The On-Line Encyclopedia of Integer Sequences (OEIS)

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A352615

Decimal expansion of Integral_{0<=x,y<=Pi/2} sqrt(1-cos^2(x)*cos^2(y)) dx dy.
(history; published version)

#24 by N. J. A. Sloane at Tue May 03 23:51:12 EDT 2022

STATUS

proposed

approved

#23 by Bernard Schott at Fri Mar 25 02:34:54 EDT 2022

STATUS

editing

proposed

#22 by Bernard Schott at Fri Mar 25 02:33:14 EDT 2022

LINKS

Robert Ferréol, <a href="https://mathcurve.com/surfaces.gb/boheme/boheme.shtml">Bohemian Domedome</a>, Mathcurve.

Discussion

Fri Mar 25

02:34

Bernard Schott: Put Bohemian dome link.

#21 by Bernard Schott at Fri Mar 25 02:28:47 EDT 2022

LINKS

Robert Ferréol, <a href="https://mathcurve.com/surfaces.gb/boheme/boheme.shtml">Bohemain Bohemian Dome</a>, Mathcurve.

#20 by Bernard Schott at Fri Mar 25 02:26:26 EDT 2022

LINKS

Robert Ferréol, <a href="https://mathcurve.com/surfaces.gb/boheme/boheme.shtml">Bohemain Dome</a>, Mathcurve.

STATUS

proposed

editing

#19 by Jon E. Schoenfield at Thu Mar 24 19:38:32 EDT 2022

STATUS

editing

proposed

#18 by Jon E. Schoenfield at Thu Mar 24 19:38:29 EDT 2022

NAME

Decimal expansion of Integral_{0<=x,y<=Pi/2} sqrt(1-cos^2(x)*cos^2(y))dxdy dx dy.

FORMULA

Equals Sum _{n>=0} (Pi^2/(4*(2*n-1))*(binomial(2*n,n)/4^n)^3).

Equals (E(a) - K(a))^2 + E(a)^2 where a = 1/sqrt(2) and E (resp. K) is the complete elliptic integral of the second (resp. first) kind.

CROSSREFS

Cf. A091670 ((1/Pi^2)*Integral_{0<=x,y<=Pi} 1/sqrt(1-cos^2(x)*cos^2(y))dxdy/Pi^2 dx dy).

STATUS

proposed

editing

#17 by Robert FERREOL at Thu Mar 24 05:34:44 EDT 2022

STATUS

editing

proposed

#16 by Robert FERREOL at Thu Mar 24 05:33:26 EDT 2022

MAPLE

a:=1/sqrt(2):evalf((EllipticK(a)-EllipticE(a)-EllipticK(a))^2+EllipticE(a)^2, 50);

CROSSREFS

Cf. A091670 (Integral_{0<=x,y<=piPi}1/sqrt(1-cos^2(x)*cos^2(y))dxdy/Pi^2).

STATUS

proposed

editing

Discussion

Thu Mar 24

05:34

Robert FERREOL: I forgot a pi to Pi

#15 by Amiram Eldar at Thu Mar 24 05:26:10 EDT 2022

STATUS

editing

proposed