In mathematics, the Glaisher–Kinkelin constant or Glaisher's constant, typically denoted A, is a mathematical constant, related to the K-function and the Barnes G-function. The constant appears in a number of sums and integrals, especially those involving Gamma functions and zeta functions. It is named after mathematicians James Whitbread Lee Glaisher and Hermann Kinkelin.
Its approximate value is:
The Glaisher–Kinkelin constant can be given by the limit:
which shows that just as π is obtained from approximation of the function , A can also be obtained from a similar approximation to the function .
An equivalent definition for A involving the Barnes G-function, given by where is the gamma function is:
The Glaisher–Kinkelin constant also appears in evaluations of the derivatives of the Riemann zeta function, such as:
The following are some integrals that involve this constant:
A series representation for this constant follows from a series for the Riemann zeta function given by Helmut Hasse.
- Guillera, Jesus; Sondow, Jonathan (2005). "Double integrals and infinite products for some classical constants via analytic continuations of Lerch's transcendent". The Ramanujan Journal. 16 (3): 247–270. arXiv:math.NT/0506319. doi:10.1007/s11139-007-9102-0.
- Guillera, Jesus; Sondow, Jonathan (2008). "Double integrals and infinite products for some classical constants via analytic continuations of Lerch's transcendent". Ramanujan Journal. 16 (3): 247&ndash, 270. arXiv:math/0506319. doi:10.1007/s11139-007-9102-0. (Provides a variety of relationships.)
- Weisstein, Eric W. "Glaisher–Kinkelin Constant". MathWorld.
- Weisstein, Eric W. "Riemann Zeta Function". MathWorld.