A244921 -id:A244921 - cp's OEIS frontend

A254133 Decimal expansion of Lamb's integral K_0.

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

Offset: 0

Jean-François Alcover, Jan 26 2015

			0.490772728958345159162717253203382640381923347758584656...

D. H. Bailey, J. M. Borwein, and R. E. Crandall, Advances in the theory of box integrals (2010) p. 18.
Eric Weisstein's World of Mathematics, Inverse Tangent Integral.

Cf. A244920, A244921, A244922, A254134, A254135.

evalf(int(arctanh(1/sqrt(3 + x^2))/(1 + x^2), x=0..1), 120); # Vaclav Kotesovec, Jan 26 2015

Ti2[x_] := (I/2)*(PolyLog[2, -I*x] - PolyLog[2, I*x]); K0 = (3/2)*Ti2[3 - 2 Sqrt[2]] + Pi/4*Log[1 + Sqrt[2]] - Catalan/2 // Re; RealDigits[K0, 10, 103] // First

K_0 = integral_[0..1] arctanh(1/sqrt(3 + x^2))/(1 + x^2) dx.

K_0 = 3/2*Ti_2(3 - 2*sqrt(2)) + Pi/4*log(1 + sqrt(2)) - Catalan/2, where Ti_2 is Lewin's arctan integral, Ti_2(x) = (i/2)*(Li_2(-i*x) - Li_2(i*x)).

A254134 Decimal expansion of Lamb's integral K_1.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Jan 26 2015

			1.6619078747381233774065816861630594973488686732512589...

D. H. Bailey, J. M. Borwein, R. E. Crandall, Advances in the theory of box integrals (2010) p. 18.
Eric Weisstein's MathWorld, Clausen's Integral

Cf. A244920, A244921, A244922, A254133, A254135.

evalf(int(arcsec(x)/sqrt(x^2 - 4*x + 3), x=3..4), 120); # Vaclav Kotesovec, Jan 26 2015

Cl2[x_] := (I/2)*(PolyLog[2, Exp[-I*x]] - PolyLog[2, Exp[I*x]]); th = (ArcTan[(16 - 3*Sqrt[15])/11] + Pi)/3; K1 = Cl2[th] - Cl2[th + Pi/3] - Cl2[th - Pi/2] + Cl2[th - Pi/6] - Cl2[3*th + Pi/3] + Cl2[3*th + 2*(Pi/3)] - Cl2[3*th - 5*(Pi/6)] + Cl2[3*th + 5*(Pi/6)] + (6*th - 5*(Pi/2))*Log[2 - Sqrt[3]] // Re; RealDigits[K1, 10, 103] // First

K_1 = integral_[3..4] arcsec(x)/sqrt(x^2 - 4*x + 3) dx.

K_1 = Cl_2(th) - Cl_2(th + Pi/3) - Cl_2(th - Pi/2) + Cl_2(th - Pi/6) - Cl_2(3*th + Pi/3) + Cl_2(3*th + 2*(Pi/3)) - Cl_2(3*th - 5*(Pi/6)) + Cl_2(3*th + 5*(Pi/6)) + (6*th - 5*(Pi/2))*log(2 - sqrt(3)), where Cl_2 is the Clausen function and th = (arctan((16 - 3*sqrt(15))/11) + Pi)/3.

A254135 Decimal expansion of Lamb's integral K_2.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Jan 26 2015

			0.69266081515264750650943118588427245846713483280766884258...

D. H. Bailey, J. M. Borwein, and R. E. Crandall, Advances in the theory of box integrals (2010) p. 18.
Eric Weisstein's World of Mathematics, Inverse Tangent Integral.

Cf. A244920, A244921, A244922, A254133, A254134.

evalf(int(sqrt(1 + sec(x)^2)*arctan(1/sqrt(1 + sec(x)^2)), x=0..Pi/4), 120); # Vaclav Kotesovec, Jan 26 2015

Ti2[x_] := (I/2)* (PolyLog[2, -I *x] - PolyLog[2, I *x]); K2 = (1/2)*Ti2[-2 + Sqrt[3]] + (Pi/8)*Log[2 + Sqrt[3]] + Pi^2/32 // Re; RealDigits[K2, 10, 103] // First

K_2 = integral_[0..Pi/4] sqrt(1 + sec(x)^2)*arctan(1/sqrt(1 + sec(x)^2)) dx.

K_2 = (1/2)*Ti_2(-2 + sqrt(3)) + (Pi/8)*log(2 + sqrt(3)) + Pi^2/32, where Ti_2 is Lewin's arctan integral, Ti_2(x) = (i/2)*(Li_2(-i*x) - Li_2(i*x)).

