A257957 -id:A257957 - cp's OEIS frontend

A257955 Decimal expansion of Gamma(1/Pi).

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

Offset: 1

Iaroslav V. Blagouchine, May 14 2015

The reference gives an interesting product representation in terms of rational multiple of 1/Pi for Gamma(1/Pi).

			2.8112975146708616421227908037104816935281655223291765...

G. C. Greubel, Table of n, a(n) for n = 1..5000
Iaroslav V. Blagouchine, Two series expansions for the logarithm of the gamma function involving Stirling numbers and containing only rational coefficients for certain arguments related to 1/Pi, Mathematics of Computation (AMS), 2015.

Cf. A257957, A257958, A257959, A002161, A073005, A068466, A175380, A175379, A220086, A203142, A256190, A256191, A256192, A203140, A203139, A203138, A203137.

Maple
```
evalf(GAMMA(1/Pi), 117);
```
Mathematica
```
RealDigits[Gamma[1/Pi], 10, 117][[1]]
```

default(realprecision, 117); gamma(1/Pi)

A257958 Decimal expansion of the Digamma function at 1/Pi, negated.

Original entry on oeis.org

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

Offset: 1

Iaroslav V. Blagouchine, May 14 2015

The reference gives an interesting series representation with rational coefficients for Psi(1/Pi) = -log(Pi) - Pi/2 - 1/2 - 1/8 - 1/72 + 1/64 +7/400 + 7/576 + 643/94080 + 103/30720 + ...

			-3.2902139601732240908430908455400190374021932820070161...

Iaroslav V. Blagouchine, Two series expansions for the logarithm of the gamma function involving Stirling numbers and containing only rational coefficients for certain arguments related to 1/Pi, Mathematics of Computation (AMS), 2015.

Cf. A257955, A257957, A257959, A155968, A256165, A256166, A256167, A255888, A256609, A255306, A256610, A256612, A256611, A256066, A256614, A256615, A256616.

Maple
```
evalf(Psi(1/Pi), 120);
```

RealDigits[PolyGamma[1/Pi], 10, 120][[1]]

PARI
```
default(realprecision, 120); psi(1/Pi)
```

Int_0^infinity x*dx/[(x^2+1)(exp(2x)-1)] = -Pi/2-Psi(1/Pi) = -1.5707...+ 3.2902.. = 1.71941... - R. J. Mathar, Aug 14 2023

A257959 Decimal expansion of the Digamma function at 1/2 + 1/Pi, negated.

Original entry on oeis.org

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

Offset: 0

Iaroslav V. Blagouchine, May 14 2015

The reference gives an interesting series representation with rational coefficients for Psi(1/2 + 1/Pi) = -log(Pi) + 1/4 + 1/16 - 5/576 - 13/512 - 569/25600 -539/36864 - 98671/12042240 - 16231/3932160 - ...

			-0.9236326759613377346000263347486747139894893215261027...

Iaroslav V. Blagouchine, Two series expansions for the logarithm of the gamma function involving Stirling numbers and containing only rational coefficients for certain arguments related to 1/Pi, Mathematics of Computation (AMS), 2015.

Cf. A257955, A257957, A257958, A155968, A256165, A256166, A256167, A255888, A256609, A255306, A256610, A256612, A256611, A256066, A256614, A256615, A256616.

Maple
```
evalf(Psi(1/2+1/Pi), 120);
```

RealDigits[PolyGamma[1/2+1/Pi], 10, 120][[1]]

default(realprecision, 120); psi(1/2+1/Pi)

A269545 Decimal expansion of Gamma(Pi).

Original entry on oeis.org

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

Offset: 1

Iaroslav V. Blagouchine, Feb 29 2016

			2.2880377953400324179595889090602339228896881533562224...

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A269546, A269547, A269557, A269558, A269559, A257955, A257957, A257958, A257959, A002161, A073005, A068466, A175380, A175379, A220086, A203142, A256190, A256191, A256192, A203140, A203139, A203138, A203137.

MATLAB
```
format long; gamma(pi)
```
Maple
```
evalf(GAMMA(Pi), 120);
```
Mathematica
```
RealDigits[Gamma[Pi], 10, 120][[1]]
```
PARI
```
default(realprecision, 120); gamma(Pi)
```

Equals Integral_{x >= 0} x^(Pi-1)/e^x dx (Euler integral of the second kind).

A269546 Decimal expansion of log(Gamma(Pi)).

Original entry on oeis.org

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

Offset: 0

Iaroslav V. Blagouchine, Feb 29 2016

Gamma(x) is the Gamma function (Euler's integral of the second kind).

			0.8276945923234371015295785584523599511535017341207373...

G. C. Greubel, Table of n, a(n) for n = 0..5000

Cf. A269545, A269547, A269557, A269558, A269559, A257955, A257957, A257958, A257959, A002161.

Cf. A073005, A068466, A175380, A175379, A220086, A203142, A256190, A256191, A256192, A203140, A203139, A203138, A203137.

MATLAB
```
format long; log(gamma(pi))
```
Maple
```
evalf(lnGAMMA(Pi), 120);
```
Mathematica
```
RealDigits[LogGamma[Pi], 10, 120][[1]]
```

default(realprecision, 120); lngamma(Pi)

A269547 Decimal expansion of Psi(Pi).

Original entry on oeis.org

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

Offset: 0

Iaroslav V. Blagouchine, Feb 29 2016

Psi(x) is the digamma function (logarithmic derivative of the Gamma function).

			0.9772133079420067332920694864061823436408346099943256...

G. C. Greubel, Table of n, a(n) for n = 0..5000

Cf. A269545, A269546, A269557, A269558, A269559, A257955, A257957, A257958, A257959, A002161.

