Conference rusure::math

1473.0. "the ART in MATH manipulations" by SMAUG::ABBASI () Thu Jul 25 1991 10:33

sometimes i find working out a problem, that if i postpone a certine
    simplifications/substituion, i can get the answer much easier, 
    and sometimes if i simplify early, i get the correct answer, 
    and i simplify later i either a hard form, or even an invalid answer, 
    a very simple example is:

                     Pi
  we all know that  /  cos (mx) cos(nx) dx   = Pi  when n=m
                    |
                   /
                   -Pi


first method: I= /                     /
                |  1/2 cos(n-m)x dx + |  1/2 cos(n+m)x dx      (step *)
                /                     /

           
I= 1   1                                  1  1
   - -----  ( sin(n-m)Pi + sin(n-m)Pi ) + - ----- ( sin(n+m)Pi + sin(n+m)Pi)
   2  n-m                                 2  n-m

when n=m  I= 0/0  + 0  where 0/0 undefined.

BUT if I substitute n=m at step * before I integrate, i get 

       /                     /
   I= | 1/2 cos(0)x dx  +   |  1/2  cos(2nx) dx
     /                      /

  I= 1/2 (Pi - (-Pi))  +   0  = Pi the correct answer.

one can some with opposite examples, of where it is better to postpone
simplifications/substitutions, until the right stage, which would result
in much simpler solution.

I know such things are learned with experience and a lot of practise, but 
iam interested on how to program into the computer such things, as in 
computer algebra systems, since it seems that depending on the input data
the program could end with expression swelling if it simplify/substitute
at the wrong stage.

i think this is an interesting problem, i understand from what little
reading I've done on the subject of computer algebra, that one tries
to arrive at what is called 'canonical' (meaning most efficient/optimal form ?)
of the expression, and that data representation of expression is critical.

i've been kicking myself lately on this, since on one last exam, i got my self
in big trouble on one problem, where i simplified an expression too early
only to end up with integral that is too hard to solve in that 'form', and
if i've left it in original form many steps earlier, i would have been able
to see the solution to the integral much easier (plus the fact i did not
    study too hard :-)

iam sure we all faced these things, i thing there is an artistic side
to algebraic manipulation that cant be really taught, and must be gained
by practice.

from what i read , old mathematicians where particular good at this, 
(old, i mean before computer age, and befor), one only has to look at how Newton
wrote and manipulated his equations to see the artiestic (sp?) of what
i mean.

any views of this subject, experience, insight is welcome.

ok, i bragged to long, good nite.
/Nasser