Catalan's constant

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In mathematics, Catalan's constant G, which occasionally appears in estimates in combinatorics, is defined by

G = \beta(2) = \sum_{n=0}^{\infty} \frac{(-1)^{n}}{(2n+1)^2} = \frac{1}{1^2} - \frac{1}{3^2} + \frac{1}{5^2} - \frac{1}{7^2} + \cdots \!

where β is the Dirichlet beta function. Its numerical value [1] is approximately (sequence A006752 in OEIS)

G = 0.915 965 594 177 219 015 054 603 514 932 384 110 774 …

It is not known whether G is irrational, let alone transcendental.

Catalan's constant was named after Eugène Charles Catalan.

Integral identities[edit]

Some identities include

G = \int_0^1 \int_0^1 \frac{1}{1+x^2 y^2} \,dx\, dy \!
G = -\int_0^1 \frac{\ln t}{1 + t^2} \,dt \!
G = \int_{0}^{\pi/4} \frac{t}{\sin t \cos t} \;dt  \!
G = \frac{1}{4} \int_{-\pi/2}^{\pi/2} \frac{t}{\sin t} \;dt \!
G = \int_0^{\pi/4} \ln ( \cot(t) ) \,dt \!
G = \int_0^\infty \arctan (e^{-t}) \,dt \!

along with

 G = \frac{1}{2} \int_0^1 \mathrm{K}(t)\,dt \!

where K(t) is a complete elliptic integral of the first kind.

And with Gamma function Γ(x+1) = x!

 G = \frac{\pi}{2} \int_0^\tfrac12\Gamma(1+x)\Gamma(1-x)\,dx

The integral:

 G = \text{Ti}_2(1)=\int_0^1 \frac{\arctan t}{t}\,dt. \!

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

Uses[edit]

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

 \psi_1 \left(\tfrac14\right) = \pi^2 + 8G
 \psi_1 \left(\tfrac34\right) = \pi^2 - 8G.

Simon Plouffe gives an infinite collection of identities between the trigamma function, π2 and Catalan's constant; these are expressible as paths on a graph.

It also appears in connection with the hyperbolic secant distribution.

Relation to other special function[edit]

Catalan's constant occurs frequently in relation to the Clausen function, the Inverse tangent integral, the Inverse sine integral, Barnes 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 G-function, the following expression is easily obtained (N.B. all the relevant relations for this derivation have been added to the page for the Clausen function):

G=4\pi \log\left( \frac{ G(\tfrac{3}{8}) G(\tfrac{7}{8}) }{ G(\tfrac{1}{8}) G(\tfrac{5}{8}) } \right) +4 \pi \log \left( \frac{ \Gamma(\tfrac{3}{8}) }{ \Gamma(\tfrac{1}{8}) } \right) +\frac{\pi}{2} \log \left( \frac{1+\sqrt{2} }{2 \, (2-\sqrt{2})} \right).

If one defines the Lerch transcendent, \Phi(z, s, \alpha), (related to the Lerch zeta function) by,

\Phi(z, s, \alpha) = \sum_{n=0}^\infty
\frac { z^n} {(n+\alpha)^s}.,

then it is clear that

 G = \tfrac{1}{4}\,\Phi(-1, 2, \tfrac{1}{2}).

Quickly converging series[edit]

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


\begin{align}
G & =
3 \sum_{n=0}^\infty \frac{1}{2^{4n}}
\left(
-\frac{1}{2(8n+2)^2}
+\frac{1}{2^2(8n+3)^2}
-\frac{1}{2^3(8n+5)^2}
+\frac{1}{2^3(8n+6)^2}
-\frac{1}{2^4(8n+7)^2}
+\frac{1}{2(8n+1)^2}
\right) \\
& {}\quad -2 \sum_{n=0}^\infty \frac{1}{2^{12n}}
\left(
\frac{1}{2^4(8n+2)^2}
+\frac{1}{2^6(8n+3)^2}
-\frac{1}{2^9(8n+5)^2}
-\frac{1}{2^{10} (8n+6)^2}
-\frac{1}{2^{12} (8n+7)^2}
+\frac{1}{2^3(8n+1)^2}
\right)
\end{align}

and

G = \tfrac18\pi \log(2 + \sqrt{3}) + \tfrac38 \sum_{n=0}^\infty \frac{(n!)^2}{(2n)!(2n+1)^2}.

The theoretical foundations for such series is given by Broadhurst (the first formula)[1] and Ramanujan (the second formula).[2] The algorithms for fast evaluation of the Catalan constant is constructed by E. Karatsuba.[3][4]

Known digits[edit]

The number of known digits of Catalan's constant 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.[5]

Number of known decimal digits of Catalan's constant 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 20,000 Greg J. Fee
1996 50,000 Greg J. Fee
August 14, 1996 100,000 Greg J. Fee & Simon Plouffe
September 29, 1996 300,000 Thomas Papanikolaou
1996 1,500,000 Thomas Papanikolaou
1997 3,379,957 Patrick Demichel
January 4, 1998 12,500,000 Xavier Gourdon
2001 100,000,500 Xavier Gourdon & Pascal Sebah
2002 201,000,000 Xavier Gourdon & Pascal Sebah
October 2006 5,000,000,000 Shigeru Kondo & Steve Pagliarulo[6]
August 2008 10,000,000,000 Shigeru Kondo & Steve Pagliarulo[7]
January 31, 2009 15,510,000,000 Alexander J. Yee & Raymond Chan[8]
April 16, 2009 31,026,000,000 Alexander J. Yee & Raymond Chan[8]
April 6, 2013 100,000,000,000 Robert J. Setti[9]

See also[edit]

Notes[edit]

  1. ^ Broadhurst, D.J. (1998). "Polylogarithmic ladders, hypergeometric series and the ten millionth digits of ζ(3) and ζ(5)". arXiv:math.CA/9803067.
  2. ^ B.C. Berndt, Ramanujan's Notebook, Part I., Springer Verlag (1985)
  3. ^ E.A. Karatsuba, Fast evaluation of transcendental functions, Probl. Inf. Transm. Vol.27, No.4, pp. 339–360 (1991)
  4. ^ E.A. Karatsuba, Fast computation of some special integrals of mathematical physics. Scientific Computing, Validated Numerics, Interval Methods, W.Krämer, J.W.von Gudenberg, eds.; pp. 29–41, (2001)
  5. ^ Gourdon, X., Sebah, P; Constants and Records of Computation
  6. ^ Shigeru Kondo's website
  7. ^ Constants and Records of Computation
  8. ^ a b Large Computations
  9. ^ 100 Billion Digits Catalan's Constant Complete

References[edit]

External links[edit]