A254968 Decimal expansion of the mean reciprocal Euclidean distance from a point in a unit cube to a given vertex of the cube (named B_3(-1) in Bailey's paper).

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Feb 11 2015

			1.1900386819897767533219086751420769449911806073574982644...

D. H. Bailey, J. M. Borwein and R. E. Crandall, Box Integrals, J. Comp. Appl. Math., Vol. 206, No. 1 (2007), pp. 196-208.
D. H. Bailey, J. M. Borwein, and R. E. Crandall, Advances in the theory of box integrals, Math. Comp. 79 (271) (2010) 1839-1866, Table 2.
Eric Weisstein's MathWorld, Box Integral.

Cf. A130590, A244920, A244921.

RealDigits[(3/2)*Log[2 + Sqrt[3]] - Pi/4, 10, 105] // First

Equals B_3(-1) = (3/2)*log(2 + sqrt(3)) - Pi/4.

Equals log(7 + 4*sqrt(3)) - Pi/4 - arcsinh(1/sqrt(2)).

Name corrected by Amiram Eldar, Jun 04 2023

A355184 Decimal expansion of the perimeter of the region that represents the set of points in a unit square that are closer to the center of the square than to the closest edge.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Jun 23 2022

The shape is formed by the intersection of four parabolas. Its area is given in A355183.

			1.70308249665895322783584912274920315719803442295049...

Amiram Eldar, Illustration.
Missouri State University, Problem #5, The Area and Perimeter of a Certain Region, Advanced Problem Archive; Solution to Problem #5, by John Shonder.

Cf. A103712, A244921, A254140, A352453, A355183 (area).

RealDigits[2*Log[Sqrt[4 - 2*Sqrt[2]] + Sqrt[2] - 1] - Sqrt[16 - 8*Sqrt[2]] + Sqrt[32 - 16*Sqrt[2]], 10, 100][[1]]

Equals 2*log(sqrt(4-2*sqrt(2))+sqrt(2)-1) - sqrt(16-8*sqrt(2)) + sqrt(32-16*sqrt(2)).

A254980 Decimal expansion of the mean reciprocal Euclidean distance from a point in a unit 4D cube to a given vertex of the cube (named B_4(-1) in Bailey's paper).

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Feb 11 2015

			0.96741202124116589866183643817815839013593700929996...

D. H. Bailey, J. M. Borwein and R. E. Crandall, Box Integrals, J. Comp. Appl. Math., Vol. 206, No. 1 (2007), pp. 196-208.
D. H. Bailey, J. M. Borwein, and R. E. Crandall, Advances in the theory of box integrals, Math. Comp. 79 (271) (2010) 1839-1866, Table 2.
Eric Weisstein's MathWorld, Inverse Tangent Integral.
Eric Weisstein's MathWorld, Polylogarithm.
Eric Weisstein's MathWorld, Box Integral.

Cf. A130590, A244920, A244921, A254968, A254979.

Ti2[x_] := (I/2)*(PolyLog[2, -I*x] - PolyLog[2, I*x]); B4[-1] = 2*Log[3] - (2/3) * Catalan + 2*Ti2[3 - 2*Sqrt[2]] - Sqrt[8]*ArcTan[1/Sqrt[8]] // Re; RealDigits[ B4[-1], 10, 104] // First

from mpmath import *
mp.dps=105
x=3 - 2*sqrt(2)
Ti2x=(j/2)*(polylog(2, -j*x) - polylog(2, j*x))
C = 2*log(3) - (2/3)*catalan + 2*Ti2x - sqrt(8) * atan(1/sqrt(8))
print([int(n) for n in list(str(C.real)[2:-1])]) # Indranil Ghosh, Jul 03 2017

B_4(-1) = 2*log(3) - (2/3)*Catalan + 2*Ti_2(3-2*sqrt(2)) - sqrt(8) * arctan( 1/sqrt(8) ), where Ti_2(x) = (i/2)*(polylog(2, -i*x) - polylog(2, i*x)) (Ti_2 is the inverse tangent integral function).

Name corrected by Amiram Eldar, Jun 04 2023

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A254133 Decimal expansion of Lamb's integral K_0.

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A254134 Decimal expansion of Lamb's integral K_1.

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A254135 Decimal expansion of Lamb's integral K_2.

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A254968 Decimal expansion of the mean reciprocal Euclidean distance from a point in a unit cube to a given vertex of the cube (named B_3(-1) in Bailey's paper).

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A355184 Decimal expansion of the perimeter of the region that represents the set of points in a unit square that are closer to the center of the square than to the closest edge.

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A254980 Decimal expansion of the mean reciprocal Euclidean distance from a point in a unit 4D cube to a given vertex of the cube (named B_4(-1) in Bailey's paper).

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