A175474 -id:A175474 - cp's OEIS frontend

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

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

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].

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).

A344965 Decimal expansion of the sum of the reciprocals of the cubes of the zeros of the digamma function (negated).

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Jun 03 2021

The sum is Sum_{k>=0} 1/x_k^3, 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.

			-7.84898826280450630489883732716055067110164127911638...

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. A344964, A344966, A344967, A344968.
Cf. A001620, A002117, A102753.
Cf. A030169, A175472, A175473, A175474, A256681, A256682, A256683, A256684, A256685, A256686, A256687.

RealDigits[EulerGamma*Pi^2/2 + 4*Zeta[3]  + EulerGamma^3, 10, 100][[1]]

Equals -gamma*Pi^2/2 - 4*zeta(3) - gamma^3, where gamma is Euler's constant (A001620).

A344966 Decimal expansion of the sum of the reciprocals of the fourth powers of the zeros of the digamma function.

Original entry on oeis.org

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

Offset: 2

Amiram Eldar, Jun 03 2021

The sum is Sum_{k>=0} 1/x_k^4, 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.

			15.90184703322349156972084557358425176519256672643402...

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. A344964, A344965, A344967, A344968.
Cf. A000796, A001620, A002117.
Cf. A030169, A175472, A175473, A175474, A256681, A256682, A256683, A256684, A256685, A256686, A256687.

RealDigits[Pi^4/9 + 2*EulerGamma^2*Pi^2/3 + 4*EulerGamma*Zeta[3] + EulerGamma^4, 10, 100][[1]]

Equals Pi^4/9 + 2*gamma^2*Pi^2/3 + 4*gamma*zeta(3) + gamma^4, where gamma is Euler's constant (A001620).

A344967 Decimal expansion of Sum_{k>=0} 1/(x_k^2 - 1), where x_k is the k-th zero of the digamma function.

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, Jun 03 2021

The zeros of the digamma function, i.e., the roots of psi(x) = 0 are x_0 = 1.461632... (A030169), the only positive root, x_1 = -0.504083... (A175472), etc.

			0.71349472210996816769933594441333563665531893958512...

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. A344964, A344965, A344966, A344968.
Cf. A000796, A001620, A072691.
Cf. A030169, A175472, A175473, A175474, A256681, A256682, A256683, A256684, A256685, A256686, A256687.

RealDigits[Pi^2/(12*EulerGamma) + EulerGamma/2 - 1, 10, 100][[1]]

Equals Pi^2/(12*gamma) + gamma/2 - 1, where gamma is Euler's constant (A001620).

A344968 Decimal expansion of Sum_{k>=0} 1/(x_k^2 - x_k), where x_k is the k-th zero of the digamma function.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Jun 03 2021

The zeros of the digamma function, i.e., the roots of psi(x) = 0 are x_0 = 1.461632... (A030169), the only positive root, x_1 = -0.504083... (A175472), etc.

			3.42698944421993633539867188882667127331063787917025...

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. A344964, A344965, A344966, A344967.
Cf. A001620, A013661.
Cf. A030169, A175472, A175473, A175474, A256681, A256682, A256683, A256684, A256685, A256686, A256687.

RealDigits[Pi^2/(6*EulerGamma) + EulerGamma, 10, 100][[1]]

Equals Pi^2/(6*gamma) + gamma, where gamma is Euler's constant (A001620).

A257433 Decimal expansion of the smallest negative real root of the equation Gamma(x) = -1 (negated).

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Apr 23 2015

			-2.45702473822080062303945414765117954323659790903378442...

Philippe Flajolet, Stefan Gerhold and Bruno Salvy, Lindelöf Representations and (Non-)Holonomic Sequences, Electronic Journal of Combinatorics, vol 17(1):R3, 2010, p. 10.
Eric Weisstein's MathWorld, Gamma Function

Cf. A175474, A257434.

x1 = x /. FindRoot[Gamma[x] == -1, {x, -5/2}, WorkingPrecision -> 104]; RealDigits[x1] // First

-3 < A257434 = -2.747682... < A175474 = -2.61072... < A257433 = -2.457024... < -2.

A257434 Decimal expansion of the second smallest negative real root of the equation Gamma(x) = -1 (negated).

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Apr 23 2015

			-2.747682646727412601391488482691499695861639395132355512...

Philippe Flajolet, Stefan Gerhold and Bruno Salvy, Lindelöf Representations and (Non-)Holonomic Sequences, Electronic Journal of Combinatorics, vol 17(1):R3, 2010, p. 10.
Eric Weisstein's MathWorld, Gamma Function

Cf. A175474, A257433.

x2 = x /. FindRoot[Gamma[x] == -1, {x, -8/3}, WorkingPrecision -> 101]; RealDigits[x2] // First

-3 < A257434 = -2.747682... < A175474 = -2.61072... < A257433 = -2.457024... < -2.

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

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

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A344965 Decimal expansion of the sum of the reciprocals of the cubes of the zeros of the digamma function (negated).

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A344966 Decimal expansion of the sum of the reciprocals of the fourth powers of the zeros of the digamma function.

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A344967 Decimal expansion of Sum_{k>=0} 1/(x_k^2 - 1), where x_k is the k-th zero of the digamma function.

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A344968 Decimal expansion of Sum_{k>=0} 1/(x_k^2 - x_k), where x_k is the k-th zero of the digamma function.