Special Values
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 |
Read more about this topic: Dirichlet Beta Function
Famous quotes containing the words special and/or values:
“... [the] special relation of women to children, in which the heart of the world has always felt there was something sacred, serves to impress upon women certain tendencies, to endow them with certain virtues ... which will render them of special value in public affairs.”
—Mary Putnam Jacobi (1842–1906)
“Postmodernity is the simultaneity of the destruction of earlier values and their reconstruction. It is renovation within ruination.”
—Jean Baudrillard (b. 1929)