If it is a real-valued function f[z] of a complex-valued variable x=x+I*y, then you really have to consider separately D[f,x] and D[f,y]. You are, in effect, not distinguishing (I was tempted to say "differentiating") between these, and that is messing things up. It will also be necessary to write the function in such a way that dependencies on {x,y} are explicit.

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/Cauchy–Riemann_equations">http://en.wikipedia.org/wiki/Cauchy–Riemann_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.