# Catalan constant

Named after its inventor, E.Ch. Catalan (1814–1894), the Catalan constant (which is denoted also by ) is defined by

(a1) |

If, in terms of the Digamma (or Psi) function , defined by

(a2) |

or

one puts

(a3) |

where

then

(a4) |

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

(a5) |

and the Hurwitz zeta-function , which is defined, when , by

(a6) |

Thus,

(a7) |

Since

(a8) |

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,

(a9) |

(a10) |

and

(a11) |

where denotes the familiar Riemann zeta-function.

## Euler–Mascheroni constant.

Another important mathematical constant is the Euler–Mascheroni constant (which is denoted also by ), defined by

(a12) |

It is named after L. Euler (1707–1783) and L. Mascheroni (1750–1800). Indeed, one also has

(a13) |

and

(a14) |

where an empty sum is interpreted, as usual, to be zero. In terms of the Riemann zeta-function , Euler's classical results state:

(a15) |

#### References

[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) |

**How to Cite This Entry:**

Catalan constant.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Catalan_constant&oldid=12431