Drexel dragonThe Math ForumDonate to the Math Forum



Search All of the Math Forum:

Views expressed in these public forums are not endorsed by Drexel University or The Math Forum.


Math Forum » Discussions » sci.math.* » sci.math.symbolic.independent

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

Advanced Search

Back to Topic List Back to Topic List Jump to Tree View Jump to Tree View   Messages: [ Previous | Next ]
clicliclic@freenet.de

Posts: 988
Registered: 4/26/08
Re: The Charlwood Fifty
Posted: Jun 1, 2013 10:33 AM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply


clicliclic@freenet.de schrieb:
>
> This is my antiderivative for problem #45 from Charlwood's appendix:
>
> SQRT(2)*SQRT(SEC(x)+1)*SQRT(SEC(x)-1)*COT(x)*(SQRT(SQRT(2)-1)*AT~
> AN(SQRT(SQRT(2)-1)*(SQRT(SEC(x)+1)-SQRT(SEC(x)-1)-SQRT(2))/(SQRT~
> (2)*SQRT(SQRT(SEC(x)+1)-SQRT(SEC(x)-1))))-SQRT(SQRT(2)+1)*ATAN(S~
> QRT(SQRT(2)+1)*(SQRT(SEC(x)+1)-SQRT(SEC(x)-1)-SQRT(2))/(SQRT(2)*~
> SQRT(SQRT(SEC(x)+1)-SQRT(SEC(x)-1))))+SQRT(SQRT(2)-1)*ATANH(SQRT~
> (2*SQRT(2)+2)*SQRT(SQRT(SEC(x)+1)-SQRT(SEC(x)-1))/(SQRT(SEC(x)+1~
> )-SQRT(SEC(x)-1)+SQRT(2)))-SQRT(SQRT(2)+1)*ATANH(SQRT(2*SQRT(2)-~
> 2)*SQRT(SQRT(SEC(x)+1)-SQRT(SEC(x)-1))/(SQRT(SEC(x)+1)-SQRT(SEC(~
> x)-1)+SQRT(2))))
>
> which holds on the entire complex plane.
>


And here are versions of the antiderivatives for problems #12 and #13
from Charlwood's appendix that do not involve the imaginary unit:

x*ATAN(SQRT(1 - x^2) + x) - 1/2*ASIN(x)
- SQRT(3)/4*ATAN((2*x^2 - 1)/SQRT(3))
+ SQRT(3)/4*ATAN((SQRT(3)*x - 1)/SQRT(1 - x^2))
+ SQRT(3)/4*ATAN((SQRT(3)*x + 1)/SQRT(1 - x^2))
- 1/8*LN(x^4 - x^2 + 1) - 1/4*ATANH(x*SQRT(1 - x^2))

- SQRT(1 - x^2)*ATAN(SQRT(1 - x^2) + x) - 1/2*ASIN(x)
- SQRT(3)/4*ATAN((2*x^2 - 1)/SQRT(3))
+ SQRT(3)/4*ATAN((SQRT(3)*x - 1)/SQRT(1 - x^2))
+ SQRT(3)/4*ATAN((SQRT(3)*x + 1)/SQRT(1 - x^2))
+ 1/8*LN(x^4 - x^2 + 1) + 1/4*ATANH(x*SQRT(1 - x^2))

While the discontinuity of Albert's antiderivatives at x = -1 and x = +1
is not serious, the above are more compact and also fully continuous.

The elliptic integral #49 in the appendix should be taken for a misprint
in view of Charlwood's statement that he considers "the ability of three
computer algebra systems (CAS) to evaluate [integrals] in closed-form,
appealing only to the class of real, elementary functions", so I see no
reason to remove the imaginary unit here. It can be done though.

The other antiderivatives in Albert's file are real already.

Martin.


