A217481 -id:A217481 - cp's OEIS frontend

A204067 Decimal expansion of the Fresnel Integral, Integral_{x >= 0} cos(x^3) dx.

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

Offset: 0

R. J. Mathar, Jan 10 2013

			0.7733429420779898501961016...

Muniru A Asiru, Table of n, a(n) for n = 0..2000
R. J. Mathar, Series expansion of generalized Fresnel integrals, arXiv:1211.3963 [math.CA], 2012, eq. (3.8).
Wikipedia, Fresnel Integral.

Cf. A010469, A019670, A073005, A073006, A204068, A217481.

evalf(int(cos(x^3),x=0..infinity),120); # Muniru A Asiru, Sep 26 2018

RealDigits[Gamma[1/3]/(2*Sqrt[3]), 10, 120][[1]] (* Amiram Eldar, May 26 2023 *)

Pi/(3*gamma(2/3)) \\ Gheorghe Coserea, Sep 26 2018

intnum(x=[0, -2/3], [oo, I], cos(x)/x^(2/3))/3 \\ Gheorghe Coserea, Sep 26 2018

Equals Pi/(3*Gamma(2/3)) = A019670 / A073006.

Equals Gamma(1/3)/(2*sqrt(3)) = A073005 / A010469. - Amiram Eldar, May 26 2023

A143149 Decimal expansion of 5sqrt(2Pi)/4.

Original entry on oeis.org

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

Offset: 1

Jonathan Vos Post, Jul 27 2008

Upper bound using Shannon entropy arising in randomly-projected hypercubes.

			3.13328534328875...

David L. Donoho and Jared Tanner, Counting the faces of randomly-projected hypercubes and orthants, with applications, Discrete & computational geometry, Vol. 43 (2010), pp. 522-541, see p. 533; arXiv preprint, arXiv:0807.3590 [math.MG], 2008, see p. 10.
Wikipedia, Fresnel Integral.
Index entries for transcendental numbers.

Apart from possible scaling sqrt(A019692/2^n) for n=0..7 are A019727, A002161, A069998, A019704, A217481, A019706, this sequence, A019710.

Cf. A143148 (lower bound).

RealDigits[5*Sqrt[2*Pi]/4, 10, 120][[1]] (* Amiram Eldar, Jun 13 2023 *)

5*sqrt(2*Pi)/4 \\ Michel Marcus, Mar 06 2020

Equals 10*Integral_{x>=0} x*sin(x^4) dx or 10*Integral_{x>=0} x*cos(x^4) dx (Fresnel integrals).

Edited and a(100) corrected by Georg Fischer, Jul 16 2021

A387213 Decimal expansion of Integral_{x>=0} sin(x) * sin(x^2) dx.

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, Aug 22 2025

			0.49169967769382111771654625416890810022151027126875...

Manjul Bhargava, Kiran Kedlaya, and Lenny Ng, Solutions to the 61st William Lowell Putnam Mathematical Competition Saturday, December 2, 2000, The Putnam Archive. See Problem A-4.
Editors of the Mathematics Magazine, The 61st Annual William Lowell Putnam Examination, News and Letters, Mathematics Magazine, Vol. 74, No. 1 (2001), pp. 75-83. See Problem A4, pages 77 and 79-80.
Jihyung Kang and others, Convergence of I = Integral_0^oo sinx sin(x2) dx, Mathematics StackExchange, 2017.
Leonard F. Klosinski, Gerald L. Alexanderson, and Loren C. Larson, The Sixty-first William Lowell Putnam Mathematical Competition, The American Mathematical Monthly, Vol. 108, No. 9 (2001), pp. 841-850. See pages 843 and 846, Problem A4.
Eric Weisstein's World of Mathematics, Fresnel Integrals.
Wikipedia, Fresnel integral.

Cf. A069998, A217481, A231863.

RealDigits[Integrate[Sin[x]*Sin[x^2], {x, 0, Infinity}], 10, 120][[1]]
(* or *)
RealDigits[Sqrt[Pi/2] * (Cos[1/4] * FresnelC[1/Sqrt[2*Pi]] + Sin[1/4] * FresnelS[1/Sqrt[2*Pi]]), 10, 120][[1]]

Equals sqrt(Pi/2) * (cos(1/4) * FresnelC(1/sqrt(2*Pi)) + sin(1/4) * FresnelS(1/sqrt(2*Pi))), where FresnelC(x) and FresnelS(x) are the Fresnel integrals C(x) and S(x), respectively.

Equals Integral_{x=0..1/2} cos(x^2 - 1/4) dx.

cp's OEIS Frontend

A204067 Decimal expansion of the Fresnel Integral, Integral_{x >= 0} cos(x^3) dx.

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A143149 Decimal expansion of 5sqrt(2Pi)/4.

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A387213 Decimal expansion of Integral_{x>=0} sin(x) * sin(x^2) dx.

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cp's OEIS Frontend

A204067 Decimal expansion of the Fresnel Integral, Integral_{x >= 0} cos(x^3) dx.

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A143149 Decimal expansion of 5*sqrt(2*Pi)/4.

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A387213 Decimal expansion of Integral_{x>=0} sin(x) * sin(x^2) dx.

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A143149 Decimal expansion of 5sqrt(2Pi)/4.