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MathGroup Archive 2006

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ComplexExpand, Piecewise, and simplification

  • To: mathgroup at smc.vnet.net
  • Subject: [mg66659] ComplexExpand, Piecewise, and simplification
  • From: Andrew Moylan <andrew.moylan at anu.edu.au>
  • Date: Fri, 26 May 2006 04:17:19 -0400 (EDT)
  • Sender: owner-wri-mathgroup at wolfram.com

Thanks for your help regarding "Simplification and Arg[]", Andrzej and 
others. I have written a function that determines which independent 
variables in an expression are not known to be real (according to 
$Assumptions), and thus must be passed as the second argument to 
ComplexExpand. This makes ComplexExpand safe to call on any expression.

I have since noticed that FullSimplify is also comparitively poor at 
handling Piecewise functions. There are two aspects to this that I have 
found:

(1) In an expression like a+Piecewise[{{b,condition1},{c,condition2}}], 
it is frequently useful to try calling PiecewiseExpand; it will bring 
the "a+" inside the Piecewise function and allow for 
simplification/cancellation of it with the pieces. This is a very cheap 
operation, but FullSimplify does not appear to take advantage of it.

(2) In an expression like 
Piecewise[{{b,condition1},{c,condition2}},defaultvalue], it is 
frequently useful to append condition1 to $Assumptions and then call 
FullSimplify on piece b, followed by instead appending condition2 to 
$Assumptions and calling FullSimplify on piece c, then finally appending 
Not[Or[condition1,condition2]] to $Assumptions and then calling 
FullSimplify on the defaultvalue.

As Andrezj suggested, I have written extended versions of Simplify and 
FullSimplify. They expose an indentical interface to the built-in 
functions, except they expose additional options specifying whether to 
try various ComplexExpand transformation functions (with various values 
for the TargetFunctions option and automatically-determined list of 
complex variables); and whether to use the simple Piecewise 
simplification methods described above.

  -Andrew


Andrzej Kozlowski wrote:
 > On 22 May 2006, at 11:30, Andrew Moylan wrote:
 >
 >
 >>Should Mathematica be able to simplify the following expression?
 >>(It is
 >>easily seen to be zero under the given condition, x > 0.)
 >>
 >>FullSimplify[
 >>	Arg[1 + I * x] + Arg[1 - I * x],
 >>	{x > 0}
 >>]
 >>
 >>In particular, I would have expected the following to yield ArcTan[b /
 >>a], from which the above expression is easily reduced to zero:
 >>
 >>FullSimplify[
 >>	Arg[a + I b],
 >>	{a > 0, b > 0}
 >>]
 >>
 >>Any ideas?
 >>
 >>Cheers,
 >>
 >>Andrew
 >>
 >>P.S. Apologies if I have sent this twice; my original message seems
 >>not
 >>to have worked.
 >>
 >
 >
 >
 > You do not even need the condition x>0: it is enough that x is real.
 >
 > ComplexExpand[Arg[1 + I * x] + Arg[1 - I * x],TargetFunctions->{Re,Im}]
 >
 > 0
 >
 > Simplify and FullSimplify by default do not make use of ComplexExpand.
 > Of course, if you wish you can make Simplify use ComplexExpand:
 >
 > Simplify[Arg[1 + I*x] + Arg[1 - I*x],
 >    TransformationFunctions ->
 >     {ComplexExpand[#1, TargetFunctions -> {Re, Im}] & ,
 >      Automatic}]
 >
 > 0
 >
 > Note that doing this automatically involves the assumption that x is
 > real, so it would not be a good idea to permanently append
 > ComplexExpand to the  TransformationFunctions, except when the
 > assumptions imply that the variables involved are real. However, it
 > is not difficult to write a version of Simplify or FullSimplify which
 > will make use ComplexExpand in a way that is compatible with the
 > assumptions about the variables (real or complex). That it is not
 > done by default is probably due to the facts that, on the one hand,
 > ComplexExpand is a high complexity function, and on the other, it
 > usually leads to more rather than less complex expressions (after all
 > it "Expands").
 >
 > Note also that:
 >
 >
 > ComplexExpand[Arg[a + I*b], TargetFunctions -> {Re, Im}]
 >
 >
 > ArcTan[a, b]
 >
 > so the assumption about a and b being positive are again not needed.
 >
 >
 > Andrzej Kozlowski
 > Tokyo, Japan
 >


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