A175472 -id:A175472 - cp's OEIS frontend

A175473 Decimal expansion of the absolute value of the abscissa of the local minimum of the Gamma function in the interval [ -2,-1].

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

Offset: 1

R. J. Mathar, May 25 2010

Also the location of the zero of the digamma function in the same interval.

			Gamma(-1.5734984731623904587782860437..) = 2.3024072583396801358235820396..

Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 44, page 427.

Eric Weisstein's World of Mathematics, Gamma Function.
Wikipedia, Particular values of the Gamma function.

Cf. A030169, A030171, A175472, A175474.

x /. FindRoot[ PolyGamma[0, x] == 0, {x, -3/2}, WorkingPrecision -> 110] // Abs // RealDigits // First // Take[#, 105]& (* Jean-François Alcover, Jan 21 2013 *)

solve(x=1.5,1.6,psi(-x)) \\ Charles R Greathouse IV, Jul 19 2013

A256687 Decimal expansion of the [negated] abscissa of the Gamma function local minimum in the interval [-10,-9].

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Apr 08 2015

			Gamma(-9.7026725400018637360844267648942153185775505998219124864...)
= 0.00000215741610452285054050313702063056774903546226316...

Eric Weisstein's MathWorld, Gamma Function
Wikipedia, Particular values of the Gamma Function

Cf. A030169, A030171, A175472, A175473, A256681-A256686.

digits = 103; x0 = x /. FindRoot[PolyGamma[0, x] == 0, {x, -19/2}, WorkingPrecision -> digits + 5]; RealDigits[x0, 10, digits] // First

Solution to PolyGamma(x) = 0 in the interval [-10,-9].

A256681 Decimal expansion of the [negated] abscissa of the Gamma function local minimum in the interval [-4,-3].

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Apr 08 2015

			Gamma(-3.6352933664369010978391815669460177139484238611935307...)
= 0.245127539834366250438230088857478287588513028833668283...

Eric Weisstein's MathWorld, Gamma Function
Wikipedia, Particular values of the Gamma Function

Cf. A030169, A030171, A175472, A175473, A175474, A256682-A256687.

digits = 104; x0 = x /. FindRoot[PolyGamma[x] == 0, {x, -7/2}, WorkingPrecision -> digits + 5]; RealDigits[x0, 10, digits] // First

Solution to PolyGamma(x) = 0 in the interval [-4,-3].

A175474 Decimal expansion of the absolute value of the abscissa of the local maximum of the Gamma function in the interval [ -3,-2].

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, May 25 2010

Also the location of the zero of the digamma function in the same interval.

			Gamma(-2.6107208684441446500015377157..) = -0.8881363584012419200955280294..

Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 44, page 427.

Eric Weisstein's World of Mathematics, Gamma Function.
Wikipedia, Particular values of the Gamma function.

Cf. A030169, A030171, A175472, A175473.

x /. FindRoot[ PolyGamma[0, x] == 0, {x, -5/2}, WorkingPrecision -> 105] // Abs // RealDigits // First (* Jean-François Alcover, Jan 21 2013 *)

solve(x=2.6,2.7,psi(-x)) \\ Charles R Greathouse IV, Jul 19 2013

A256682 Decimal expansion of the [negated] abscissa of the Gamma function local maximum in the interval [-5,-4].

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Apr 08 2015

			Gamma(-4.653237761743142441714598151148207363719069416133868555...)
= -0.05277963958731940076048357076290307426383130501056893...

Eric Weisstein's MathWorld, Gamma Function
Wikipedia, Particular values of the Gamma Function

Cf. A030169, A030171, A175472, A175473, A256681, A256683-A256687.

digits = 105; x0 = x /. FindRoot[PolyGamma[0, x] == 0, {x, -9/2}, WorkingPrecision -> digits + 5]; RealDigits[x0, 10, digits] // First

Solution to PolyGamma(x) = 0 in the interval [-5,-4]

A256683 Decimal expansion of the [negated] abscissa of the Gamma function local minimum in the interval [-6,-5].

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Apr 08 2015

			Gamma(-5.66716244155688553584947417451815542471179578769484889367...)
= 0.00932459448261485052171192379918266310927330588790814482...

