A020759 -id:A020759 - cp's OEIS frontend

A250129 Decimal expansion of the negated value of the digamma function at 1/8.

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

Offset: 1

Jean-François Alcover, Jan 15 2015

			Psi(1/8) = -8.388492663295854867802742923086343000051446...

Eric Weisstein's MathWorld, Gauss's Digamma Theorem.
Wikipedia, Digamma function: special values.
Index entries for sequences related to the digamma function

Cf. A020759, A020777, A047787, A200064, A222457, A222458.

RealDigits[PolyGamma[1/8], 10, 105] // First

-psi(1/8) \\ Charles R Greathouse IV, Jan 15 2015

Psi(1/8) = -gamma - (1/2)*(1+sqrt(2))*Pi - sqrt(2)*arccoth(sqrt(2)) - 4*log(2).

A306716 Decimal expansion of the negated value of the digamma function at 1/10.

Original entry on oeis.org

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

Offset: 2

Vaclav Kotesovec, Aug 22 2019

			Equals 10.4237549404110767951682162190100254042916425624441889203263920841...

Eric Weisstein's World of Mathematics, Digamma Function
Wikipedia, Digamma function
Index entries for sequences related to the digamma function

Cf. A020759, A047787, A020777, A200135, A222457, A250129.

Maple
```
evalf(-Psi(1/10), 102);
```

RealDigits[-PolyGamma[1/10], 10, 105][[1]]

PARI
```
-psi(1/10)
```

Psi(1/10) = -gamma - Pi*5^(1/4)*(sqrt(2 + sqrt(5))/2) - 2*log(2) - 5*log(5)/4 - 3*sqrt(5)*log((1 + sqrt(5))/2)/2, where gamma is the Euler-Mascheroni constant A001620.

Equals gamma - H(-9/10), H(z) the harmonic number. - Peter Luschny, Aug 22 2019

A200136 Decimal expansion of the negated value of the digamma function at 2/5.

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Nov 13 2011

			Psi(2/5) = -2.5613845445851161457306754820475...

Index entries for sequences related to the digamma function

Cf. A020759, A020777, A047787, A200064, A200135-A200138.

-gamma-Pi*sqrt(1-2/sqrt(5))/2-5*log(5)/4+sqrt(5)*log((3+sqrt(5))/2)/4 ; evalf(%) ;

RealDigits[ PolyGamma[2/5], 10, 84] // First (* Jean-François Alcover, Feb 21 2013 *)

-psi(2/5) \\ Charles R Greathouse IV, Jul 19 2013

Psi(2/5) = -gamma -Pi*sqrt(1-2/sqrt 5)/2 -5*log(5)/4 +sqrt(5)*log((3+sqrt 5)/2)/4.

A200137 Decimal expansion of the negated digamma function at 3/5.

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Nov 13 2011

			Psi(3/5) = -1.540619213893190414760663948806231941510...

Index entries for sequences related to the digamma function

Cf. A020759, A047787, A020777, A200134-A200138

-gamma+Pi*sqrt(1-2/sqrt(5))/2-5*log(5)/4+sqrt(5)/4*log(3/2+sqrt(5)/2) ; evalf(%) ;

RealDigits[-PolyGamma[3/5], 10, 105] // First (* Jean-François Alcover, Feb 11 2013 *)

-psi(3/5) \\ Charles R Greathouse IV, Jul 19 2013

Psi(3/5) = -gamma +Pi*sqrt( 1-2/sqrt 5)/2 -5*log(5)/4 +sqrt(5)*log((3+sqrt 5)/2)/4.

More terms from Jean-François Alcover, Feb 11 2013

A131265 Decimal expansion of the negative of the first derivative of the Gamma Function at 1/2.

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Sep 28 2007

			3.4802309069132620269385951981443497500324293345037602151543...

G. C. Greubel, Table of n, a(n) for n = 1..10000
T. Amdeberhan, L. Medina and V. H. Moll, The integrals in Gradshteyn and Ryzhik. Part 5: Some trigonometric integrals, arXiv:0705.2379 [math.CA], 2007, example 3.1.

Cf. A020759, A002161.

RealDigits[ Sqrt[Pi]*PolyGamma[0, 1/2], 10, 59] // First (* Jean-François Alcover, Feb 20 2013 *)

PARI
```
print(sqrt(Pi)*(Euler+2*log(2)));
```

Equals A020759 * A002161.

A248176 Decimal expansion of psi(-1/2).

Original entry on oeis.org

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

Offset: 0

Stanislav Sykora, Oct 03 2014

Psi denotes the digamma function.

			0.0364899739785765205590236670012444328068403953395658929528727461...

Stanislav Sykora, Table of n, a(n) for n = 0..2000
Michael I. Shamos, A catalog of the real numbers, (2007). See p. 69.
Wikipedia, Digamma function.

Cf. A001620, A002162, A020759.

SetDefaultRealField(RealField(100)); R:= RealField(); 2 - (2*Log(2) + EulerGamma(R)); // G. C. Greubel, Aug 30 2018

RealDigits[2 - (EulerGamma + 2Log[2]), 10, 100][[1]] (* Alonso del Arte, Oct 03 2014 *)

PARI
```
psi(-1/2)
```

Equals 2 + psi(1/2) = 2 - 2*log(2) - EulerGamma = 2 - A020759, (since psi(1 + x) = psi(x) + 1/x).
Equals PolyGamma(3/2). - Peter Luschny, Apr 14 2020
Equals Sum_{k>=1} (zeta(2*k+1)-1)/((k+1)*(2*k+1)). - Amiram Eldar, May 24 2021
Equals 4*Pi*Integral_{x>=0} (log(1 + i*x) / (exp(-Pi*x) + exp(Pi*x))^2). - Peter Luschny, Aug 04 2021
Equals lim_{n->oo} (log(n) - Sum_{k=1..n} 1/(k+1/2)). - Amiram Eldar, Mar 04 2023

A247017 Decimal expansion of integral_{0..infinity} exp(-x^2)*log(x) dx.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Sep 09 2014

			-0.87005772672831550673464879953608743750810733362594...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 1.5 Euler-Mascheroni constant, p. 31, 1.5.2 Integrals.

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

Cf. A020759.

RealDigits[-(1/4)*Sqrt[Pi]*( EulerGamma + 2*Log[2]), 10, 102] // First
RealDigits[NIntegrate[Exp[-x^2]Log[x],{x,0,\[Infinity]},WorkingPrecision->120],10,120][[1]] (* Harvey P. Dale, Mar 29 2024 *)

-(1/4)*sqrt(Pi)*(Euler + 2*log(2)) \\ Michel Marcus, Sep 09 2014

Equals -(1/4)*sqrt(Pi)*(EulerGamma + 2*log(2)).

cp's OEIS Frontend

A250129 Decimal expansion of the negated value of the digamma function at 1/8.

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A306716 Decimal expansion of the negated value of the digamma function at 1/10.

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Maple

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A200136 Decimal expansion of the negated value of the digamma function at 2/5.

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A200137 Decimal expansion of the negated digamma function at 3/5.

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A131265 Decimal expansion of the negative of the first derivative of the Gamma Function at 1/2.

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A248176 Decimal expansion of psi(-1/2).

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Magma

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A247017 Decimal expansion of integral_{0..infinity} exp(-x^2)*log(x) dx.

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