A204067 -id:A204067 - cp's OEIS frontend

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

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)).

A204068 Decimal expansion of the Fresnel Integral Integral_{x>=0} sin(x^3) dx.

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Jan 10 2013

Imaginary part associated with A204067.

			0.446489755784624605609282...

R. J. Mathar, Series expansion of generalized Fresnel integrals, arXiv:1211.3963 [math.CA], 2012, eq. (3.8).
Wikipedia, Fresnel Integral.

Cf. A073005, A073010, A073006, A202623, A204067

Maple
```
evalf(Pi/GAMMA(2/3)/3^(3/2) ) ;
```

RealDigits[Gamma[1/3]/6, 10, 120][[1]] (* Amiram Eldar, May 26 2023 *)

Equals Pi/(Gamma(2/3)* 3^(3/2)) = A073010 / A073006.
(this value)^2 + A204067^2 = A202623^2.
Equals Gamma(1/3)/6 = A073005 / 6. - Amiram Eldar, May 26 2023

A206161 Decimal expansion of the Fresnel integral int_{x=0..infinity} cos(x^4) dx.

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Jan 10 2013

			0.83740669676908648308360272218083226...

Wikipedia, Fresnel Integral

Cf. A204067.

evalf(Pi*cos(Pi/8)/GAMMA(3/4)/2^(3/2)) ;

RealDigits[ Sqrt[2 + Sqrt[2]]*Gamma[1/4]/8, 10, 72] // First (* Jean-François Alcover, Feb 20 2013 *)

Equals A093954 * A144981 / A068465 .

cp's OEIS Frontend

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

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A204068 Decimal expansion of the Fresnel Integral Integral_{x>=0} sin(x^3) dx.

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A206161 Decimal expansion of the Fresnel integral int_{x=0..infinity} cos(x^4) dx.

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