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Re: Teaching Complex Analysis

  • To: mathgroup at
  • Subject: Re: Teaching Complex Analysis
  • From: bert at (Roberto Sierra)
  • Date: Sun, 20 Feb 1994 13:54:00 -0800

In[2]:=w = (x + I y)^3  //Expand

 3        2          2      3
x  + 3 I x  y - 3 x y  - I y


    3        2          2      3
Re[x  + 3 I x  y - 3 x y  - I y ]

The desired expansion can be forced by using ComplexExpand
after using Re[w]  (or Im[w]).

In[14]:=Re[w]  //ComplexExpand

 3        2
x  - 3 x y

Your explanation for why Re[w] and Im[w] do not expand the result is
not entirely correct -- Mma does not assume that x and y must be
real-valued.  Compare, for example, the different results you get
when you assign x and y complex values into your expressions:

In[5]:= %3 /. {x->3I,y->2+I}

Out[5]= 74

In[6]:= %4 /. {x->3I,y->2+I}

Out[6]= 36 - 54 I

Clearly, the longer Re[...] is correct for these values of x and y,
which is why Mma left them evaluated.  The simplified expression will
only work if x and y are limited to real values.

A minor point, but this may help you understand exactly why Re[w] is
left unevaluated.

QUESTION.  Are there other interesting symbolic manipulations
for complex numbers that are not well known ?

If you send e-mail to mathsource at containing the single

	find complex

you'll receive information about a number of packages which are
probably of interest to you.  In particular, I see there's a package
called ReIm.m which allows you specify that certain variables and
expressions are pure-real or pure-imaginary, making Re[...] and Im[...]
automatically recognize these facts without the use of ComplexExpand.
Instructions for obtaining these items will be mailed to you by the
MathSource server.  You can also obtain items via FTP or gopher to

  - -                          "Dyslexics UNTIE!  Dyslexics LURE!!"
  o o                                     -- Elbo Room ex-graffito
   J   roberto sierra
   O   tempered microdesigns    NOTICE:
  \_/  san francisco, ca        The ideas and opinions expressed
       bert at          herein are not those of the author.

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