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

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clicliclic@freenet.de

Posts: 1,245
Registered: 4/26/08
Re: first anniversary of the IITS
Posted: Jan 9, 2015 1:06 PM
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Albert Rich schrieb:
>
> On Friday, December 19, 2014 9:13:37 AM UTC-10, clicl...@freenet.de wrote:
>

> >> [...] A systematic check against the evaluations in Timofeev's book,
> >> however, remains to be made.

> >
> > This referred to the trigonometric integration examples in Chapter 5
> > of Timofeev's 1948 book, as provided in digitized form at
> > <http://www.apmaths.uwo.ca/~arich/>. Such a check has now been made,
> > with the result that seven integrands need to be corrected because they
> > were typeset incorrectly or transcribed incorrectly or both.

>
> [...]
>
> I revised the 7 problems in the Timofeev test suite you flagged for
> correction pretty much as you suggested.
>

> > While the next two integrals and their evaluations are correct, the
> > evaluations can be improved in some respects.
> >
> > 1. Either of the following evaluations of Example 19.m (p. 220) is
> > shorter than the version currently shown, the two differ in being
> > discontinuous where SIN(x)^2 = 1 and where COS(x)^2 = 1, respectively:
> >
> > INT(SIN(x)^(2*m)*COS(x)^(2*m),x)=SIN(x)^(2*m+1)*COS(x)^(2*m+1)/(~
> > 2*m+1)*F21(1,1+2*m,3/2+m,SIN(x)^2)=-SIN(x)^(2*m+1)*COS(x)^(2*m+1~
> > )/(2*m+1)*F21(1,1+2*m,3/2+m,COS(x)^2)

>
> Although shorter, and probably correct, I am unable to get Mathematica
> to verify these antiderivatives.
>

> > 2. The evaluation of Example 45 (p. 254) can be continuitized in the
> > customary manner:
> >
> > INT(1/(4-3*COS(x)^2+5*SIN(x)^2),x)=x/3+1/3*ATAN(2*SIN(x)*COS(x)/(2*SIN(x)^2+1))

>
> I tweaked Rubi to give the continuous antiderivative you suggested.
>

> > The remaining 29 items involve minor variations from the versions
> > currently shown, such as a modified term order in the integrand and/or
> > different trigonometric functions in the evaluation; they are offered
> > here for inspection without much further comment.

>
> [...]
>
> The only suggestion I incorporated was to example 54 (p. 260).
> Although some of your suggestions resulted in slightly more compact
> results, they were not really simpler.
>

> > I expect repercussions of the above on Rubi to be small.
>
> Yes the repercussions were small, but resulted in a few nice
> improvements. The revised Timofeev test suite is now available at
> http://www.apmaths.uwo.ca/~arich.
>


Looking at Example 114 again, I find that both the integrand involving
1 + (1-8*TAN(x)^2)^(1/3) and that involving 1 - (1-8*TAN(x)^2)^(1/3)
necessitate the correction of exactly one sign in the printed evaluation
and the printed integral, respectively. This means that Timofeev's
original intention just cannot be deduced here. Actually, the integrand
involving 1 + (1-8*TAN(x)^2)^(1/3) might then be preferred to emphasize
the similarity with Example 115.

Martin.



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