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词条 Catalan's constant
释义

  1. Integral identities

  2. Uses

  3. Relation to other special functions

  4. Quickly converging series

  5. Known digits

  6. See also

  7. References

  8. External links

In mathematics, Catalan's constant {{mvar|G}}, which appears in combinatorics, is defined by

where {{mvar|β}} is the Dirichlet beta function. Its numerical value[1] is approximately {{OEIS|A006752}}

{{math|G {{=}} {{val|0.915965594177219015054603514932384110774}}…}}

{{unsolved|mathematics|Is Catalan's constant irrational? If so, is it transcendental?}}

It is not known whether {{mvar|G}} is irrational, let alone transcendental.[2]

Catalan's constant was named after Eugène Charles Catalan.

The similar but apparently more complicated series

can be evaluated exactly and is π3/32.

Integral identities

Some identities involving definite integrals include

where the last three formulas are related to Malmsten's integrals [3].

If {{math|K(t)}} is a complete elliptic integral of the first kind, then

With the gamma function {{math|Γ(x + 1) {{=}} x!}}

The integral

is a known special function, called the inverse tangent integral, and was extensively studied by Srinivasa Ramanujan.

Uses

{{mvar|G}} appears in combinatorics, as well as in values of the second polygamma function, also called the trigamma function, at fractional arguments:

Simon Plouffe gives an infinite collection of identities between the trigamma function, {{pi}}2 and Catalan's constant; these are expressible as paths on a graph.

In low-dimensional topology, Catalan's constant is a rational multiple of the volume of an ideal hyperbolic octahedron, and therefore of the hyperbolic volume of the complement of the Whitehead link.[4]

It also appears in connection with the hyperbolic secant distribution.

Relation to other special functions

Catalan's constant occurs frequently in relation to the Clausen function, the inverse tangent integral, the inverse sine integral, the Barnes {{mvar|G}}-function, as well as integrals and series summable in terms of the aforementioned functions.

As a particular example, by first expressing the inverse tangent integral in its closed form – in terms of Clausen functions – and then expressing those Clausen functions in terms of the Barnes {{mvar|G}}-function, the following expression is obtained (see Clausen function for more):

.

If one defines the Lerch transcendent {{math|Φ(z,s,α)}} (related to the Lerch zeta function) by

then

Quickly converging series

The following two formulas involve quickly converging series, and are thus appropriate for numerical computation:

and

The theoretical foundations for such series are given by Broadhurst, for the first formula,[5] and Ramanujan, for the second formula.[6] The algorithms for fast evaluation of the Catalan constant were constructed by E. Karatsuba.[7][8]

Known digits

The number of known digits of Catalan's constant {{mvar|G}} has increased dramatically during the last decades. This is due both to the increase of performance of computers as well as to algorithmic improvements.[9]

G}}
Date Decimal digits Computation performed by
1832 16 Thomas Clausen
1858 19 Carl Johan Danielsson Hill
1864 14 Eugène Charles Catalan
1877 20 James W. L. Glaisher
1913 32 James W. L. Glaisher
1990 {{val|20000}} Greg J. Fee
1996 {{val|50000}} Greg J. Fee
August 14, 1996 {{val|100000}} Greg J. Fee & Simon Plouffe
September 29, 1996 {{val|300000}} Thomas Papanikolaou
1996 {{val|1500000}} Thomas Papanikolaou
1997 {{val|3379957}} Patrick Demichel
January 4, 1998 {{val|12500000}} Xavier Gourdon
2001 {{val|100000500}} Xavier Gourdon & Pascal Sebah
2002 {{val|201000000}} Xavier Gourdon & Pascal Sebah
October 2006 {{val|5000000000}} Shigeru Kondo & Steve Pagliarulo[10]
August 2008 {{val|10000000000}} Shigeru Kondo & Steve Pagliarulo[11]
January 31, 2009 {{val|15510000000}} Alexander J. Yee & Raymond Chan[12]
April 16, 2009 {{val|31026000000}} Alexander J. Yee & Raymond Chan[12]
June 7, 2015 {{val|200000001100}} Robert J. Setti[13]
April 12, 2016 {{val|250000000000}} Ron Watkins[14]
February 16, 2019 {{val|300000000000}} Tizian Hanselmann[15]

