In mathematics, the Barnes G-function G(z) 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. [1] It can be written in terms of the double gamma function.
Formally, the Barnes G-function is defined in the following Weierstrass product form:
where is the Euler–Mascheroni constant, exp(x) = ex is the exponential function, and Π denotes multiplication ( capital pi notation).
The integral representation, which may be deduced from the relation to the double gamma function, is
As an entire function, G is of order two, and of infinite type. This can be deduced from the asymptotic expansion given below.
The Barnes G-function satisfies the functional equation
with normalisation G(1) = 1. Note the similarity between the functional equation of the Barnes G-function and that of the Euler gamma function:
The functional equation implies that G takes the following values at integer arguments:
(in particular, ) and thus
where denotes the gamma function and K denotes the K-function. The functional equation uniquely defines the Barnes G-function if the convexity condition,
is added. [2] Additionally, the Barnes G-function satisfies the duplication formula, [3]
where is the Glaisher–Kinkelin constant.
Similar to the Bohr–Mollerup theorem for the gamma function, for a constant , we have for [4]
and for
as .
The difference equation for the G-function, in conjunction with the functional equation for the gamma function, can be used to obtain the following reflection formula for the Barnes G-function (originally proved by Hermann Kinkelin):
The log-tangent integral on the right-hand side can be evaluated in terms of the Clausen function (of order 2), as is shown below:
The proof of this result hinges on the following evaluation of the cotangent integral: introducing the notation for the log-cotangent integral, and using the fact that , an integration by parts gives
Performing the integral substitution gives
The Clausen function – of second order – has the integral representation
However, within the interval , the absolute value sign within the integrand can be omitted, since within the range the 'half-sine' function in the integral is strictly positive, and strictly non-zero. Comparing this definition with the result above for the logtangent integral, the following relation clearly holds:
Thus, after a slight rearrangement of terms, the proof is complete:
Using the relation and dividing the reflection formula by a factor of gives the equivalent form:
Adamchik (2003) has given an equivalent form of the reflection formula, but with a different proof. [5]
Replacing z with 1/2 − z in the previous reflection formula gives, after some simplification, the equivalent formula shown below (involving Bernoulli polynomials):
By Taylor's theorem, and considering the logarithmic derivatives of the Barnes function, the following series expansion can be obtained:
It is valid for . Here, is the Riemann zeta function:
Exponentiating both sides of the Taylor expansion gives:
Comparing this with the Weierstrass product form of the Barnes function gives the following relation:
Like the gamma function, the G-function also has a multiplication formula: [6]
where is a constant given by:
Here is the derivative of the Riemann zeta function and is the Glaisher–Kinkelin constant.
It holds true that , thus . From this relation and by the above presented Weierstrass product form one can show that
This relation is valid for arbitrary , and . If , then the below formula is valid instead:
for arbitrary real y.
The logarithm 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 [7] 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.
The parametric log-gamma can be evaluated in terms of the Barnes G-function: [5]
The proof is somewhat indirect, and involves first considering the logarithmic difference of the gamma function and Barnes G-function:
where
and is the Euler–Mascheroni constant.
Taking the logarithm of the Weierstrass product forms of the Barnes G-function and gamma function gives:
A little simplification and re-ordering of terms gives the series expansion:
Finally, take the logarithm of the Weierstrass product form of the gamma function, and integrate over the interval to obtain:
Equating the two evaluations completes the proof:
And since then,
In mathematics, the Barnes G-function G(z) 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. [1] It can be written in terms of the double gamma function.
Formally, the Barnes G-function is defined in the following Weierstrass product form:
where is the Euler–Mascheroni constant, exp(x) = ex is the exponential function, and Π denotes multiplication ( capital pi notation).
The integral representation, which may be deduced from the relation to the double gamma function, is
As an entire function, G is of order two, and of infinite type. This can be deduced from the asymptotic expansion given below.
The Barnes G-function satisfies the functional equation
with normalisation G(1) = 1. Note the similarity between the functional equation of the Barnes G-function and that of the Euler gamma function:
The functional equation implies that G takes the following values at integer arguments:
(in particular, ) and thus
where denotes the gamma function and K denotes the K-function. The functional equation uniquely defines the Barnes G-function if the convexity condition,
is added. [2] Additionally, the Barnes G-function satisfies the duplication formula, [3]
where is the Glaisher–Kinkelin constant.
Similar to the Bohr–Mollerup theorem for the gamma function, for a constant , we have for [4]
and for
as .
The difference equation for the G-function, in conjunction with the functional equation for the gamma function, can be used to obtain the following reflection formula for the Barnes G-function (originally proved by Hermann Kinkelin):
The log-tangent integral on the right-hand side can be evaluated in terms of the Clausen function (of order 2), as is shown below:
The proof of this result hinges on the following evaluation of the cotangent integral: introducing the notation for the log-cotangent integral, and using the fact that , an integration by parts gives
Performing the integral substitution gives
The Clausen function – of second order – has the integral representation
However, within the interval , the absolute value sign within the integrand can be omitted, since within the range the 'half-sine' function in the integral is strictly positive, and strictly non-zero. Comparing this definition with the result above for the logtangent integral, the following relation clearly holds:
Thus, after a slight rearrangement of terms, the proof is complete:
Using the relation and dividing the reflection formula by a factor of gives the equivalent form:
Adamchik (2003) has given an equivalent form of the reflection formula, but with a different proof. [5]
Replacing z with 1/2 − z in the previous reflection formula gives, after some simplification, the equivalent formula shown below (involving Bernoulli polynomials):
By Taylor's theorem, and considering the logarithmic derivatives of the Barnes function, the following series expansion can be obtained:
It is valid for . Here, is the Riemann zeta function:
Exponentiating both sides of the Taylor expansion gives:
Comparing this with the Weierstrass product form of the Barnes function gives the following relation:
Like the gamma function, the G-function also has a multiplication formula: [6]
where is a constant given by:
Here is the derivative of the Riemann zeta function and is the Glaisher–Kinkelin constant.
It holds true that , thus . From this relation and by the above presented Weierstrass product form one can show that
This relation is valid for arbitrary , and . If , then the below formula is valid instead:
for arbitrary real y.
The logarithm 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 [7] 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.
The parametric log-gamma can be evaluated in terms of the Barnes G-function: [5]
The proof is somewhat indirect, and involves first considering the logarithmic difference of the gamma function and Barnes G-function:
where
and is the Euler–Mascheroni constant.
Taking the logarithm of the Weierstrass product forms of the Barnes G-function and gamma function gives:
A little simplification and re-ordering of terms gives the series expansion:
Finally, take the logarithm of the Weierstrass product form of the gamma function, and integrate over the interval to obtain:
Equating the two evaluations completes the proof:
And since then,