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
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 differentiation.
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.