Barnes G-function
Encyclopedia
In mathematics
, the Barnes G-function G(z) is a function
that is an extension of superfactorials to the complex number
s. It is related to the Gamma function
, the K-function
and the Glaisher-Kinkelin constant
, and was named after mathematician
Ernest William Barnes
. Up to elementary factors, it is a special case of the double gamma function.
Formally, the Barnes G-function is defined (in the form of a Weierstrass product) as
where γ is the Euler-Mascheroni constant
, exp(x) = ex, and ∏ is capital pi notation.
with normalisation G(1)=1. The difference equation implies that G takes the following values at integer
arguments:
and thus
where Γ denotes the Gamma function
and K denotes the K-function
. The difference equation uniquely defines the G function if the convexity condition:
is added.
The difference equation for the G function and the functional equation
for the Gamma function
yield the following functional equation
for the G function, originally proved by Hermann Kinkelin:
where
is a constant given by:
Here
is the derivative of the Riemann zeta function and 
is the Glaisher-Kinkelin constant
.
of G(z + 1) has the following asymptotic expansion, as established by Barnes:
Here the
are the Bernoulli numbers and 
is the Glaisher-Kinkelin constant
. (Note that somewhat confusingly at the time of Barnes the Bernoulli number
would have been written as 
, but this convention is no longer current.) This expansion is valid for 
in any sector not containing the negative real axis with 
large.
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 Barnes G-function G(z) is a function
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
that is an extension of superfactorials to the complex number
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
s. It is related to the 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...
, 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 Glaisher-Kinkelin constant
Glaisher-Kinkelin constant
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...
, and was 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....
Ernest William Barnes
Ernest William Barnes
Ernest William Barnes FRS was an English mathematician and scientist who later became a theologian and bishop....
. Up to elementary factors, it is a special case of the double gamma function.
Formally, the Barnes G-function is defined (in the form of a Weierstrass product) as
where γ is the Euler-Mascheroni constant
Euler-Mascheroni constant
The Euler–Mascheroni constant is a mathematical constant recurring in analysis and number theory, usually denoted by the lowercase Greek letter ....
, exp(x) = ex, and ∏ is capital pi notation.
Difference equation, functional equation and special values
The Barnes G-function satisfies the difference equationwith normalisation G(1)=1. The difference equation implies that G takes the following values at integer
Integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
arguments:
and thus
where Γ denotes the 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...
and K denotes 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...
. The difference equation uniquely defines the G function if the convexity condition:
is added.The difference equation for the G function and the functional equation
Functional equation
In mathematics, a functional equation is any equation that specifies a function in implicit form.Often, the equation relates the value of a function at some point with its values at other points. For instance, properties of functions can be determined by considering the types of functional...
for the 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...
yield the following functional equation
Functional equation
In mathematics, a functional equation is any equation that specifies a function in implicit form.Often, the equation relates the value of a function at some point with its values at other points. For instance, properties of functions can be determined by considering the types of functional...
for the G function, originally proved by Hermann Kinkelin:
Multiplication formula
Like the Gamma function, the G-function also has a multiplication formula:where
is a constant given by:Here
is the derivative of the Riemann zeta function and is the Glaisher-Kinkelin constantGlaisher-Kinkelin constant
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...
.
Asymptotic expansion
The logarithmLogarithm
The logarithm of a number is the exponent by which another fixed value, the base, has to be raised to produce that number. For example, the logarithm of 1000 to base 10 is 3, because 1000 is 10 to the power 3: More generally, if x = by, then y is the logarithm of x to base b, and is written...
of G(z + 1) has the following asymptotic expansion, as established by Barnes:
Here the
are the Bernoulli numbers and is the Glaisher-Kinkelin constantGlaisher-Kinkelin constant
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...
. (Note that somewhat confusingly at the time of Barnes the Bernoulli number
Bernoulli number
In mathematics, the Bernoulli numbers Bn are a sequence of rational numbers with deep connections to number theory. They are closely related to the values of the Riemann zeta function at negative integers....
would have been written as , but this convention is no longer current.) This expansion is valid for in any sector not containing the negative real axis with large.