Grossman,s Constant
المؤلف:
Ewing, J. and Foias, C
المصدر:
"An Interesting Serendipitous Real Number." In Finite versus Infinite: Contributions to an Eternal Dilemma (Ed. C. Caluse and G. Păun). London: Springer-Verlag
الجزء والصفحة:
...
25-3-2020
946
Grossman's Constant
Define the sequence 
, 
, and
 |
(1)
|
for 
. The first few values are
Janssen and Tjaden (1987) showed that this sequence converges for exactly one value 
, where 
(OEIS A085835), confirming Grossman's conjecture. However, no analytic form is known for this constant, either as the root of a function or as a combination of other constants. The plot above shows the first few iterations of 
for 
to 30, with odd 
shown in red and even 
shown in blue, for 
ranging from 0 to 1. As can be seen, the solutions alternate by parity. For each fixed 
, the red values go to 0, while the blue values go to some positive number.
Nyerges (2000) has generalized the recurrence to the functional equation
![x=[1+F(x)]F^2(x).](https://mathworld.wolfram.com/images/equations/GrossmansConstant/NumberedEquation2.gif) |
(6)
|
REFERENCES:
Ewing, J. and Foias, C. "An Interesting Serendipitous Real Number." In Finite versus Infinite: Contributions to an Eternal Dilemma (Ed. C. Caluse and G. Păun). London: Springer-Verlag, pp. 119-126, 2000.
Finch, S. R. "Grossman's Constant." §6.4 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 429-430, 2003.
Grossman, J. W. "Problem 86-2." Math. Intel. 8, 31, 1986.
Janssen, A. J. E. M. and Tjaden, D. L. A. Solution to Problem 86-2. Math. Intel. 9, 40-43, 1987.
Michon, G. P. "Final Answers: Numerical Constants." http://home.att.net/~numericana/answer/constants.htm#grossman.
Nyerges, G. "The Solution of the Functional Equation 
." Preprint, Oct. 19, 2000. http://eent3.sbu.ac.uk/Staff/nyergeg/www/etc/fneq.pdf.
Sloane, N. J. A. Sequence A085835 in "The On-Line Encyclopedia of Integer Sequences."
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