Catalan's constant - Alchetron, The Free Social Encyclopedia

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

Integral identities

Some identities involving definite integrals include

G = ∫ 0 1 ∫ 0 1 1 1 + x 2 y 2 d x d y G = − ∫ 0 1 ln ⁡ t 1 + t 2 d t G = ∫ 0 π 4 t sin ⁡ t cos ⁡ t d t G = 1 4 ∫ − π 2 π 2 t sin ⁡ t d t G = ∫ 0 π 4 ln ⁡ cot ⁡ t d t G = 1 2 ∫ 0 π 2 ln ⁡ ( sec ⁡ t + tan ⁡ t ) d t G = ∫ 0 ∞ arctan ⁡ e − t d t G = ∫ 1 ∞ ln ⁡ t 1 + t 2 d t G = 1 2 ∫ 0 ∞ t cosh ⁡ t d t

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

G = 1 2 ∫ 0 1 K ( t ) d t

With the gamma function Γ(x + 1) = x!

G = π 4 ∫ 0 1 Γ ( 1 + x 2 ) Γ ( 1 − x 2 ) d x = π 2 ∫ 0 1 2 Γ ( 1 + y ) Γ ( 1 − y ) d y

The integral

G = Ti 2 ⁡ ( 1 ) = ∫ 0 1 arctan ⁡ t t d t

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

Uses

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

ψ 1 ( 1 4 ) = π 2 + 8 G ψ 1 ( 3 4 ) = π 2 − 8 G .

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.

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.

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 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 (see Clausen function for more):

G = 4 π log ⁡ ( G ( 3 8 ) G ( 7 8 ) G ( 1 8 ) G ( 5 8 ) ) + 4 π log ⁡ ( Γ ( 3 8 ) Γ ( 1 8 ) ) + π 2 log ⁡ ( 1 + 2 2 ( 2 − 2 ) ) .

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

Φ ( z , s , α ) = ∑ n = 0 ∞ z n ( n + α ) s ,

then it is clear that

G = 1 4 Φ ( − 1 , 2 , 1 2 ) .

Quickly converging series

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

G = 3 ∑ n = 0 ∞ 1 2 4 n ( − 1 2 ( 8 n + 2 ) 2 + 1 2 2 ( 8 n + 3 ) 2 − 1 2 3 ( 8 n + 5 ) 2 + 1 2 3 ( 8 n + 6 ) 2 − 1 2 4 ( 8 n + 7 ) 2 + 1 2 ( 8 n + 1 ) 2 ) − − 2 ∑ n = 0 ∞ 1 2 12 n ( 1 2 4 ( 8 n + 2 ) 2 + 1 2 6 ( 8 n + 3 ) 2 − 1 2 9 ( 8 n + 5 ) 2 − 1 2 10 ( 8 n + 6 ) 2 − 1 2 12 ( 8 n + 7 ) 2 + 1 2 3 ( 8 n + 1 ) 2 )

and

G = π 8 log ⁡ ( 2 + 3 ) + 3 8 ∑ n = 0 ∞ 1 ( 2 n + 1 ) 2 ( 2 n n ) .

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.

Known digits

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.

References

Catalan's constant Wikipedia

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