From: colin@tri.org.au (Colin Rose)
Subject: Re: conditions in integration results, Mathematica, Macsyma
Date: Sun, 20 Jun 1999 02:58:17 +1000
Newsgroups: sci.math.symbolic

Richard J. Fateman <fateman@peoplesparc.cs.berkeley.edu> wrote:

    If [ Im[a]==0 && Im[2a-2b]==0 && Im[b]==0 && Im[2a+2b]==0 ...

> Have others found the conditionalized results of the
> Mathematica integration program useful in further computation??


No. 

I personally find it incredibly irritating.
Five issues come to mind:

1. Superfluous:  The conditional statements are often superfluous,
   and sometimes wrong. Under v3, for instance, integrating
   a Normal(mu, sigma^2) distribution yields the conditional:

     Re[Mu/Sigma^2 > 0 && Re[Sigma^2] > 0 && X <= 0

   which is clearly nonsense. Fortunately, this example
   has been fixed in the new v4 release.


2.  Syntax:    
    Under the new v4, the Assumption syntax used by Integrate is
    different to the excellent NEW Assumptions technology used by
    Simplify, FullSimplify and Expand.
    This means that one has to work with two assumptions technologies
    that have completely different syntaxes and do not know how
    to talk to each other. 


3. Peculiar:   For example... 

      Under v2:  Integrate did NOT give conditional results.
                 Mma knew how to integrate x^-c:

                 In[]:=   Integrate[x^(-c), {x, a, Infinity}]

                 Out[]=   a^(1-c)/(-1 + c)


     Under v4:   With Conditions turned off, Mathematica
                 no longer knows the answer:

                 In[]:=   Integrate[x^(-c), {x, a, Infinity},        
                                GenerateConditions -> False]
 
                 Out[]=   Zilch

     even though it does know the answer if you allow it
     to present the conditions. This is very strange indeed.

     In other situations, supplying some of the required assumptions,
     but not all of them, is actually worse than providing none at all 
     with GenerateConditions set to False.
   

4. Funtionally poor:  
 
     Normally, If[] structures are used in situations such as:

       If[x < 0, x^2, x^3]

     Here, different things happen depending on whether x is
     smaller or greater than 0. 

     By contrast, Integrate always returns answers of form:

        If[ cond,  answer,  ELSE I don't know]

     and NEVER EVER the more interesting case:

        If[ cond, answer1,  answer2 ]    
      
         (two different answers, depending on whether the
          condition is satisfied or not.)

    By using a conditional If[] statement to always display a
    single answer has no functional purpose. Worse, it has many
    disadvantages, including making it more difficult to see the
    answer, and ... 




5. Post-processing results:

      So, Mma returns:

        If[ cond, ANSWER, ELSE I don't know]
      
      Richard asked whether this output is useful in
      further computation ... now suppose you find that 
      the conditions specified are suitable, and you now want to
      do further work with ANSWER, such as Simplify it or Expand it etc. 
      To do this, you have to grab the ANSWER out of the If statement,
      create a new expression with it, and then Simplify it.  Eg:
      
         sol =  If[a, b, c];     sol[[2]] // Simplify

      But things get messy when the solution is say:

         sol = 2 If[a, b, c];

      because now: 
          
         sol[[2]]  ==   If[a, b, c],    NOT b.


--


Under the circumstances, I find the best situation is to simply 
avoid conditional results altogether by setting:

    SetOptions[Integrate, GenerateConditions -> False]

which causes the conditional statements to disappear all together.
This is quite delicious. 
On the downside, one doesn't get to see the assumptions
Mathematica is making in generating the answer, which can
be useful. This information [ the condition ] could easily
be displayed in the Message box, or pop up in a special 
info text cell or similar ... this would be an easy way to
convey the necessary information to the user without using the
irksome If approach.

Cheers

Colin

-- 
Colin Rose 
tr(I)    -  Theoretical Research Institute
__________________________________________
colin@tri.org.au    http://www.tri.org.au/