 at 13:49
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    $\begingroup$ From the help: Arg[z] is left unevaluated if z is not a numeric quantity. $\endgroup$ – Andrew Jun 10 '16 at 15:52
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    $\begingroup$ Do you have a reason to think that the sine and cosine terms can be written in a way that does't involve transcendental functions? If not, perhaps you can't simplify it any further. $\endgroup$ – mikado Jun 10 '16 at 17:56

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