From: fateman@cs.berkeley.edu   
      
   Just to be clear on the  s1 = cot(2*x)/cot(x)  issue:   
   there are various ways of writing this.   
   s2= (2*cos(x)^2-1)/(2*cos(x)^2)   is one.   
   here is another   
   s3= (exp(4*i*x)+1)/ (exp(2*i*x+1)^2   
       (if you are unfamiliar with the complex exponential, read up about it.)   
      
   s1=s2=s3 always.   
      
   If you take sqrt(s1), sqrt(s2), sqrt(3),  and simplify, you would expect   
   them to be equal always.  Oh, you aren't going to fall for that   
   trick, you have to pick the right sign.   
   In particular, for s3, you see the denominator is (something)^2.   
      
   so you can do this:   
      
   sqrt(s3)  simplifies to  sqrt( (exp(4*i*x+1)) / (exp(2*i*x+1)   
      
   and that should be equal to sqrt(s1).  Or it might have the wrong sign,   
   That's easy to fix.   
      
   We welcome your experimentation, but for the usual guesses of sqrt,   
      
   sqrt(s1) and sqrt(s3) are not only equal but have the SAME SIGN for   
   every x between 0 and pi/4.   
      
   sqrt(s1) and simplified sqrt(s3) are both imaginary for x between pi/4   
   and pi/2 and have equal magnitude but DIFFERENT SIGNS.   
      
   How can you choose a sign for that sqrt in the absence of any context?   
   Or given the context of indefinite integration, where the symbolic value   
   "x" explicitly is an arbitrary parameter with no context...   
   you have to be very careful.   
   RJF   
      
      
      
      
      
   Simple algebraOn 6/20/2015 9:55 AM, Richard Fateman wrote:   
   > On 6/17/2015 4:51 PM, clicliclic@freenet.de wrote:   
   >>   
   > ?   
   >>   
   >> Surely, a consistent solution to this problem would be to have Maxima   
   >> itself query the user when in doubt: "There are two square roots. Which   
   >> sqrt(cot(2*x)/cot(x)) did you have in mind?"   
   >   
   > This is not particularly an integration problem, but a sqrt problem. Any   
   > time a user types sqrt( ) she could be asked "do you want to specify a   
   > particular branch, or would you like to continue computing with an   
   > ambiguous specification, or would you like the system to choose one   
   > branch based on some criterion?"   
   >   
   > It is also not so easy to specify the choice.  How would you specify   
   > sqrt(1-cos(x)^2) ?  The "positive one"  would be sin(x)  maybe??  But   
   > sin(x) oscillates between pos and neg, so it is not "positive".   
   > And what if was some complicated expression?   
   >   
   > Better perhaps to leave  Root(...f(x), 2)   and keep it ambiguous, or   
   > have a set { branch1, branch2} ?   
   >   
   >   
   > Should a CAS have free will? How else can it choose?   
   > RJF   
   >   
   > PS the bug about %i and sign -- it can be tickled without using   
   > integration.  Maxima CAN do the integral in exponential form,   
   > but if one asks to re-express that answer in rectangular form (rectform() )   
   > it has the same problem.   
   >   
   >   
   >   
   >   
   >>   
   >>>   
   >>> This example does seem to identify a bug in Maxima, though.   
   >>   
   >> Martin.   
   >>   
   >> PS @ Waldek: you are asking a lot: I will try to answer at the end of   
   >> the week :).   
   >>   
   >   
      
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