Wednesday, November 17, 2010

Feigenbaum Constant Approximations

Feigenbaum Constant Approximations
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A curious approximation to the Feigenbaum constant delta is given by


where e^pi is Gelfond's constant, which is good to 6 digits to the right of the decimal point.

M. Trott (pers. comm., May 6, 2008) noted

 delta approx 2G+3,

where G is Gauss's constant, which is good to 4 decimal digits, and

 delta approx 9/T,

where T is the tetranacci constant, which is good to 3 decimal digits.

A strange approximation good to five digits is given by the solution to


which is


where W(z) is the Lambert W-function (G. Deppe, pers. comm., Feb. 27, 2003).

 delta approx (10)/(pi-1)

gives delta to 3 digits (S. Plouffe, pers. comm., Apr. 10, 2006).

M. Hudson (pers. comm., Nov. 20, 2004) gave

delta approx (1182102)/(773825)+pi
 approx (46875)/(15934)-sqrt(2)+pi
 approx tan((1954)/(1781))+e,

which are good to 17, 13, and 9 digits respectively.

Stoschek gave the strange approximation

 delta approx 4(1+(12^2)/(163)+(4·12^2+31)/(4·163^2)+...)/(1+(10^2)/(163)+(10^2+30)/(163^2)+...),

which is good to 9 digits.

R. Phillips (pers. comm., Sept. 14, 2004-Jan. 25, 2005) gave the approximations

delta approx 3/2pi-e^(-pi)
 approx pi+e-tan^(-1)|alpha|
 approx (e^(10)-e^9)/(e^8+1)
 approx 3/2pi-(e^(-pi))/(1+exp(-8+e^(-1/2)))
 approx pi-tan^(-1)[(e-1)^(-16)-e^pi],
 approx (e(e-1))/(1+exp{8[(1+e^(-8))^(3/2)-2]})

where e is the base of the natural logarithm and e^pi is Gelfond's constant, which are good to 3, 3, 5, 7, 9, and 10 decimal digits, respectively, and

|alpha| approx (e/(e-1))^2
 approx tan(e-delta)
 approx tan[e-tan^(-1)(e^pi)]
 approx -cot(e+e^(-pi))
 approx tan[e+tan^(-1)(2/((e-1)^8e)-e^pi)]
 approx (e^2)/((e-1)^2-e^(-(3+sqrt(26))))
 approx (e^2)/((e-1)^2-exp(-8-e^(-1/lnlndelta)),)

which are good to 3, 3, 3, 4, 6, 8, and 8 decimal digits, respectively.

An approximation to mu_infty due to R. Phillips (pers. comm., Jan. 27, 2005) is obtained by numerically solving


for x, where phi is the golden ratio, which is good to 4 digits.

SEE ALSO: Almost Integer, Feigenbaum Constant