A030171 -id:A030171 - cp's OEIS frontend

A030169 Decimal expansion of the real positive number x such that Gamma(x) is a minimum.

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

Offset: 1

Eric W. Weisstein

The gamma function has a minimum at this point. 1.461632144968362341262659542325721328468196204006446351295988409... is the solution of the equation: Psi(x)*Gamma(x)=0. The point y of that function is 0.8856031944108887002788159005825887332079515336699034488712001659... - Simon Plouffe

Positive root of psi(x) = 0, where psi is the digamma function. - Charles R Greathouse IV, May 30 2012

			x = 1.461632144968362..., Gamma(x) = 0.885603194410888...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.5.4, p. 34.
Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 44, page 427.

G. C. Greubel, Table of n, a(n) for n = 1..5000
Simon Plouffe, editor, Miscellaneous Mathematical Constants Project Gutenberg, 1996 [see "Minimal y of GAMMA(x)" paragraph].
Eric Weisstein's World of Mathematics, Gamma Function.

Cf. A030171 for value of Gamma(x).

Digits:= 120; fsolve(Psi(x)=0, x); # Iaroslav V. Blagouchine, Feb 16 2016

First@ RealDigits[ FindMinimum[ Gamma[x], {x, 1.4}, WorkingPrecision -> 2^7][[2, 1, 2]]] (* Robert G. Wilson v, Aug 03 2010 *)
RealDigits[x /. FindRoot[PolyGamma[x], {x, 1}, WorkingPrecision -> 200]][[1]] (* Charles R Greathouse IV, May 30 2012 *)

solve(x=1,2,psi(x)) \\ Charles R Greathouse IV, May 30 2012

Additional comments from Francisco Salinas (franciscodesalinas(AT)hotmail.com), Dec 29 2001

Broken URL to Project Gutenberg replaced by Georg Fischer, Jan 03 2009

A175472 Decimal expansion of the absolute value of the abscissa of the local maximum of the Gamma function in the interval [ -1,0].

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, May 25 2010

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

			Gamma(-0.5040830082644554092582693045...) = -3.5446436111550050891219639933..

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, A175473, A175474.

x /. FindRoot[ PolyGamma[0, x] == 0, {x, -1/2}, WorkingPrecision -> 105] // Abs // RealDigits // First  (* Jean-François Alcover, Jan 21 2013 *)

solve(x=.5,.6,psi(-x)) \\ Charles R Greathouse IV, Jul 19 2013

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

Original entry on oeis.org

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].

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A030169 Decimal expansion of the real positive number x such that Gamma(x) is a minimum.

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A175472 Decimal expansion of the absolute value of the abscissa of the local maximum of the Gamma function in the interval [ -1,0].

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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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