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): If one defines the Lerch transcendent (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, 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, . Other quickly converging series, due to Guillera and Pilehrood and employed by the y-cruncher software, include: All of these series have time complexity . Continued fraction can be expressed in the following form The simple continued fraction is given by See also Gieseking manifold List of mathematical constants Mathematical constant Particular values of Riemann zeta function References Further reading 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) Combinatorics Mathematical constants