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Re: Eliminating common factors?

  • To: mathgroup at smc.vnet.net
  • Subject: [mg93354] Re: Eliminating common factors?
  • From: AES <siegman at stanford.edu>
  • Date: Tue, 4 Nov 2008 06:21:14 -0500 (EST)
  • Organization: Stanford University

I appreciate the effort Bob Hanlon put into listing multiple solutions 
to my original post in his response reproduced below.  My actual 
calculation, which I simplified for the initial post, was

   sx0sqr = wx0sqr/4; sy0sqr = wy0sqr/4;
   sxsqr = sx0sqr + (mx4 lamsqr (z - z0x)^2/(16 pisqr sx0sqr));
   sysqr = sy0sqr + (my4 lamsqr (z - z0y)^2/(16 pisqr sy0sqr));
   srsqr = sxsqr + sysqr;
   sr0sqr = sx0sqr + sy0sqr;
   wxsqr = 4 sxsqr; wysqr = 4 sysqr;
  {wxsqr, wysqr}

leading to the output

   {4 (wx0sqr/4 + (lamsqr mx4 (z - z0x)^2)/(4 pisqr wx0sqr)),
     4 (wy0sqr/4 + (lamsqr my4 (z - z0y)^2)/(4 pisqr wy0sqr))}

This is just the first cell of a lengthy and messy symbolic calculation 
of optical beam propagation formulas, in which the objective is to 
transform these results in x,y coordinates into comparable formulas in 
cylindrical coordinates and derive relations between rectangular and 
cylindrical beam parameters.

By accident as much as design, the outputs above, except for the 
superfluous factors of 4, come out in just the form conventionally used 
in the relevant literature, namely a "constant" or z-independent term, 
plus a separate z-dependent term containing the physically significant 
factors (z-z0x)^2 and (z-z0y)^2, which I'd like to preserve intact since 
they'll provide helpful physical insight in messier results later on.

The problem is that _every_one_ of the commands suggested below (plus 
the FactorTerms command suggested by someone else), while getting rid of 
the unnecessary factors of 4, also expands out the (z-z0)^2
factors into 3 separate terms, thus making the results longer, messier, 
and less easy to view.

One solution would be to replace each of the (z-z0)^2 factors by a 
single factor zminusz0sqr in my initial inputs, then convert back at the 
very end of the calculations.  Doing this will make any of the Hanlon 
suggestions work as desired.

I realize it's pointless to ask why Mathematica doesn't automatically 
remove the utterly superfluous integer factors of 4 in the above 
outputs.  In fact, I still wonder if there is some deep, deep reason 
that I don't grasp that says that Mathematica _should_not_ do this 
apparently obvious simplification, because doing it could possibly lead 
to trouble in some other situation if Mathematica were designed to do 
this automatically.



=============================================
>From Bob Hanlon:

In the simple form provided, just about any command that touches it will 
work. Without the actual expression there is no way to help.

expr = 4 (a/4 + b/(4 c))

4*(a/4 + b/(4*c))

expr // Simplify

a + b/c

expr // Apart

a + b/c

expr // Expand

a + b/c

expr // ExpandAll

a + b/c

expr // Cancel // Apart

a + b/c

expr // Together // Apart

a + b/c

expr // Factor // Apart

a + b/c

Bob Hanlon


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