Date Subject Author
5/23/13
Read The Charlwood Fifty
Albert D. Rich
5/23/13
Read Re: The Charlwood Fifty
Nasser Abbasi
5/23/13
Read Re: The Charlwood Fifty
clicliclic@freenet.de
6/16/13
Read Re: The Charlwood Fifty
clicliclic@freenet.de
6/16/13
Read Re: The Charlwood Fifty
Nasser Abbasi
6/17/13
Read Re: The Charlwood Fifty
clicliclic@freenet.de
6/17/13
Read Re: The Charlwood Fifty
Nasser Abbasi
6/17/13
Read Re: The Charlwood Fifty
clicliclic@freenet.de
6/17/13
Read Re: The Charlwood Fifty
Nasser Abbasi
6/17/13
Read Re: The Charlwood Fifty
Waldek Hebisch
6/18/13
Read Re: The Charlwood Fifty
Albert D. Rich
6/24/13
Read Re: The Charlwood Fifty
clicliclic@freenet.de
6/24/13
Read Re: The Charlwood Fifty
Albert D. Rich
5/23/13
Read Re: The Charlwood Fifty
Albert D. Rich
5/23/13
Read Re: The Charlwood Fifty
Waldek Hebisch
5/24/13
Read Re: The Charlwood Fifty
Andreas Dieckmann
5/25/13
Read Re: The Charlwood Fifty
clicliclic@freenet.de
6/1/13
Read Re: The Charlwood Fifty
clicliclic@freenet.de
6/3/13
Read Re: The Charlwood Fifty
Albert D. Rich
6/6/13
Read Re: The Charlwood Fifty
clicliclic@freenet.de
6/6/13
Read Re: The Charlwood Fifty
Nasser Abbasi
6/6/13
Read Re: The Charlwood Fifty
clicliclic@freenet.de
6/8/13
Read Re: The Charlwood Fifty
clicliclic@freenet.de
6/8/13
Read Re: The Charlwood Fifty
Albert D. Rich
6/8/13
Read Re: The Charlwood Fifty
clicliclic@freenet.de
6/8/13
Read Re: The Charlwood Fifty
clicliclic@freenet.de
6/9/13
Read Re: The Charlwood Fifty
Albert D. Rich
6/9/13
Read Re: The Charlwood Fifty
Nasser Abbasi
6/9/13
Read Re: The Charlwood Fifty
Albert D. Rich
6/9/13
Read Re: The Charlwood Fifty
Nasser Abbasi
6/10/13
Read Re: The Charlwood Fifty
clicliclic@freenet.de
6/10/13
Read Re: The Charlwood Fifty
Nasser Abbasi
6/10/13
Read Re: The Charlwood Fifty
Nasser Abbasi
6/12/13
Read Re: The Charlwood Fifty
Nasser Abbasi
6/13/13
Read Re: The Charlwood Fifty
clicliclic@freenet.de
6/13/13
Read Re: The Charlwood Fifty
Nasser Abbasi
6/13/13
Read Re: The Charlwood Fifty
Waldek Hebisch
6/13/13
Read Re: The Charlwood Fifty
Nasser Abbasi
6/13/13
Read Re: The Charlwood Fifty
Waldek Hebisch
6/14/13
Read Re: The Charlwood Fifty
clicliclic@freenet.de
6/14/13
Read Re: The Charlwood Fifty
Waldek Hebisch
6/14/13
Read Re: The Charlwood Fifty / Sage and sympy
Richard Fateman
6/14/13
Read Re: The Charlwood Fifty
clicliclic@freenet.de
6/9/13
Read Re: The Charlwood Fifty and Macsyma
Richard Fateman
6/9/13
Read Re: The Charlwood Fifty and Macsyma
Albert D. Rich
6/9/13
Read Re: The Charlwood Fifty, another Macsyma result
Richard Fateman
6/9/13
Read Re: The Charlwood Fifty, another Macsyma result
Albert D. Rich
6/10/13
Read Re: The Charlwood Fifty, another Macsyma result
Richard Fateman
6/10/13
Read Re: The Charlwood Fifty, another Macsyma result
clicliclic@freenet.de
6/10/13
Read Re: The Charlwood Fifty, another Macsyma result
Waldek Hebisch
6/11/13
Read Re: The Charlwood Fifty, another Macsyma result
Nasser Abbasi
6/11/13
Read Re: The Charlwood Fifty, another Macsyma result
clicliclic@freenet.de
6/11/13
Read Re: The Charlwood Fifty, another Macsyma result
Nasser Abbasi

Point your RSS reader here for a feed of the latest messages in this topic.

[Privacy Policy] [Terms of Use]

© Drexel University 1994-2014. All Rights Reserved.
The Math Forum is a research and educational enterprise of the Drexel University School of Education.