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Re: Specifying rules OR "How to do complex math in Mathematica"

>Firstly, the raw Mathematica kernel is pretty stupid at doing
>complex algebra of anything that contains symbols rather than just
>Mathematica does not know a priori if a symbol stands for a real or
>a complex number, so is assuming that both parts may be present....

  You can also use ComplexExpand. Symbols are assumed to be real  
unless explicitly put on the (optiuonal) "assumed-complex" list.  
Relevant examples are included below.
  Daniel Lichtblau, WRI

In[6]:= ComplexExpand[Re[1/a], {a}]//InputForm
Out[6]//InputForm= Re[a]/Abs[a]^2

In[7]:= ComplexExpand[Re[(a + z)^2], {a, z}]//InputForm
Out[7]//InputForm= -(Im[a] + Im[z])^2 + (Re[a] + Re[z])^2

In[8]:= ComplexExpand[Re[1/a]]//InputForm
Out[8]//InputForm= a^(-1)

In[9]:= ComplexExpand[Re[(a + z)^2]]//InputForm
Out[9]//InputForm= (a + z)^2

In[10]:= ComplexExpand[Re[(a + z)^2], {z}]//InputForm
Out[10]//InputForm= -Im[z]^2 + (a + Re[z])^2

(* We cannot specify that a symbol is positive to get further get  
simplification, but this is still reasonably good. We could get more  
simplification with careful use of PowerExpand (details left to the  
interested reader). *)
In[11]:= ComplexExpand[Abs[a]]//InputForm
Out[11]//InputForm= (a^2)^(1/2)

In[16]:= ComplexExpand[Abs[x*y]]//InputForm
Out[16]//InputForm= (x^2)^(1/2)*(y^2)^(1/2)

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