Glaisher-Kinkelin constant
Encyclopedia
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 sum
s and integral
s, especially those involving Gamma function
s and zeta functions. It is named after mathematician
s James Whitbread Lee Glaisher
and Hermann Kinkelin.
Its approximate value is:
The Glaisher-Kinkelin constant
can be given by the limit:
where
is the K-function
. An equivalent form involving the Barnes G-function
, given by
where 
is the gamma function
is:
.
The Glaisher-Kinkelin constant also appears in the Riemann zeta function, such as:
where
is the Euler-Mascheroni constant
.
Some integrals involve this constant:
A series representation for this constant follows from a series for the Riemann zeta function given by Helmut Hasse
.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the Glaisher–Kinkelin constant or Glaisher's constant, typically denoted A, is a mathematical constant
Mathematical constant
A mathematical constant is a special number, usually a real number, that is "significantly interesting in some way". Constants arise in many different areas of mathematics, with constants such as and occurring in such diverse contexts as geometry, number theory and calculus.What it means for a...
, related to the K-function
K-function
In mathematics, the K-function, typically denoted K, is a generalization of the hyperfactorial to complex numbers, similar to the generalization of the factorial to the Gamma function.Formally, the K-function is defined as...
and the Barnes G-function
Barnes G-function
In mathematics, the Barnes G-function G is a function that is an extension of superfactorials to the complex numbers. It is related to the Gamma function, the K-function and the Glaisher-Kinkelin constant, and was named after mathematician Ernest William Barnes...
. The constant appears in a number of sum
SUM
SUM can refer to:* The State University of Management* Soccer United Marketing* Society for the Establishment of Useful Manufactures* StartUp-Manager* Software User’s Manual,as from DOD-STD-2 167A, and MIL-STD-498...
s and integral
Integral
Integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...
s, especially those involving Gamma function
Gamma function
In mathematics, the gamma function is an extension of the factorial function, with its argument shifted down by 1, to real and complex numbers...
s and zeta functions. It is named after mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
s James Whitbread Lee Glaisher
James Whitbread Lee Glaisher
James Whitbread Lee Glaisher son of James Glaisher, the meteorologist, was a prolific English mathematician.He was educated at St Paul's School and Trinity College, Cambridge, where he was second wrangler in 1871...
and Hermann Kinkelin.
Its approximate value is:
The Glaisher-Kinkelin constant
can be given by the limit:where
is the K-functionK-function
In mathematics, the K-function, typically denoted K, is a generalization of the hyperfactorial to complex numbers, similar to the generalization of the factorial to the Gamma function.Formally, the K-function is defined as...
. An equivalent form involving the Barnes G-function
Barnes G-function
In mathematics, the Barnes G-function G is a function that is an extension of superfactorials to the complex numbers. It is related to the Gamma function, the K-function and the Glaisher-Kinkelin constant, and was named after mathematician Ernest William Barnes...
, given by
where is the gamma functionGamma function
In mathematics, the gamma function is an extension of the factorial function, with its argument shifted down by 1, to real and complex numbers...
is:
.The Glaisher-Kinkelin constant also appears in the Riemann zeta function, such as:
where
is the Euler-Mascheroni constantEuler-Mascheroni constant
The Euler–Mascheroni constant is a mathematical constant recurring in analysis and number theory, usually denoted by the lowercase Greek letter ....
.
Some integrals involve this constant:
A series representation for this constant follows from a series for the Riemann zeta function given by Helmut Hasse
Helmut Hasse
Helmut Hasse was a German mathematician working in algebraic number theory, known for fundamental contributions to class field theory, the application of p-adic numbers to local classfield theory and diophantine geometry , and to local zeta functions.-Life:He was born in Kassel, and died in...
.