Feigenbaum Constant Approximations

Author

Eric W. Weisstein
April 20, 2023

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Constants/FeigenbaumConstantApproximations.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/FeigenbaumConstantApproximations.html.

©2023 Wolfram Research, Inc. except for portions noted otherwise

Sources

David Broadhurst  22-Mar-1999
http://pi.lacim.uqam.ca/piDATA/feigenbaum.txt
and David Broadhurst, pers. comm., Jan. 31, 2005

Values

δ=4.6692016091029906718532038204662016172581855774757686327456513430041343302113147371386897440239480138171659848551898151344086271420279325223124429888908908599449354632367134115324817142199474556443658237932020095610583305754586176522220703854106467494942849814533917262005687556659523398756038256372256480040951071283890611844702775854285419801113440175002428585382498335715522052236087250291678860362674527213399057131606875345083433934446103706309452019115876972432273589838903794946257251289097948986768334611626889116563123474460575179539122045562472807095202198199094558581946136877445617396074115614074243754435499204869180982648652368438702799649017397793425134723808737136211601860128186102056381818354097598477964173900328936171432159878240789776614391395764037760537119096932066998361984288981837003229412030210655743295550388845849737034727532121925706958414074661841981961006129640161487712944415901405467941800198133253378592493365883070459999938375411726563553016862529032210862320550634510679399023341675;
α=-2.5029078750958928222839028732182157863812713767271499773361920567792354631795902067032996497464338341295952318699958547239421823777854451792728633149933725781121635948795037447812609973805986712397117373289276654044010306698313834600094139322364490657889951220584317250787337746308785342428535198858750004235824691874082042817009017148230518216216194131998560661293827426497098440844701008054549677936760888126446406885181552709324007542506497157047047541993283178364533256241537869395712509706638797949265462313767459189098131167524342211101309131278371609511583412308415037164997020224681219644081216686527458043026245782561067150138521821644953254334987348741335279581535101658360545576351327650181078119483694595748502373982354526256327794753972699020128915166457939420198920248803394051699686551494477396533876979741232354061781989611249409599035312899773361184984737794610842883329383390395090089140863515256268033814146692799133107433497051435452013446434264752001621384610729922641994332772918977769053802596851;
(Subscript[μ, ∞])=3.5699456718709449018420051513864989367638369115148323781079755299213628875001367775263210342163;

Continued fraction

convergents=Convergents[δ,10]

Transpose[NumeratorDenominator[convergents]]

N[Abs[δ-convergents],20]

Deppe

FullSimplify[Solve[x^x==1333,x]]

FullSimplify[(E^(ProductLog[Log[1333]]))]

CorrectDigits[{(E^(ProductLog[Log[1333]])),δ}]

Flanagan

δ

April 19, 2023

CorrectDigits[E^2-E,δ]

N[{E^2-E,δ},10]

Abs[Subtract@@%]

Hudson

Mark Hudson
22 Nov 2004

CorrectDigits[{1182102/773825+Pi,δ}]

CorrectDigits[{46875/15934-Sqrt[2]+Pi,δ}]

CorrectDigits[{Tan[1954/1781]+E,δ}]

Phillips

δ

CorrectDigits[π+E-ArcTan[Abs[α]],δ]

Jan. 10, 2005

CorrectDigits[(3/2)Pi-(E^(-π)),δ]

N[{(3/2)Pi-(E^(-π)),δ},20]//Column

Sept. 14, 2004

CorrectDigits[(((E^10)-(E^9))/((E^8)+1)),δ]

Jan. 10, 2005

CorrectDigits[(3/2)Pi-((E^(-π))/(1+Exp[-8+Exp[-1/2]])),δ]

Sept. 27, 2004

CorrectDigits[ArcTan[E^Pi-(E-1)^(-16)]+Pi,δ]

TraditionalForm[ArcTan[E^Pi-(E-1)^(-16)]+Pi]

Jan. 25, 2005

CorrectDigits[E(E-1)/(1+(E^8)^((1+E^-8)^(3/2))/E^16),δ]

E(E-1)/(1+(E^8)^((1+E^-8)^(3/2))/E^16)//TraditionalForm

α

Sept. 14, 2004

CorrectDigits[((((E/(E-1))))^2),Abs[α]]

Jan. 26, 2005

CorrectDigits[Tan[E-δ],Abs[α]]

CorrectDigits[Tan[E-ArcTan[E^Pi]],Abs[α]]

Jan. 10, 2005

FullSimplify[-Tan[(3/2)π-(E^(-π))-E]]

CorrectDigits[-Cot[E+(E^(-π))],Abs[α]]

Jan. 25

CorrectDigits[Tan[E+ArcTan[2(E-1)^-8/E-E^Pi]] ,Abs[α]]

Tan[E+ArcTan[2(E-1)^-8/E-E^Pi]] //TraditionalForm

Dec. 30, 2004

((E^2)/((((E-1))^2)-(E^(-((Sqrt[26])+3)))))

CorrectDigits[E^2/((E-1)^2-E^-(3+Sqrt[26])),Abs[α]]

Jan. 2, 2005

CorrectDigits[E^2/((E-1)^2-Exp[-(8+Exp[-1/Log[Log[δ]]])]),Abs[α]]

E^2/((E-1)^2-Exp[-(8+Exp[-1/Log[Log[d]]])])//TraditionalForm

(Subscript[μ, ∞])

FullSimplify[(1+E^-8/E^(Log[Log[x^(1/2)]]))(E^(GoldenRatio^(1/2))) ]

(E^(Sqrt[GoldenRatio])) (1+(2/((E^8) Log[x])))//TraditionalForm

Solve[(E^(Sqrt[GoldenRatio])) (1+(2/((E^8) Log[x])))== x,x]

Solve[x==Exp[((2(E^(-8)))/(x Exp[-Sqrt[GoldenRatio]]-1))],x]

CorrectDigits[{x/.FindRoot[(E^(Sqrt[GoldenRatio])) (1+(2/((E^8) Log[x])))== x,{x,3.5},WorkingPrecision->10],(Subscript[μ, ∞])}]

Plouffe

CorrectDigits[δ,(10/(π-1))]

N[(10/(π-1)),10]

Stoscheck

d[0]:=4
d[1]:=4(1+12^2/163)/(1+10^2/163)
d[2]:=4(1+12^2/163+(4 12^2+31)/4/163^2)/(1+10^2/163+(10^2+30)/163^2)

CorrectDigits[{#,δ}]&/@Array[d,2]

Trott

May 6, 2008

δ==2 G+3

GaussConstant=(((Gamma[1/4])^2)/(((2π))^(3/2)));

N[{δ,3+2GaussConstant},10]

CorrectDigits[3+2GaussConstant,δ]

δ=9/T

TetranacciConstant=Root[#^4-#^3-#^2-#-1,2];

N[{δ,9/TetranacciConstant},10]

CorrectDigits[9/TetranacciConstant,δ]

Unattributed

δ==(tan^(-1))((e^π))+π

N[{δ,π+ArcTan[E^π]},10]

CorrectDigits[{π+ArcTan[E^π],δ}]