A030171 -id:A030171 - cp's OEIS frontend

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

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].

A172081 Decimal expansion of the local minimum F(x) of the Fibonacci Function at x = A171909.

Original entry on oeis.org

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

Offset: 0

Gerd Lamprecht (gerdlamprecht(AT)googlemail.com), Jan 25 2010

Define the Fibonacci Function F(x) and its derivative as in A171909.
The derivative is dF/dx = (phi^x * log(phi) - cos(Pi*x)*log(phi)/phi^x + Pi*sin(Pi*x)/phi^x)/sqrt(5).
Set dF(x)/dx = 0 to find the local minimum.

			F(1.67668837258...) = 0.896946387424606172912600371068765...

Gerd Lamprecht, Iterationsrechner
Gerd Lamprecht, Zahlenfolgen (sequences)
E. Weisstein, Fibonacci Number, Mathworld.

Cf. A171909, A001622, A030171, A172197.

p := (1+sqrt(5))/2 ; F := (p^x - cos(Pi*x)/p^x )/sqrt(5);
Fpr := diff(F,x) ; Fpr2 := diff(Fpr,x) ;
Digits := 80 ; x0 := 1.67 ;
for n from 1 to 10 do
x0 := evalf(x0-subs(x=x0,Fpr)/subs(x=x0,Fpr2)) ;
print( evalf(subs(x=x0,F))) ;
end do : # R. J. Mathar, Feb 02 2010

digits = 104; F[x_] := (GoldenRatio^x - Cos[Pi*x]/GoldenRatio^x)/Sqrt[5]; x0 = x /. FindRoot[F'[x], {x, 2}, WorkingPrecision -> digits+1]; RealDigits[F[x0], 10, digits][[1]] (* Jean-François Alcover, Jan 28 2014 *)

Edited, offset and leading zero normalized by R. J. Mathar, Feb 02 2010

A030172 Let c be the point at which Gamma(x), x>0, is minimized; sequence gives continued fraction for Gamma(c).

Original entry on oeis.org

0, 1, 7, 1, 2, 1, 6, 1, 1, 1, 1, 6, 1, 4, 1, 1, 10, 3, 1, 10, 3, 4, 2, 1, 1, 2, 1, 6, 1, 1, 11, 1, 1, 2, 1, 2, 1, 1, 1, 3, 35, 3, 5, 2025, 1, 1, 2, 6, 3, 4, 16, 1, 1, 2, 2, 3, 1, 2, 1, 1, 3, 1, 8, 4, 13, 1, 8, 10, 3, 14, 1, 2

Offset: 0

Eric W. Weisstein

G. C. Greubel, Table of n, a(n) for n = 0..5000
Eric Weisstein's World of Mathematics, Gamma Function.

Cf. A030171.

ContinuedFraction[ Gamma[x] /. FindRoot[Gamma'[x] == 0, {x, 1}, WorkingPrecision -> 100], 72] (* Jean-François Alcover, Oct 29 2012 *)

contfrac(gamma(solve(x=1, 2, psi(x)))) \\ G. C. Greubel, Dec 28 2019

A334850 Decimal expansion of the maximal curvature of y = Gamma(x), for x>0.

Original entry on oeis.org

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

Offset: 0

Clark Kimberling, Jun 21 2020

Each branch of y = Gamma(x) has a point of maximal curvature (MC), at which the osculating circle has minimal radius (R).  The branch in Quadrant I has MC at (x, Gamma(x)), where x = 0.9757... and R = 0.77642... Details for 4 branches (shown by 1st Mathematica program):
For the branch -3 < x < -2:
MC at x=-2.6209004043183225054792567933147...
R = 0.1025411250345462193237149178953328755...
For the branch -2 < x < -1:
MC at x=-1.57452893040224357315540638154037...
R = 0.043652981140784797188517226949156690045...
For the branch -1 < x < 0:
MC at x=-0.50414409519766396393374935693160...
R = 0.0315571147317663900987190484592293666...
For the branch 0 < x:
MC at x=0.97574729311153379112462151102264...
R = 0.7764237137148324259856982062600903642...

Eric Weisstein's World of Mathematics, Gamma Function

Cf. A030171.

