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Re: Forcing Re[]'s to be Real

  • To: mathgroup at smc.vnet.net
  • Subject: [mg18258] Re: Forcing Re[]'s to be Real
  • From: Adam Strzebonski <adams at wolfram.com>
  • Date: Thu, 24 Jun 1999 14:24:41 -0400
  • Organization: Wolfram Research, Inc.
  • References: <7kpbj1$4b8@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

Anthony Foglia wrote:
> 
>         I seem to have found an interesting problem involving computing the
> real part of complex numbers.  (Interesting, in that it wasn't there a few
> weeks ago when I ran the (as-far-as-i-can-remember) exact same code.)
> 
> I have a complex function:
> 
> k[w] := Sqrt[w^2 (1 + (2 / (1 + I w)))]
> 
> I want to graph the real and imaginary parts, but Mathematica doesn't want
> to express the Re[k[w]] as a real number.  What do I mean?  Well, if I
> type:
> 
> Re[Sqrt[1+I]]
> 
> I get out
> 
> Re[Sqrt[1+I]]
> 
> Same if I do Re[ComplexExpand[Sqrt[1+I]]], or Re[(1+I)^(1/2)].  But if I
> enter:
> 
> Re[ComplexExpand[(1+I)^(1/2)]
> 
> Mathematica is kind enough to respond with:
> 
> 2^(1/4) Cos[Pi/8]
> 
> I'm certain that this is the root of my problem, but I'll be damned if I
> know why Mathematica doesn't like it now, but did a few weeks ago.  Any
> help?
> 
> --Anthony


You can use RootReduce here

In[1]:= Re[Sqrt[1+I]] // RootReduce

                      2       4
Out[1]= Root[-1 - 4 #1  + 4 #1  & , 2]

or, if you prefer radicals,


In[2]:= % // ToRadicals

             1 + Sqrt[2]
Out[2]= Sqrt[-----------]
                  2

This approach works for k[w] for any rational 
number w.

In[3]:= k[w_]:= Sqrt[w^2 (1 + (2 / (1 + I w)))]
 
 
In[5]:= Re[k[-17/3]] // RootReduce // ToRadicals

                158 + Sqrt[27565]
        17 Sqrt[-----------------]
                       298
Out[5]= --------------------------
                    3


Best Regards,

Adam Strzebonski
Wolfram Research


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