On Sat, Aug 27, 2022 at 03:15:56PM +0000, Waldek Hebisch wrote:
> On Sat, Aug 27, 2022 at 06:53:24PM +0800, Qian Yun wrote:
> > I tested some of the integrals, not all.
> >
> > It seems that for integrals contain only x (no a/b), failure is caused
> > by commit 1f42999f91ce516a8d027a61be4ecbf32ad2ada4 on Jun 13 2022,
> > "Handle some elliptic integrals".
>
> The actual problem is use of 'real' by integrator.
ANd to elaborate a bit: this is old problem and can be shown
using elementary functions. Notable example is:
(1) -> f := 1/acosh(sin(x))
1
(1) -------------
acosh(sin(x))
Type: Expression(Integer)
(2) -> integrate(D(f, x), x)
(2) 0
Interestingly:
(3) -> normalize(f)
1
(3) --------------------------------------
+---+ x 2 x +---+
\|- 1 tan(-) + 2 tan(-) - \|- 1
2 2
log(---------------------------------)
x 2
tan(-) + 1
2
Type: Expression(Integer)
(4) -> complexNormalize(f)
1
(4) ---------------------
+---+ +---+
log(\|- 1 ) - x\|- 1
Type: Expression(Integer)
which shows that there is algebraic dependency which is only
discoverd over complex numbers. That add twist to real version
of Risch structure theorem (which says that for real function
it is enough to look at dependencies over real numbers):
in Risch structure theorem function must be expressed in
terms on exp, log, tan and atan. acosh(sin(x)) looks
real, but when re-expressed in terms of basic functions
it turn out complex, so we need to use complex version to
handle it.
--
Waldek Hebisch
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