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This article is about the Dirichlet beta function. For other beta functions, see Beta function (disambiguation).

In mathematics, the **Dirichlet beta function** (also known as the **Catalan beta function**) is a special function, closely related to the Riemann zeta function. It is a particular Dirichlet L-function, the L-function for the alternating character of period four.

## Contents |

The Dirichlet beta function is defined as

or, equivalently,

In each case, it is assumed that Re(*s*) > 0.

Alternatively, the following definition, in terms of the Hurwitz zeta function, is valid in the whole complex *s*-plane:

Another equivalent definition, in terms of the Lerch transcendent, is:

which is once again valid for all complex values of *s*.

The functional equation extends the beta function to the left side of the complex plane Re(*s*)<0. It is given by

where Γ(*s*) is the gamma function.

Some special values include:

where *G* represents Catalan's constant, and

where in the above is an example of the polygamma function. More generally, for any positive integer *k*:

where represent the Euler numbers. For integer *k* ≥ 0, this extends to:

Hence, the function vanishes for all odd negative integral values of the argument.

s | approximate value β(s) | OEIS |
---|---|---|

1/5 | 0.5737108471859466493572665 | |

1/4 | 0.5907230564424947318659591 | |

1/3 | 0.6178550888488520660725389 | |

1/2 | 0.6676914571896091766586909 | A195103 |

1 | 0.7853981633974483096156608 | A003881 |

2 | 0.9159655941772190150546035 | A006752 |

3 | 0.9689461462593693804836348 | A153071 |

4 | 0.9889445517411053361084226 | A175572 |

5 | 0.9961578280770880640063194 | A175571 |

6 | 0.9986852222184381354416008 | A175570 |

7 | 0.9995545078905399094963465 | |

8 | 0.9998499902468296563380671 | |

9 | 0.9999496841872200898213589 | |

10 | 0.9999831640261968774055407 |

- Glasser, M. L. (1972). "The evaluation of lattice sums. I. Analytic procedures".
*J. Math. Phys.***14**: 409. DOI:10.1063/1.1666331. - J. Spanier and K. B. Oldham,
*An Atlas of Functions*, (1987) Hemisphere, New York. - Weisstein, Eric W., "Dirichlet Beta Function" from MathWorld.

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