In mathematics, Catalan's constant , is defined by :

G = \beta(2) = \sum_^ \frac = \frac - \frac + \frac - \frac + \frac - \cdots,

where is the Dirichlet beta function. Its numerical value is approximately : It is not known whether is

irrational Irrationality is cognition, thinking, talking, or acting without inclusion of rationality. It is more specifically described as an action or opinion given through inadequate use of reason, or through emotional distress or cognitive deficiency. T ...

, let alone transcendental. has been called "arguably the most basic constant whose irrationality and transcendence (though strongly suspected) remain unproven". Catalan's constant was named after Eugène Charles Catalan, who found quickly-converging series for its calculation and published a memoir on it in 1865.

Uses

In low-dimensional topology, Catalan's constant is 1/4 of the volume of an ideal hyperbolic

octahedron In geometry, an octahedron (plural: octahedra, octahedrons) is a polyhedron with eight faces. The term is most commonly used to refer to the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at ...

, and therefore 1/4 of the

hyperbolic volume In the mathematical field of knot theory, the hyperbolic volume of a hyperbolic link is the volume of the link's complement with respect to its complete hyperbolic metric. The volume is necessarily a finite real number, and is a topological inv ...

of the complement of the Whitehead link. It is 1/8 of the volume of the complement of the

Borromean rings In mathematics, the Borromean rings are three simple closed curves in three-dimensional space that are topologically linked and cannot be separated from each other, but that break apart into two unknotted and unlinked loops when any one of the ...

. In

combinatorics Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many ap ...

and

statistical mechanics In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. It does not assume or postulate any natural laws, but explains the macroscopic be ...

, it arises in connection with counting domino tilings, spanning trees, and

Hamiltonian cycle In the mathematical field of graph theory, a Hamiltonian path (or traceable path) is a path in an undirected or directed graph that visits each vertex exactly once. A Hamiltonian cycle (or Hamiltonian circuit) is a cycle that visits each vertex ...

s of

grid graph In graph theory, a lattice graph, mesh graph, or grid graph is a graph whose drawing, embedded in some Euclidean space , forms a regular tiling. This implies that the group of bijective transformations that send the graph to itself is a lat ...

s. In

number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mathe ...

, Catalan's constant appears in a conjectured formula for the asymptotic number of primes of the form

n^2+1

according to Hardy and Littlewood's Conjecture F. However, it is an unsolved problem (one of

Landau's problems At the 1912 International Congress of Mathematicians, Edmund Landau listed four basic problems about prime numbers. These problems were characterised in his speech as "unattackable at the present state of mathematics" and are now known as Landau ...

) whether there are even infinitely many primes of this form. Catalan's constant also appears in the calculation of the

mass distribution In physics and mechanics, mass distribution is the spatial distribution of mass within a solid body. In principle, it is relevant also for gases or liquids, but on Earth their mass distribution is almost homogeneous. Astronomy In astronomy mass d ...

spiral galaxies Spiral galaxies form a class of galaxy originally described by Edwin Hubble in his 1936 work ''The Realm of the Nebulae''\begin G &= -\frac\int_^ \ln\ln \tan x \ln \tan x \,dx \\ ptG &= \iint_ \! \frac \,dx\, dy \\ pt G &= \int_0^1\int_0^ \frac \,dy\,dx \\ ptG &= \int_1^\infty \frac \,dt \\ ptG &= -\int_0^1 \frac \,dt \\ ptG &= \int_0^\frac \frac \,dt \\ ptG &= \frac \int_^\frac \frac \,dt \\ ptG &= \int_0^\frac \ln \cot t \,dt \\ ptG &= \int_\frac^\frac \ln \tan t \,dt \\ ptG &= \frac \int_0^\frac \ln \left( \sec t +\tan t \right) \,dt \\ ptG &= \int_0^1 \frac \,dt \\ ptG &= \int_0^1 \frac \,dt \\ ptG &= \frac \int_0^\infty \frac \,dt \\ ptG &= \int_0^\infty \arccot e^ \,dt \\ ptG &= \frac \int_0^ \csc \sqrt \,dt \\ ptG &= \frac \left(\pi^2 + 4\int_1^\infty \arccsc^2 t \,dt\right) \\ ptG &= \frac \int_0^\infty \frac \,dt \\ ptG &= \frac \int_1^\infty \frac \,dt \\ ptG &= 1 + \lim_\!\left\ \\ ptG &= 1 - \frac18 \iint_\!\!\frac \,dx\,dy \\ ptG &= \int_^\int_^\fracdxdy \end where the last three formulas are related to Malmsten's integrals. If is the

complete elliptic integral of the first kind In integral calculus, an elliptic integral is one of a number of related functions defined as the value of certain integrals, which were first studied by Giulio Fagnano and Leonhard Euler (). Their name originates from their originally arising i ...

