A093341 -id:A093341 - cp's OEIS frontend

A249283 Decimal expansion of K(3/4), where K is the complete elliptic integral of the first kind.

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

Offset: 1

Jean-François Alcover, Oct 24 2014

			2.15651564749964323543867499880032202886411021649282536...

Steven R. Finch, Gergonne-Schwarz Surface, April 12, 2013. [Cached copy, with permission of the author]
Eric Weisstein's World of Mathematics, Complete Elliptic Integral of the First Kind.

Cf. A093341 (K(1/2)), A249282 (K(1/4)), A000796, A068521.

evalf(EllipticK(sqrt(3)/2), 120); # Vaclav Kotesovec, Apr 22 2015

RealDigits[EllipticK[3/4], 10, 102] // First

ellK(sqrt(3/4)) \\ Charles R Greathouse IV, Feb 04 2025

Equals Pi/agm(1, 2) = A000796 / A068521. - Amiram Eldar, Apr 28 2025

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

Original entry on oeis.org

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

Jean-François Alcover, Apr 22 2015

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

			1.3506438810476755025201747353387258413495223669243545453232537...

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

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

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

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

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

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

A263809 Decimal expansion of C_{1/2}, a constant related to Kolmogorov's inequalities.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Oct 27 2015

			2.78640785937135371836849252065073648531496243503123575794856326376...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 7.7 Riesz-Kolmogorov Constants, p. 474.

Burgess Davis, On Kolmogorov's Inequalities, Transactions of the American Mathematical Society Vol. 222 (Sep., 1976), pp. 179-192.

Cf. A068465, A068466, A093341, A242822.

RealDigits[Gamma[1/4]^2/(Pi*Gamma[3/4]^2), 10, 105] // First

gamma(1/4)^2/(Pi*gamma(3/4)^2) \\ Michel Marcus, Oct 27 2015

C_{1/2} = gamma(1/4)^2/(Pi*gamma(3/4)^2).
Equals (1/Pi^2)*(integral_{0..Pi} sqrt(csc(t)) dt)^2.
Also equals (8/Pi^2)*A093341^2.

A276627 Decimal expansion of K(3-2*sqrt(2)), where K is the complete elliptic integral of the first kind.

Original entry on oeis.org

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

Offset: 1

Benedict W. J. Irwin, Sep 07 2016

The modulus k=3-2*sqrt(2).

K(k_4) in the MathWorld link.

			1.58255172722371591183313507107040987652948814961878924349716944847...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Elliptic Integral Singular Values
I. J. Zucker and G. S. Joyce, Special values of the hypergeometric series II, Math. Proc. Camb. Phil. Soc. 131 (2001) 309-319. Table 1 N=4.

Cf. A157259 (for 3-2*sqrt(2)).

SetDefaultRealField(RealField(100)); R:= RealField(); 2*(2+Sqrt(2))*Pi(R)^(3/2)/Gamma(-1/4)^2; // G. C. Greubel, Oct 08 2018

evalf(2*(2+sqrt(2))*Pi^(3/2)/GAMMA(-1/4)^2,120); # Muniru A Asiru, Oct 08 2018

RealDigits[N[EllipticK[(3 - 2 Sqrt[2])^2], 105]][[1]]
RealDigits[2*(2+Sqrt[2])*Pi^(3/2)/Gamma[-1/4]^2, 10, 100][[1]] (* G. C. Greubel, Oct 08 2018 *)

default(realprecision, 100); 2*(2+sqrt(2))*Pi^(3/2)/gamma(-1/4)^2 \\ G. C. Greubel, Oct 08 2018

ellK(3-sqrt(8)) \\ Charles R Greathouse IV, Feb 05 2025

Equals 2*(2+sqrt(2))*Pi^(3/2)/Gamma(-1/4)^2.
Equals A174968 * A062539 / 2. - R. J. Mathar, Aug 18 2023
Equals A093341 * A201488 [Zucker]. - R. J. Mathar, Jun 24 2024

A371859 Decimal expansion of Integral_{x=0..oo} 1 / sqrt(1 + x^5) dx.

Original entry on oeis.org

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

Offset: 1

Ilya Gutkovskiy, Apr 09 2024

			1.54969627774735302956219538317088212891969758...

Decimal expansions of Integral_{x=0..oo} 1 / sqrt(1 + x^k) dx: A118292 (k=3), A093341 (k=4), this sequence (k=5).

Cf. A001622, A087197, A340723, A352324.

RealDigits[Gamma[3/10] Gamma[6/5]/Sqrt[Pi], 10, 105][[1]]
RealDigits[2^(2/5) * Gamma[1/5]^2 / (5*GoldenRatio*Gamma[2/5]), 10, 105][[1]] (* Vaclav Kotesovec, Apr 09 2024 *)

Equals Gamma(3/10) * Gamma(6/5) / sqrt(Pi).

Equals 2^(2/5) * Gamma(1/5)^2 / (5 * phi * Gamma(2/5)), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Apr 09 2024

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A249283 Decimal expansion of K(3/4), where K is the complete elliptic integral of the first kind.

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

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A263809 Decimal expansion of C_{1/2}, a constant related to Kolmogorov's inequalities.

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A276627 Decimal expansion of K(3-2*sqrt(2)), where K is the complete elliptic integral of the first kind.

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A371859 Decimal expansion of Integral_{x=0..oo} 1 / sqrt(1 + x^5) dx.

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