Eric Weisstein's MathWorld, Gamma Function
Wikipedia, Particular values of the Gamma Function

Cf. A030169, A030171, A175472, A175473, A256681, A256682, A256684-A256687.

digits = 104; x0 = x /. FindRoot[PolyGamma[0, x] == 0, {x, -11/2}, WorkingPrecision -> digits + 5]; RealDigits[x0, 10, digits] // First

Solution to PolyGamma(x) = 0 in the interval [-6,-5]

A256684 Decimal expansion of the [negated] abscissa of the Gamma function local maximum in the interval [-7,-6].

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Apr 08 2015

			Gamma(-6.6784182130734267428298558886022000992046860101507601433975...)
= -0.0013973966089497673013074886687985785170487809897563228...

Eric Weisstein's MathWorld, Gamma Function
Wikipedia, Particular values of the Gamma Function

Cf. A030169, A030171, A175472, A175473, A256681-A256683, A256685-A256687.

digits = 104; x0 = x /. FindRoot[PolyGamma[0, x] == 0, {x, -13/2}, WorkingPrecision -> digits + 5]; RealDigits[x0, 10, digits] // First

Solution to PolyGamma(x) = 0 in the interval [-7,-6]

A256685 Decimal expansion of the [negated] abscissa of the Gamma function local minimum in the interval [-8,-7].

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Apr 08 2015

			Gamma(-7.6877883250316260374400988918437049536838217978826433594...)
= 0.0001818784449094041881014174426244626530404358160668...

Eric Weisstein's MathWorld, Gamma Function
Wikipedia, Particular values of the Gamma Function

Cf. A030169, A030171, A175472, A175473, A256681-A256684, A256686, A256687.

digits = 106; x0 = x /. FindRoot[PolyGamma[0, x] == 0, {x, -15/2}, WorkingPrecision -> digits + 5]; RealDigits[x0, 10, digits] // First

Solution to PolyGamma(x) = 0 in the interval [-8,-7].

A256686 Decimal expansion of the [negated] abscissa of the Gamma function local maximum in the interval [-9,-8].

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Apr 08 2015

			Gamma(-8.695764163816401266488776160804645820272438084966728783...)
= -0.00002092529044652666875369728468060738117860083247673665...

Eric Weisstein's MathWorld, Gamma Function
Wikipedia, Particular values of the Gamma Function

Cf. A030169, A030171, A175472, A175473, A256681-A256685, A256687.

digits = 104; x0 = x /. FindRoot[PolyGamma[0, x] == 0, {x, -17/2}, WorkingPrecision -> digits + 5]; RealDigits[x0, 10, digits] // First

Solution to PolyGamma(x) = 0 in the interval [-9,-8].

A344964 Decimal expansion of the sum of the reciprocals of the squares of the zeros of the digamma function.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Jun 03 2021

The sum is Sum_{k>=0} 1/x_k^2, where x_k is the k-th zero of the digamma function, i.e., root of psi(x) = 0: x_0 = 1.461632... (A030169) is the only positive root, x_1 = -0.504083... (A175472), etc.

			5.26798012435239798373562163629331979431626684387002...

István Mező and Michael E. Hoffman, Zeros of the digamma function and its Barnes G-function analogue, Integral Transforms and Special Functions, Vol. 28, No. 11 (2017), pp. 846-858.
Wikipedia, Digamma function.

Cf. A344965, A344966, A344967, A344968.
Cf. A001620, A102753.
Cf. A030169, A175472, A175473, A175474, A256681, A256682, A256683, A256684, A256685, A256686, A256687.

RealDigits[Pi^2/2 + EulerGamma^2, 10, 100][[1]]

Equals Pi^2/2 + gamma^2 = A102753 + A155969, where gamma is Euler's constant (A001620).

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A175473 Decimal expansion of the absolute value of the abscissa of the local minimum of the Gamma function in the interval [ -2,-1].

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A256687 Decimal expansion of the [negated] abscissa of the Gamma function local minimum in the interval [-10,-9].

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A256681 Decimal expansion of the [negated] abscissa of the Gamma function local minimum in the interval [-4,-3].

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A175474 Decimal expansion of the absolute value of the abscissa of the local maximum of the Gamma function in the interval [ -3,-2].

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A256682 Decimal expansion of the [negated] abscissa of the Gamma function local maximum in the interval [-5,-4].

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A256683 Decimal expansion of the [negated] abscissa of the Gamma function local minimum in the interval [-6,-5].

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Formula

A256684 Decimal expansion of the [negated] abscissa of the Gamma function local maximum in the interval [-7,-6].

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Formula

A256685 Decimal expansion of the [negated] abscissa of the Gamma function local minimum in the interval [-8,-7].

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