A204068 -id:A204068 - 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

A265011 Decimal expansion of Integral_{x=0..1} sin(log(x))/((x+1)*log(x)) dx.

Original entry on oeis.org

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

Offset: 0

John M. Campbell, Apr 06 2016

This integral has an elegant evaluation in terms of the gamma function (see below formula). There is an interesting "symmetry" between the expressions involving the gamma function in this evaluation.

			This integral is equal to 0.50667090321662298198525580478358151247...

John M. Campbell, An Algorithm for Trigonometric-Logarithmic Definite Integrals, in the Mathematica Journal, Vol. 19.10 (2017).
W. M. Gosper, Material from Bill Gosper's Computers & Math talk, M.I.T., 1989, i+38+1 pages, annotated and scanned, included with the author's permission. (There are many blank pages because about half of the original pages were two-sided, half were one-sided.) See page 8.

Decimal expansions of definite integrals over elementary functions: A256127, A256128, A256129, A204067, A204068, A205885, A206161, A206160, A206769, A229174, A083648, A094691, A098687, A177218, A188141, A233382, A256273, A258086.

Cf. A309209 (continued fraction of the negation of this constant).

Print[RealDigits[Re[Log[2] + Log[((Gamma[1 - I/2]^2 Gamma[1 + I])/(Gamma[1 + I/2]^2 Gamma[1 - I]))^(I/2)]], 10, 100]] ;
NIntegrate[Sin[Log[x]]/(x + 1)/Log[x], {x, 0, 1}]

PARI
```
intnum(x=0,1,sin(log(x))/(x+1)/log(x))
```

Equals log(2) + log(((Gamma(1 - i/2)^2*Gamma(1 + i))/(Gamma(1 + i/2)^2*Gamma(1 - i)))^(i/2)), where i = sqrt(-1) denotes the imaginary unit.

Equals Sum_{n >= 0} (-1)^n*arctan(1/(n+1)).

A206160 Decimal expansion of the Fresnel integral Integral_{x=0..oo} x*sin(x^3) dx.

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Jan 10 2013

The imaginary part associated with A205885.

			0.3909001784211668179621792998...

Wikipedia, Fresnel Integral.

Cf. A002194, A204068, A205885.

Maple
```
evalf(sqrt(3)*GAMMA(2/3)/6) ;
```

RealDigits[Sqrt[3] * Gamma[2/3] / 6, 10, 120][[1]] (* Amiram Eldar, Aug 23 2024 *)

Equals A205885 times A002194.

cp's OEIS Frontend

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

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A265011 Decimal expansion of Integral_{x=0..1} sin(log(x))/((x+1)*log(x)) dx.

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A206160 Decimal expansion of the Fresnel integral Integral_{x=0..oo} x*sin(x^3) dx.

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