See also

  • Particular values of Riemann zeta function
  • Mathematical constant

References

1. ^{{cite web|last1=Papanikolaou|first1=Thomas|title=Catalan's Constant to 1,500,000 Places|url=http://www.gutenberg.org/etext/812|website=Gutenberg.org|date=March 1997}}
2. ^{{citation | last = Nesterenko | first = Yu. V. | date = January 2016 | doi = 10.1134/s0081543816010107 | issue = 1 | journal = Proceedings of the Steklov Institute of Mathematics | pages = 153–170 | title = On Catalan's constant | volume = 292}}.
3. ^{{Cite journal|first1=Iaroslav |last1=Blagouchine|title=Rediscovery of Malmsten's integrals, their evaluation by contour integration methods and some related results|year=2014|doi=10.1007/s11139-013-9528-5|url=https://iblagouchine.perso.centrale-marseille.fr/publications/Blagouchine-Malmsten-integrals-and-their-evaluation-by-contour-integration-methods-(Ramanujan-J-2014).pdf|volume=35|journal=The Ramanujan Journal|pages=21–110}}
4. ^{{citation | last = Agol | first = Ian | authorlink = Ian Agol | doi = 10.1090/S0002-9939-10-10364-5 | issue = 10 | journal = Proceedings of the American Mathematical Society | mr = 2661571 | pages = 3723–3732 | title = The minimal volume orientable hyperbolic 2-cusped 3-manifolds | volume = 138 | year = 2010| arxiv = 0804.0043}}.
5. ^{{cite arXiv|first1=D. J. |last1=Broadhurst|eprint=math.CA/9803067 |title=Polylogarithmic ladders, hypergeometric series and the ten millionth digits of {{math|ζ(3)}} and {{math|ζ(5)}}| year=1998}}
6. ^{{cite book|first=B. C.|last=Berndt|title=Ramanujan's Notebook, Part I|publisher=Springer Verlag|date=1985}} {{ISBN missing}}
7. ^{{cite journal|first=E. A.|last=Karatsuba|title=Fast evaluation of transcendental functions|journal=Probl. Inf. Transm.|volume=27|issue=4|pages=339–360|date=1991|zbl=0754.65021|mr=1156939}}
8. ^{{cite book|first=E. A.|last=Karatsuba|contribution=Fast computation of some special integrals of mathematical physics|title=Scientific Computing, Validated Numerics, Interval Methods|editor1-first=W.|editor1-last=Krämer|editor2-first=J. W.|editor2-last=von Gudenberg|pages=29–41|date=2001}} {{ISBN missing}}
9. ^{{cite web|last1=Gourdon|first1=X.|last2=Sebah|first2=P.|url=http://numbers.computation.free.fr/Constants/constants.html|title=Constants and Records of Computation}}
10. ^{{Cite web |url=http://ja0hxv.calico.jp/pai/ecatalan.html# |title=Shigeru Kondo's website |access-date=2008-01-31 |archive-url=https://web.archive.org/web/20080211185703/http://ja0hxv.calico.jp/pai/ecatalan.html# |archive-date=2008-02-11 |dead-url=yes |df= }}
11. ^Constants and Records of Computation
12. ^1 Large Computations
13. ^Catalan's constant records using YMP
14. ^Catalan's constant records using YMP
15. ^Catalan's constant records using YMP

External links

  • Victor Adamchik, 33 representations for Catalan's constant (undated)
  • {{cite journal

|first1=Victor
|last1= Adamchik,
|year=2002
|journal=Zeitschrift für Analysis und ihre Anwendungen
|volume=21
|issue=3
|pages=1–10
|url=http://www-2.cs.cmu.edu/~adamchik/articles/csum.html
|title=A certain series associated with Catalan's constant
|mr=1929434
|doi=10.4171/ZAA/1110
}}
  • {{cite web

|first1=Simon
|last1=Plouffe
|url=http://www.lacim.uqam.ca/~plouffe/IntegerRelations/identities3a.html
|title= A few identities (III) with Catalan
|year=1993 }} (Provides over one hundred different identities).
  • Simon Plouffe, A few identities with Catalan constant and Pi^2, (1999) (Provides a graphical interpretation of the relations)
  • {{MathWorld|title=Catalan's Constant|urlname=CatalansConstant}}
  • Catalan constant: Generalized power series at the Wolfram Functions Site
  • Greg Fee, Catalan's Constant (Ramanujan's Formula) (1996) (Provides the first 300,000 digits of Catalan's constant.).
  • {{citation

|first1=Greg
|last1=Fee
|year=1990
|title=Computation of Catalan's constant using Ramanujan's Formula
|pages=157–160
|series=Proceedings of the ISSAC '90
|doi=10.1145/96877.96917
}}
  • {{Cite journal

|first1=David M.
|last1=Bradley
|title=A class of series acceleration formulae for Catalan's constant
|doi=10.1023/A:1006945407723
|year=1999
|journal=The Ramanujan Journal
|volume=3
|issue=2
|pages=159–173
|mr=1703281
|arxiv=0706.0356
}}
  • {{Cite journal

|first1=David M.
|last1=Bradley
|title=A class of series acceleration formulae for Catalan's constant
|arxiv=0706.0356
|year=2007
|doi=10.1023/A:1006945407723
|volume=3
|journal=The Ramanujan Journal
|pages=159–173
}}
  • {{Citation

|first1=David M.
|last1=Bradley
|title = Representations of Catalan's constant
|citeseerx = 10.1.1.26.1879
|year=2001
}}
  • {{springer|title=Catalan constant|id=p/c130040}}
  • Catalan's Constant — from Wolfram MathWorld
  • [https://archive.org/details/ctcst10a Catalan's Constant (Ramanujan's Formula)]
  • catalan's constant — www.cs.cmu.edu

2 : Combinatorics|Mathematical constants
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