Re: Complex Oddity

• To: mathgroup at smc.vnet.net
• Subject: [mg57475] Re: Complex Oddity
• From: "Carl K. Woll" <carlw at u.washington.edu>
• Date: Sun, 29 May 2005 01:03:32 -0400 (EDT)
• Organization: University of Washington
• References: <d79enu\$lbl\$1@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

```John,

See below.

"John Reed" <nospamjreed at alum.mit.edu> wrote in message
news:d79enu\$lbl\$1 at smc.vnet.net...
>I was trying to separate the real and imaginary parts of a complicated
> expression, and ended up with something strange.  Here is a short version
> of
> what happened.
>
> Let z = x + I y, then realPart = z /. {Complex[a_,b_]->a} gives realPart =
> x.  Great!
>
> Now, try imagPart = z /. {Complex[a_,b_]->b}  returns with imagPart = x +
> y.
> Oops
>

Whenever you have unexpected output, it's usually a good idea to use
FullForm on your input to see if it matches what you expect.

In[4]:=
FullForm[x + I y]
Out[4]//FullForm=
Plus[x, Times[Complex[0, 1], y]]

Now, you can see the problem, and why your expected output doesn't occur.
Basically, you shouldn't think of Complex[a,b] as an expression with head
Complex and a and b as it's arguments. Rather, Complex[a,b] is only defined
when a and b are numbers, and even then it isn't an expression, its an atom.

> In my original expression, it was harder to see, but the same error was
> occuring.  What I tried first was using Re[z] and Im[z], but then I have
> to
> work with Im[y] and Im[x].  It seems to me two things need to be done
> here.
> First, be able to assign variables so that they always stay real or else
> indicate an error is occuring if they turn out to be complex, and second
> do
> something to avoid errors like the above.  I have to say that I don't
> trust
> Mathematica's answers as much as I did before this came up. Now I feel
> like
> I better have a good idea of what the answer is before I  trust
> Mathematica.
>

The function you should look into is ComplexExpand. ComplexExpand assumes
that variables are real and performs simplifications that are possible in
that case.

In[8]:=
ComplexExpand[Re[x + I y]]
ComplexExpand[Im[x + I y]]
Out[8]=
x
Out[9]=
y

> John Reed
>

Carl Woll

```

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