In Mathematica, function Abs is not differentiable unless you make an extra assumption that the argument is real
FullSimplify[D[Abs[x], x], Assumptions->{x \[Element] Real}]
This code will give the result Sign[x].
However, what should I do if the argument is complex. Let me give an example.(Where F is a matrix function of 4 variables Omega, omega, qx and qy, I don't show its explicit form here, which is not important, just focus on the result)
FullSimplify[Normalize[Eigenvectors[F[omega, Omega, qx, qy]][[3]]
/.{Omega->0.4}][[1]],Assumptions->{{omega, qx, qy} \[Element] Reals}]
The above code will give a tedious expression
Look at its denominator, which contains an abs function
Abs[(1+E^{I qx})/(1+E^{I qy})]^2
As we know, when the argument is complex, Abs means modulus.
So the above expression should be equal to(1+Cos[x])/(1+Cos[y])
But why Mathematica cannot simplify it?
Someone told me to add ComplexExpand within the FullSimplify function, it does work for a simple expression like Abs[1+E^{I qx}].But when applying this to a complicated expression like I gave above, it failed. It took my computer more than 12 hours to evaluate, and it is still running now... ...
So what can I do now? I must get the fully simplified version of the abs function. Because I need to calculate the gradient of that tedious expression in {qx,qy} space. If I just do this to that expression, it will give me something like Abs', which is a desirable result.
If someone is willing to help, I will be very grateful, and I can provide more details of this problem if needed.
