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Barnes G-function

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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

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.

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.