, as a function of the elliptic modulus , then

G = \tfrac \int_0^1 \mathrm(k)\,dk

If is the complete elliptic integral of the second kind, as a function of the elliptic modulus , then

G = -\tfrac+\int_0^1 \mathrm(k)\,dk

With the

gamma function In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except t ...

\begin
G &= \frac \int_0^1 \Gamma\left(1+\frac\right)\Gamma\left(1-\frac\right)\,dx \\
&= \frac \int_0^\frac12\Gamma(1+y)\Gamma(1-y)\,dy
\end

The integral

G = \operatorname_2(1)=\int_0^1 \frac\,dt

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

Srinivasa Ramanujan Srinivasa Ramanujan (; born Srinivasa Ramanujan Aiyangar, ; 22 December 188726 April 1920) was an Indian mathematician. Though he had almost no formal training in pure mathematics, he made substantial contributions to mathematical analysis, ...

Relation to other special functions

appears in values of the second

polygamma function In mathematics, the polygamma function of order is a meromorphic function on the complex numbers \mathbb defined as the th derivative of the logarithm of the gamma function: :\psi^(z) := \frac \psi(z) = \frac \ln\Gamma(z). Thus :\psi^(z) ...

, also called the

trigamma function In mathematics, the trigamma function, denoted or , is the second of the polygamma functions, and is defined by : \psi_1(z) = \frac \ln\Gamma(z). It follows from this definition that : \psi_1(z) = \frac \psi(z) where is the digamma functi ...

, at fractional arguments:

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

Simon Plouffe Simon Plouffe (born June 11, 1956) is a mathematician who discovered the Bailey–Borwein–Plouffe formula (BBP algorithm) which permits the computation of the ''n''th binary digit of π, in 1995. His other 2022 formula allows extracting the ' ...

gives an infinite collection of identities between the trigamma function, ² and Catalan's constant; these are expressible as paths on a graph. Catalan's constant occurs frequently in relation to the Clausen function, the inverse tangent integral, the inverse sine integral, the Barnes -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 -function, the following expression is obtained (see Clausen function for more):

G=4\pi \log\left( \frac \right) +4 \pi \log \left( \frac \right) +\frac \log \left( \frac \right).

If one defines the Lerch transcendent (related to the

Lerch zeta function In mathematics, the Lerch zeta function, sometimes called the Hurwitz–Lerch zeta function, is a special function that generalizes the Hurwitz zeta function and the polylogarithm. It is named after Czech mathematician Mathias Lerch, who publis ...

) by

\Phi(z, s, \alpha) = \sum_^\infty \frac  ,

then

G = \tfrac\Phi\left(-1, 2, \tfrac\right).

Quickly converging series

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

\begin
G & = 3 \sum_^\infty \frac
\left(-\frac+\frac-\frac+\frac-\frac+\frac\right)- \\
& \qquad -2 \sum_^\infty \frac
\left(\frac+\frac-\frac-\frac-\frac+\frac\right)
\end

and

G = \frac\log\left(2 + \sqrt\right) + \frac\sum_^\infty \frac.

The theoretical foundations for such series are given by Broadhurst, for the first formula, and Ramanujan, for the second formula. The algorithms for fast evaluation of the Catalan constant were constructed by E. Karatsuba. Using these series, calculating Catalan's constant is now about as fast as calculating Apery's constant,

\zeta(3)

. Other quickly converging series, due to Guillera and Pilehrood and employed by the y-cruncher software, include: :

G = \frac\sum_^\frac

G = \frac\sum_^\frac

G = -\frac\sum_^\frac\left( \frac \right)

All of these series have

time complexity In computer science, the time complexity is the computational complexity that describes the amount of computer time it takes to run an algorithm. Time complexity is commonly estimated by counting the number of elementary operations performed by t ...

O(n\log(n)^3)

Continued fraction

can be expressed in the following form :

G=\cfrac

:The simple continued fraction is given by :

G=\cfrac

References

External links

* * (Provides over one hundred different identities). * (Provides a graphical interpretation of the relations) * (Provides the first 300,000 digits of Catalan's constant) * * * * * * {{springer , title = Catalan constant , id = p/c130040 , mode=cs1 Combinatorics Mathematical constants