Named after its inventor, E.Ch. Catalan (1814–1894), the Catalan constant (which is denoted also by ) is defined by
If, in terms of the Digamma (or Psi) function , defined by
which provides a relationship between the Catalan constant and the Digamma function .
The Catalan constant is related also to other functions, such as the Clausen function , defined by
and the Hurwitz zeta-function , which is defined, when , by
the last expression in (a7) would follow also from (a4) in light of the definition in (a3).
A fairly large number of integrals and series can be evaluated in terms of the Catalan constant . For example,
where denotes the familiar Riemann zeta-function.
Another important mathematical constant is the Euler–Mascheroni constant (which is denoted also by ), defined by
It is named after L. Euler (1707–1783) and L. Mascheroni (1750–1800). Indeed, one also has
where an empty sum is interpreted, as usual, to be zero. In terms of the Riemann zeta-function , Euler's classical results state:
|[a1]||A. Erdélyi, W. Magnus, F. Oberhettinger, F.G. Tricomi, "Higher transcendental functions" , I , McGraw-Hill (1953)|
|[a2]||L. Lewin, "Polylogarithms and associated functions" , Elsevier (1981)|
|[a3]||H.M. Srivastava, J. Choi, "Series associated with the zeta and related functions" , Kluwer Acad. Publ. (2001)|
Catalan constant. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Catalan_constant&oldid=12431