oldk1331 wrote:
>
> But your example makes me realized that in general this
> simplification is not an algorithm (impossible to solve all input):
> for example "g x == sqrt(x+1)+sqrt(x-1)",
> then the kernels are [x,sqrt(x+1),sqrt(x-1)] and you can't
> figure out the "real variable" should be "sqrt(x+1)+sqrt(x-1)".
>
> So for a trival class of integrands, there can't be an algorithm
> that can compute their continuous integral. Is that a theorem?
> Or common knowledge?
Well, there is a theorem which essentially say that if 'x'
exists and is resonably specified then there is algorithm
to find 'x'. This algorithm may be completely impractical
(enumerate all 'x' till you find one which satisifes
specification), but is considerd to be an algorithm.
In case of rational functions problem is that branch cuts
of logarithm are articial thing and argument of logarithm
may wind up several times around 0. AFAICS one can
get "continuous" version only in quite artifical way
or not at all. Namely, if argument factors in linear
factors then each of them makes ony half turn and separate
log (in real domain atan) terms are OK. You can evan allow
quadratics and make full turn. But irreducible factors
of higer order can make more turns... The only way around
this is to approxinate irreducible higher order factors
by factorizable ones. This can be done when there is
no real zeros. You than get one term which is of high
order but has values in narrow range and the rest is
sum of terms that can be qured. But the result contains
terms which make no sense from symbolic point of view.
In case of parametic integrals clearly there are integrals
where any rearrangement helping with branch cuts with
respect to argument will be discontinuous with respect
to parameter.
My personal conclusion: do not try to hard. If there
is simple obvoius way to avoid branch cuts on real
line, then do it. But many proposed approches have
limited applicability and some make integral worse
from symbolic point of view.
--
Waldek Hebisch
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