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Topic: first anniversary of the IITS
Replies: 21   Last Post: Jan 10, 2015 4:01 PM

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Posts: 1,039
Registered: 4/26/08
Re: first anniversary of the IITS
Posted: Mar 21, 2014 2:42 PM
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Albert Rich schrieb:
> On Saturday, March 8, 2014 4:22:30 AM UTC-10, clicl...@freenet.de wrote:

> > My interest in Example 44 (like the similar but simpler 62, 64, 66, and
> > 118) was mostly in the asymmetric LN evaluation. Please fuse the ATAN's
> > in numbers 62, 64, 66 as you see fit; I find no continuity problems:
> >
> > ATAN(1 + SQRT(2)*SQRT(TAN(x))) - ATAN(1 - SQRT(2)*SQRT(TAN(x)))
> > = ATAN((TAN(x) - 1)/(SQRT(2)*SQRT(TAN(x)))) + pi/2
> >
> > [...]

> I revised the optimal antiderivatives for examples 62, 64, 66, 90, 91,
> 112 and 118 of Chapter 5 of the Timofeev integration test suite
> unifying arctangents as you suggested, and posted the results on the
> Rubi website at
> http://www.apmaths.uwo.ca/~arich/
> Note that the derivative of each side of the above equation are equal,
> so the unified arctangents can be used for antiderivatives. However,
> the two sides are not always equal assuming standard CCW
> (counter-clockwise) continuity is used along the arctangent's branch
> cut on the imaginary line. Interestingly the equation IS everywhere
> valid in Derive, since Derive uses CW continuity on the arctangent's
> branch cut...

It should be pointed out that paired ATAN's cannot always be fused
without continuity problems; consider

ATAN((1 - SQRT(3)*x)/SQRT(1-x^2)) - ATAN((1 + SQRT(3)*x)/SQRT(1-x^2))
= ATAN((1 - 2*x^2)/(SQRT(3)*x*SQRT(1-x^2))) ± pi/2

at x=0, for example.

The additive constants in the ATAN - ATAN = ATAN equations I gave apply
in Derive 6.10 on the real axis; often - and the more so on the complex
plane - they will be piecewise constants only. However, as you say, for
antiderivatives it suffices that the relations hold everywhere after

Just cosmetic defects are left to report in the Chapter 5 evaluations:
in Example 100, different versions of the radicand are employed in
parallel. And COS(2*x) could be used to advantage in the evaluations of
Examples 93-94, 96 and 117. A systematic check against the evaluations
in Timofeev's book, however, remains to be made.


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