Newsgroups: sci.math,sci.math.symbolic
From: Joe Keane <jgk@netcom.com>
Subject: Re: Integral challenges
Message-ID: <jgkDAJ0Fu.HCt@netcom.com>
Date: Wed, 21 Jun 1995 14:04:42 GMT

In article <3s3j8f$37d@alecto.algonet.se>
Bengt Mansson <bengtmn@algonet.se> writes:
>                                            inf
>                                             /
>1. Calculate the Cauchy principal value of   |  1/(x^5+x+1) dx
>                                             /
>                                           -inf 

In article <3s4l5a$d4h@nntp.ucs.ubc.ca>
Robert Israel <israel@math.ubc.ca> writes:
>                      4                         1
>  cpv := I Pi (-------------- + --------------------------------
>                          1/2     /    1/3      1         \4
>               - 3 + 5 I 3      5 |- %1    - ------- + 1/3|  + 1
>                                  |              1/3      |
>                                  \          9 %1         /
>
>                                           2
>       + --------------------------------------------------------------------)
>           /      1/3       1                   1/2 /    1/3      1   \\4
>         5 |1/2 %1    + -------- + 1/3 - 1/2 I 3    |- %1    + -------||  + 1
>           |                 1/3                    |              1/3||
>           \            18 %1                       \          9 %1   //
>
>                            25           1/2  1/2
>%1 :=                      ---- + 1/18 23    3
>                            54

I played with this for a while and simplified it to the following:

5/21*pi*sqrt(3)*{1+1/2*(sqrt(3)+5/sqrt(23))*cbrt(1/2*(sqrt(27)-sqrt(23)))
                  -1/2*(sqrt(3)-5/sqrt(23))*cbrt(1/2*(sqrt(27)+sqrt(23)))}

~= 1.5834249205602550720406091783245876180769193286348342610383720568

I think that's better, at least it's more symmetrical. :-)

--
Joe Keane, amateur mathematician