Cf. A073005, A068466, A175380, A175379, A220086, A203142, A256190, A256191, A256192, A203140, A203139, A203138, A203137.

MATLAB
```
format long; psi(pi)
```
Maple
```
evalf(Psi(Pi), 120)
```
Mathematica
```
RealDigits[PolyGamma[Pi], 10, 120][[1]]
```
PARI
```
default(realprecision, 120); psi(Pi)
```

A269557 Decimal expansion of Gamma(log(2)).

Original entry on oeis.org

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

Offset: 1

Iaroslav V. Blagouchine, Feb 29 2016

Gamma(x) is the Gamma function (Euler's integral of the second kind).

			1.3090409112814812698245325213959295756125890319181890...

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A269558, A269559, A269545, A269546, A269547, A257955, A257957, A257958, A257959, A002161.

Cf. A073005, A068466, A175380, A175379, A220086, A203142, A256190, A256191, A256192, A203140, A203139, A203138, A203137.

MATLAB
```
format long; gamma(log(2))
```
Maple
```
evalf(GAMMA(ln(2)), 120);
```
Mathematica
```
RealDigits[Gamma[Log[2]], 10, 120][[1]]
```

default(realprecision, 120); gamma(log(2))

A269558 Decimal expansion of log(Gamma(log(2))).

Original entry on oeis.org

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

Offset: 0

Iaroslav V. Blagouchine, Feb 29 2016

Gamma(x) is the Gamma function (Euler's integral of the second kind).

			0.2692947402831312429499165832117128248889035102111661...

G. C. Greubel, Table of n, a(n) for n = 0..5000

Cf. A269557, A269559, A269545, A269546, A269547, A257955, A257957, A257958, A257959, A002161.

Cf. A073005, A068466, A175380, A175379, A220086, A203142, A256190, A256191, A256192, A203140, A203139, A203138, A203137.

MATLAB
```
format long; log(gamma(log(2)))
```
Maple
```
evalf(lnGAMMA(ln(2)), 120);
```

RealDigits[LogGamma[Log[2]], 10, 120][[1]]

default(realprecision, 120); lngamma(log(2))

A269559 Decimal expansion of Psi(log(2)), negated.

Original entry on oeis.org

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

Offset: 1

Iaroslav V. Blagouchine, Feb 29 2016

Psi(x) is the digamma function (logarithmic derivative of the Gamma function).

			-1.2395972796176185082441275516860842454332895226874208...

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A269557, A269558, A269545, A269546, A269547, A257955, A257957, A257958, A257959, A002161.

Cf. A073005, A068466, A175380, A175379, A220086, A203142, A256190, A256191, A256192, A203140, A203139, A203138, A203137.

MATLAB
```
format long; psi(log(2))
```
Maple
```
evalf(Psi(ln(2)), 120);
```

RealDigits[PolyGamma[Log[2]], 10, 120][[1]]

default(realprecision, 120); psi(log(2))

A257963 Decimal expansion of the Integral_{x=0..1} arctan(arctanh(x))/x.

Original entry on oeis.org

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

Offset: 1

Robert G. Wilson v, May 14 2015

"The arctangent of the hyperbolic arctangent is analytic in the whole disk |x| < 1, and therefore, can be expanded into the MacLaurin series", see the first reference.

			= 1.02576051093133045039854866096955279533487185621506939422338684401585192089...

Iaroslav V. Blagouchine, Two series expansions for the logarithm of the gamma function involving Stirling numbers and containing only rational coefficients for certain arguments related to 1/pi, Journal of Mathematical Analysis and Applications (Elsevier), 2016. arXiv version, arXiv:1408.3902 [math.NT], 2014-2016.
Iaroslav V. Blagouchine, Rediscovery of Malmsten's integrals, their evaluation by contour integration methods and some related results, The Ramanujan Journal, Volume 35, Issue 1, pp. 21-110, 2014, DOI: 10.1007/s11139-013-9528-5 PDF file

Cf. A257957, A257955, A257958, A257959, A155968, A049541, A000796.

evalf(Pi*(log(GAMMA(1/Pi)) - log(GAMMA(1/2 + 1/Pi)) - log(Pi)/2),120); # Vaclav Kotesovec, May 17 2015

nn = 111; RealDigits[ NIntegrate[ ArcTan[ ArcTanh[ x]]/x, {x, 0, 1}, AccuracyGoal -> nn, WorkingPrecision -> nn], 10, nn][[1]] (* or *)
RealDigits[Pi (Log[Gamma[1/Pi]] - Log[Gamma[1/2 + 1/Pi]] - Log[Pi]/2), 10, 111][[1]] (* Robert G. Wilson v, May 14 2015 *)

The integral is equivalent to Pi*(log(Gamma(1/Pi)) - log(Gamma(1/2 + 1/Pi)) - log(Pi)/2), see page 82 of the second reference.

cp's OEIS Frontend

A257955 Decimal expansion of Gamma(1/Pi).

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A257958 Decimal expansion of the Digamma function at 1/Pi, negated.

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A257959 Decimal expansion of the Digamma function at 1/2 + 1/Pi, negated.

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Maple

Mathematica

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A269545 Decimal expansion of Gamma(Pi).

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MATLAB

Maple

Mathematica

PARI

Formula

A269546 Decimal expansion of log(Gamma(Pi)).

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MATLAB

Maple

Mathematica

PARI

A269547 Decimal expansion of Psi(Pi).

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Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

MATLAB

Maple

Mathematica

PARI

A269557 Decimal expansion of Gamma(log(2)).

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Author