Feigenbaum Constant Approximations
المؤلف:
Friedman, E.
المصدر:
"Problem of the Month (August 2004)." http://www.stetson.edu/~efriedma/mathmagic/0804.html.
الجزء والصفحة:
...
24-2-2020
1293
Feigenbaum Constant Approximations
A curious approximation to the Feigenbaum constant 
is given by
 |
(1)
|
where 
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
 |
(2)
|
where 
is Gauss's constant, which is good to 4 decimal digits, and
 |
(3)
|
where 
is the tetranacci constant, which is good to 3 decimal digits.
A strange approximation good to five digits is given by the solution to
 |
(4)
|
which is
 |
(5)
|
where 
is the Lambert W-function (G. Deppe, pers. comm., Feb. 27, 2003).
 |
(6)
|
gives 
to 3 digits (S. Plouffe, pers. comm., Apr. 10, 2006).
M. Hudson (pers. comm., Nov. 20, 2004) gave
which are good to 17, 13, and 9 digits respectively.
Stoschek gave the strange approximation
 |
(10)
|
which is good to 9 digits.
R. Phillips (pers. comm., Sept. 14, 2004-Jan. 25, 2005) gave the approximations
where e is the base of the natural logarithm and 
is Gelfond's constant, which are good to 3, 3, 5, 7, 9, and 10 decimal digits, respectively, and
which are good to 3, 3, 3, 4, 6, 8, and 8 decimal digits, respectively.
An approximation to 
due to R. Phillips (pers. comm., Jan. 27, 2005) is obtained by numerically solving
 |
(24)
|
for 
, where 
is the golden ratio, which is good to 4 digits.
REFERENCES:
Friedman, E. "Problem of the Month (August 2004)." http://www.stetson.edu/~efriedma/mathmagic/0804.html.
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