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Topic: The Charlwood Fifty
Replies: 52   Last Post: Jun 24, 2013 10:24 PM

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 clicliclic@freenet.de Posts: 1,245 Registered: 4/26/08
Re: The Charlwood Fifty
Posted: Jun 6, 2013 12:23 PM

Albert Rich schrieb:
>
> I just posted a revised pdf file of the Charlwood Fifty integration
> test-suite at
>
> http://www.apmaths.uwo.ca/~arich/CharlwoodIntegrationProblems.pdf
>
> that shows improved antiderivatives for problems #12, #13 (that Martin
> suggested) and #41. Please disregard the previous version of the file.
> Also posted is a Mathematica package file of the test-suite in machine
>
> http://www.apmaths.uwo.ca/~arich/CharlwoodProblems.m
>
> Not having heard back from Professor Charlwood, I took the liberty of
> changing the integrand of problem #49 from arcsin(x*sqrt(1-x^2)) to
> arcsin(x/sqrt(1-x^2)). This was done so all the integrands and
> antiderivatives in the test-suite would involve only elementary
> functions and operators.
>
> Hopefully all the antiderivatives in Charlwood Fifty test-suite are
> now optimal...
>

It would have been nice if Prof. Charlwood could have thrown light on
problem #49 from his appendix: Did he really want his students to work
on an elementary evaluation of INT(ASIN(x*SQRT(1-x^2)), x) and fail?
Anyway, here is a real version of the elliptic result:

x*ASIN(x*SQRT(1-x^2))+2*SQRT(1-x^2)*SQRT(x^4-x^2+1)/(2-x^2)+SQRT~
(x^4/(2-x^2)^2)*SQRT(x^4-x^2+1)/(2*x^2*SQRT((x^4-x^2+1)/(2-x^2)^~
2))*(EL_F(ASIN(2*SQRT(1-x^2)/(2-x^2)),SQRT(3)/2)-4*EL_E(ASIN(2*S~
QRT(1-x^2)/(2-x^2)),SQRT(3)/2))

where the incomplete elliptic integrals are defined as:

EL_F(phi, k) := INT(1/SQRT(1 - k^2*SIN(t)^2), t, 0, phi)

EL_E(phi, k) := INT(SQRT(1 - k^2*SIN(t)^2), t, 0, phi)

Martin.