[seqfan] Re: A140133

[Richard Mathar]

dmn> From: Douglas McNeil
dmn> Date: Tue, 1 Feb 2011 16:55:28 +0800
dmn> To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
dmn> 
dmn> http://oeis.org/A140133
dmn> "Decimal expansion of the area enclosed in the lens-shaped region of
dmn> the Laplace Limit.
dmn> Roughly 1.85298.
dmn> 
dmn> Could someone else check this?  I seem to converge on a value beginning
dmn> 
dmn> 1.8532684487079870332219364034397278879469653896325464013557810020678797365085166271171334885569025885

Supposed the parametrization with u is correct is correct on the MathWorld page,

int_0^1 dx int _0^y(x) dy
= int _0^ ustar (dx/du)*dy *y(u)
= int _ustar^ 0 du (1/2)* (coth u -u -u coth^2u )/sqrt(u*coth u -u^2) *sqrt(u^2 -u tanh u) ;

multiplied by 4 to get all 4 quadrant is roughly
1.853268449
agreeing with the modified value.

In Maple

# A085984, ustar:
umax := 1.19967864025773383391636984864114194426145878841860720891547778391812472522384747999908699214650937 ;

xu := sqrt(u*(coth(u)-u)) ;
yu := sqrt(u*(u-tanh(u))) ;
du := diff(xu,u) ;
intu := %*yu ;
intu := expand(%) ;
4*int(%,u=0..umax) ;
evalf(%) ;

In rare cases, Eric's numbers have been incorrect (A014571, A086465, A143524, A086240)
and this might be another one.