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

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 Albert D. Rich Posts: 311 From: Hawaii Island Registered: 5/30/09
Re: first anniversary of the IITS
Posted: Dec 21, 2014 2:30 AM

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.

Albert