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Re: The A. F. Timofeev symbolic integration test suite
Posted:
Nov 13, 2013 6:56 AM
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Albert Rich schrieb:
>
> On Tuesday, November 12, 2013 7:33:53 AM UTC-10, clicl...@freenet.de wrote:
>
> > I noticed these points by scanning your file visually and checking
> > against the book only where I had doubts; no systematic comparison was
> > made - but should eventually be made.
> >
>
> All your proposed changes are incorporated into the revised pdf file
> at
>
> http://www.apmaths.uwo.ca/~arich/TimofeevChapter8TestResults.pdf
>
> with the exception of adding generic exponents for examples 5a and
> 5b, for which I could not find closed-form antiderivatives...
>
Oops. Integrals 5a and 6a for a generic exponent have the evaluations:
INT((a^(k*x) + a^(l*x))^n, x) =
(a^(k*x) + a^(l*x))^n/(k*n*LN(a)*(1 + a^((l - k)*x))^n)
*F21(-n, - k*n/(k - l), - k*n/(k - l) + 1, - a^((l - k)*x))
INT((a^(k*x) - a^(l*x))^n, x) =
(a^(k*x) - a^(l*x))^n/(k*n*LN(a)*(1 - a^((l - k)*x))^n)
*F21(-n, - k*n/(k - l), - k*n/(k - l) + 1, a^((l - k)*x))
The hypergeometric series here (and in Example 3) terminate for a
non-negative integer exponent n, while the series for your evaluation of
Examples 6a,b terminate (after applying Euler's transformation) for a
negative integer exponent n. The latter type of representation is
somewhat more compact; a disadvantage is that the antiderivative 6a for
positive a,m,x ends up on the branch cut of the hypergeometric function.
I therefore propose to normalize to the former type:
INT((1 + a^(m*x))^n, x) =
(1 + a^(m*x))^n/(m*n*LN(a)*(1 + a^(- m*x))^n)
*F21(-n, -n, 1 - n, - a^(- m*x))
INT((1 - a^(m*x))^n, x) =
(1 - a^(m*x))^n/(m*n*LN(a)*(1 - a^(- m*x))^n)
*F21(-n, -n, 1 - n, a^(- m*x))
I also notice now that the evaluations proposed for Examples 36a,b,c,d
involve TAN(x/2) and friends, whereas your evaluations for Examples 38,
40, 41, 43 involve SIN(x)/(1+COS(x)) and friends instead. I propose to
normalize in accordance with your policy for the entire Timofeev suite.
(In fact, I like the latter choice more; in Example 36 I simply followed
Timofeev, p. 355.)
Martin.
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