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The facts about some of the algorithms set up by Maple and Mathematica
are as follows:
- They are not reliable for treating functions with singularities in
higher dimensions.
- We know that the double integral of the function if x2+y2>0 and f(x,y)=0 if x2+y2=0 in the region does not exist
and yet the value of its repeated integrals is 0. Both Maple V Release 4
and Mathematica 2.01 give the ''wrong'' answer 0.
- Both Mathematica and Maple can't handle singularities which lie on
the diagonal of a region, examples will be given later.
These facts motivate me to develop new algorithms which shall use my
theoretical integration backgrounds.