(* FIRST program *)
g[x_] := Gamma[x]; p[k_, x_] := PolyGamma[k, x]
solns = Map[#[[1]][[1]] &, GatherBy[Map[{#[[2]], Rationalize[#[[2]], 10^-30]} &,
    Select[Table[{nn, #, Accuracy[#]} &[x /. FindRoot[
         0 == (2 g[x]^2 p[0, x]^5 + 3 p[0, x] p[1, x] (-1 + g[x]^2 p[1, x]) +
            p[0, x]^3 (-1 + 3 g[x]^2 p[1, x]) - (1 +  g[x]^2 p[0, x]^2) p[2, x]), {x, nn},
         WorkingPrecision -> 100]], {nn, -2.8, 2.5, .101}], #[[3]] > 40 &]], #[[2]] &]]
{coords, rads} = Chop[Transpose[Map[{{(-p[0, x] + x p[0, x]^2 - g[x]^2 p[0, x]^3 +
           x p[1, x])/(p[0, x]^2 + p[1, x]), (1 + g[x]^2 (2 p[0, x]^2 + p[1, x]))/(g[x] (p[0, x]^2 + p[1, x]))}, Sqrt[(1 + g[x]^2 p[0, x]^2)^3/(g[x]^2 (p[0, x]^2 + p[1, x])^2)]} /. x -> # &, solns]]]
Show[Plot[g[x], {x, -3, 2}], Map[{Graphics[Circle[coords[[#]], rads[[#]]]],
    Graphics[Point[coords[[#]]]]} &, Range[Length[rads]]],
AspectRatio -> Automatic, PlotRange -> {-4, 4}, ImageSize -> 600]
(* Peter J. C. Moses, Jun 17 2020 *)
(* Graphics output:: 4 osculating circles;
Numerical output: first 4 numbers are x-coordinates of touchpoints of osculating circles with graph of gamma function; next 8 numbers are in pairs: (x,y) for the centers of the four circles; last 4 numbers are radii of the 4 circles *)
(* SECOND program: animation of osculating circle *)
Animate[Show[cent = {(-PolyGamma[0, x] + x PolyGamma[0, x]^2 -
       Gamma[x]^2 PolyGamma[0, x]^3 + x PolyGamma[1, x])/(PolyGamma[0, x]^2 + PolyGamma[1, x]), (1 + Gamma[x]^2 (2 PolyGamma[0, x]^2 + PolyGamma[1, x]))/(Gamma[x] (PolyGamma[0, x]^2 + PolyGamma[1, x]))}; rad = Sqrt[(1 +
        Gamma[x]^2 PolyGamma[0, x]^2)^3/(Gamma[x]^2 (PolyGamma[0, x]^2 + PolyGamma[1, x])^2)]; Plot[Gamma[x], {x, 0, 4}],
  Graphics[{PointSize[Large], Point[{x, Gamma[x]}]}],
  Graphics[{PointSize[Large], Point[cent]}],
  Graphics[Circle[cent, rad]], AxesOrigin -> {0, 0},
  PlotRange -> {{0, 4}, {0, 6}}, ImageSize -> 400,
  AspectRatio -> Automatic], {x, 0.4, 3.5}, AnimationRunning -> True]
(* Peter J. C. Moses, Jun 18 2020 *)

A362810 Define G(n, k) to be the n-th derivative of Gamma(x) at k. a(n)=floor(min(G(2n, x))), where min(f) is the local minimum of f in [0,oo).

Original entry on oeis.org

0, 0, 1, 6, 30, 173, 1138, 8386, 67951, 596745, 5618916, 56249658, 594648335, 6602123630, 76631632344, 926329705808, 11623455427764, 150970962492188, 2024773236657401, 27980260971851306, 397645587914766071, 5801999753304428181, 86784442260270596447, 1328924296505789704631, 20807559990139289975657, 332753116291423840918784

Offset: 0

Jodi Spitz, May 04 2023

nonn

Appears to grow factorially (superexponentially).

Conjecture: limit_{n->oo} log(a(n)) / log(n!) < 1. - Vaclav Kotesovec, Nov 17 2023

			a(5) = 173 since the local minimum in [0,oo) of the 10th derivative of Gamma(x) is 173.195...

Cf. A030171.

Join[{0}, Floor[Table[d = Simplify[D[Gamma[x], {x, 2 n}]]; d /. FindRoot[D[d, x] == 0, {x, n/2}, WorkingPrecision -> 50], {n, 1, 10}]]] (* Vaclav Kotesovec, Nov 17 2023 *)

a(7)-a(25) from Vaclav Kotesovec, Nov 18 2023

A365797 Decimal expansion of smallest positive number x such that Gamma(x) = 2.

Original entry on oeis.org

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

Offset: 0

David Ulgenes, Sep 19 2023

Second branch (i.e., the first after the principal branch) of the inverse gamma function Gamma(y) = x at x=2. See for instance Uchiyama.
Since 1 - x = 0.55712260351... (approximately equal to A249649), we can obtain the interesting approximation Gamma(zeta(2) - zeta(3)) ≈ 2.000001... - David Ulgenes, Feb 19 2024
x is the least positive real number where 1+Gamma(1+Gamma(1+Gamma...(x)...)) converges; it converges to 3. - Colin Linzer, Nov 25 2024

			0.4428773964847274374520325165206056717103645380663664...

K. Amenyo Folitse, David J. Jeffrey, and Robert M. Corless, Properties and Computation of the Functional Inverse of Gamma, 2017 19th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing (SYNASC). International Symposium on Symbolic and Numeric Algorithms for Scientific Computing (SYNASC). p. 65.
Mitsuru Uchiyama, The principal inverse of the gamma function, Proc. Amer. Math. Soc. 140 (2012), 1343-1348.

Cf. A030169, A030171.

Digits:= 140:
with(RootFinding):
NextZero(x -> (x - 1)! - 2, 0);

FindRoot[-2 + (-1 + x)! == 0, {x, 0, 1}, WorkingPrecision -> 15]

solve(x=0.1, 1, gamma(x)-2) \\ Michel Marcus, Sep 19 2023

Equals ((((1/2)!/2)!/2)!/2)!/2...
Proof: Since y = y! / x we substitute the expression into itself to obtain an iterative scheme for the inverse gamma function.
Equals (1/(2*Pi))*Integral_{x=-oo..oo} log((2-Gamma(i*x))/(2-Gamma(1+i*x))) dx. Proof: Follows from writing the inverse gamma function using the Lagrange inversion theorem together with Cauchy's formula for differentiation. - David Ulgenes, Feb 11 2024

cp's OEIS Frontend

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

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A172081 Decimal expansion of the local minimum F(x) of the Fibonacci Function at x = A171909.

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Maple

Mathematica

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A030172 Let c be the point at which Gamma(x), x>0, is minimized; sequence gives continued fraction for Gamma(c).

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Keywords

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Programs

Mathematica

PARI

A334850 Decimal expansion of the maximal curvature of y = Gamma(x), for x>0.

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Mathematica

A362810 Define G(n, k) to be the n-th derivative of Gamma(x) at k. a(n)=floor(min(G(2n, x))), where min(f) is the local minimum of f in [0,oo).

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Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Extensions

A365797 Decimal expansion of smallest positive number x such that Gamma(x) = 2.

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Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula