Khinchin

Khinchin

is Khinchin's constant, with numerical value .

Details

  • Mathematical constant treated as numeric by NumericQ and as a constant by D.
  • Khinchin can be evaluated to any numerical precision using N.
  • Khinchin's constant (sometimes called Khintchine's constant) is given by .

Background & Context

  • Khinchin is the symbol representing Khinchin's constant , also known as Khintchine's constant. Khinchine is defined as the limiting value for the geometric mean of the terms of a simple continued fraction expansion of a real number , where the value of is independent of the choice of . Khinchin has a numerical value and a closed form product is given by . Khinchin arises most commonly in the theory of continued fractions and in ergodic theory.
  • When Khinchin is used as a symbol, it is propagated as an exact quantity.
  • It is not currently known if Khinchin is rational (meaning it can be expressed as a ratio of integers), algebraic (meaning it is the root of some integer polynomial), or normal (meaning the digits in its base- expansion are equally distributed) to any base.
  • Khinchin can be numerically evaluated using N. However, no efficient formulas for computing large numbers of its digits are currently known. RealDigits can be used to return a list of digits of Khinchin and ContinuedFraction to obtain terms of its continued fraction expansion.

Examples

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Basic Examples  (1)

Evaluate to any precision:

Scope  (2)

Do an exact numerical computation:

TraditionalForm formatting:

Applications  (1)

Geometric mean of the first 1000 continued fraction terms in π:

Properties & Relations  (2)

Various symbolic relations are automatically used:

Various products give results that can be expressed using Khinchin:

Neat Examples  (1)

Terms in the continued fraction:

Wolfram Research (1999), Khinchin, Wolfram Language function, https://reference.wolfram.com/language/ref/Khinchin.html.

Text

Wolfram Research (1999), Khinchin, Wolfram Language function, https://reference.wolfram.com/language/ref/Khinchin.html.

BibTeX

@misc{reference.wolfram_2021_khinchin, author="Wolfram Research", title="{Khinchin}", year="1999", howpublished="\url{https://reference.wolfram.com/language/ref/Khinchin.html}", note=[Accessed: 24-October-2021 ]}

BibLaTeX

@online{reference.wolfram_2021_khinchin, organization={Wolfram Research}, title={Khinchin}, year={1999}, url={https://reference.wolfram.com/language/ref/Khinchin.html}, note=[Accessed: 24-October-2021 ]}

CMS

Wolfram Language. 1999. "Khinchin." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/Khinchin.html.

APA

Wolfram Language. (1999). Khinchin. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Khinchin.html