As I mention above, the absolute value function is not complex differentiable. This is because it does not satisy the Cauchy Riemann Equations:
In[1]:= u = ComplexExpand[Re[Abs[x + I y]]]
Out[1]= Sqrt[x^2 + y^2]
In[2]:= v = Im[Abs[x + I y]]
Out[2]= 0
In[3]:= D[u, x]
Out[3]= x/Sqrt[x^2 + y^2]
<a href="http://en.wikipedia.org/wiki/CauchyRiemann_equations">http://en.wikipedia.org/wiki/CauchyRiemann_equations
http://mathworld.wolfram.com/Cauchy-RiemannEquations.html
Thus in your attached notebook (to avoid differentiating the Abs function via the chain rule) you need to do ComplexExpand before doing any differentiations.
Eg:
ComplexExpand[F[A, B, f]];
D[F[A, B, 1] , A] /. {A -> 1 , B -> 1}
should be replaced by
D[ComplexExpand [F[A, B, 1]] , A] /. {A -> 1 , B -> 1}
An additional bit of advice: you should define your functions using a delayed evaluation (:=) rather than an immediate evaluation (=).
(Also in your attached notebook Re1 and Im1 are not defined.)
You get a Abs'[...] in your result because of the chain rule, but Abs['[...] cannot be interpreted in an unambiguous manner. Therefore you need to avoid it.
