A257407 Decimal expansion of E(1/sqrt(2)) = 1.35064..., where E is the complete elliptic integral.

1, 3, 5, 0, 6, 4, 3, 8, 8, 1, 0, 4, 7, 6, 7, 5, 5, 0, 2, 5, 2, 0, 1, 7, 4, 7, 3, 5, 3, 3, 8, 7, 2, 5, 8, 4, 1, 3, 4, 9, 5, 2, 2, 3, 6, 6, 9, 2, 4, 3, 5, 4, 5, 4, 5, 3, 2, 3, 2, 5, 3, 7, 0, 8, 8, 5, 7, 8, 7, 7, 8, 9, 0, 8, 3, 6, 1, 2, 7, 3, 6, 9, 0, 4, 0, 2, 3, 6, 0, 7, 7, 8, 2, 2, 4, 9, 1, 5, 6, 3, 6, 0, 9, 9, 4, 7

Offset: 1

Author

Jean-François Alcover, Apr 22 2015

Keywords

• nonn
• cons
• easy

Comments

This constant is sometimes expressed as E(1/2), with a different convention of argument (Cf. Mathematica).

Examples

			1.3506438810476755025201747353387258413495223669243545453232537...
		

References

• Jonathan Borwein, David H. Bailey, Mathematics by Experiment, 2nd Edition: Plausible Reasoning in the 21st Century, CRC Press (2008), p. 145.

Links

• G. C. Greubel, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Elliptic Integral of the Second Kind
• Wikipedia, Complete elliptic integral of the second kind

Crossrefs

Cf. A010466, A053004, A062539, A076390, A093341, A105419.

Programs

Maple

evalf(EllipticE(1/sqrt(2)),120); # Vaclav Kotesovec, Apr 22 2015

Mathematica

RealDigits[EllipticE[1/2], 10, 106] // First

Formula

Equals (4*B^2 + Pi)/(4*sqrt(2)*B), where B is the lemniscate constant A076390.

Equals Pi^(3/2)/Gamma(1/4)^2 + Gamma(1/4)^2/(8*Pi^(1/2)).

Equals (agm(1,sqrt(2))+Pi/agm(1,sqrt(2)))/sqrt(8) = (A053004+A062539)/A010466. - Gleb Koloskov, Jun 29 2021