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.

Plot of the Barnes G function G(z) in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D
Plot of the Barnes G aka double gamma function G(z) in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D
The Barnes G function along part of the real axis

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.

Functional equation and integer arguments

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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.

Characterisation

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Similar to the Bohr–Mollerup theorem for the gamma function, for a constant  , we have for  [4]

 

and for  

 

as  .

Reflection formula

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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):

 

Taylor series expansion

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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:

 

Multiplication formula

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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.

Absolute value

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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.

Asymptotic expansion

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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.

Relation to the log-gamma integral

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The parametric log-gamma can be evaluated in terms of the Barnes G-function:[5]

 
A proof of the formula

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,

 

References

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  1. ^ E. W. Barnes, "The theory of the G-function", Quarterly Journ. Pure and Appl. Math. 31 (1900), 264–314.
  2. ^ M. F. Vignéras, L'équation fonctionelle de la fonction zêta de Selberg du groupe mudulaire SL , Astérisque 61, 235–249 (1979).
  3. ^ Park, Junesang (1996). "A duplication formula for the double gamma function $Gamma_2$". Bulletin of the Korean Mathematical Society. 33 (2): 289–294.
  4. ^ Marichal, Jean Luc. A Generalization of Bohr-Mollerup's Theorem for Higher Order Convex Functions (PDF). Springer. p. 218.
  5. ^ a b Adamchik, Viktor S. (2003). "Contributions to the Theory of the Barnes function". arXiv:math/0308086.
  6. ^ I. Vardi, Determinants of Laplacians and multiple gamma functions, SIAM J. Math. Anal. 19, 493–507 (1988).
  7. ^ E. T. Whittaker and G. N. Watson, "A Course of Modern Analysis", CUP.