Barnes zeta function

Barnes zeta function

In mathematics, a Barnes zeta function is a generalization of the Riemann zeta function introduced by E. W. Barnes (1901). It is further generalized by the Shintani zeta function.

Definition

The Barnes zeta function is defined by

\zeta_N(s,w|a_1,...,a_N)=\sum_{n_1,\dots,n_N\ge 0}\frac{1}{(w+n_1a_1+\cdots+n_Na_N)^s}

where w and aj have positive real part and s has real part greater than N.

It has a meromorphic continuation to all complex s, whose only singularities are simple poles at s = 1, 2, ..., N. For N = w = a1 = 1 it is the Riemann zeta function.

References

  • Barnes, E. W. (1899), "The Theory of the Double Gamma Function. [Abstract]", Proceedings of the Royal Society of London (The Royal Society) 66: 265–268, ISSN 0370-1662, JSTOR 116064 
  • Barnes, E. W. (1901), "The Theory of the Double Gamma Function", Philosophical Transactions of the Royal Society of London. Series A, Containing Papers of a Mathematical or Physical Character (The Royal Society) 196: 265–387, ISSN 0264-3952, JSTOR 90809 
  • Barnes, E. W. (1904), "On the theory of the multiple gamma function", Trans. Cambridge Philos. Soc. 19: 374–425 
  • Friedman, Eduardo; Ruijsenaars, Simon (2004), "Shintani–Barnes zeta and gamma functions", Advances in Mathematics 187 (2): 362–395, doi:10.1016/j.aim.2003.07.020, ISSN 0001-8708, MR2078341 
  • Ruijsenaars, S. N. M. (2000), "On Barnes' multiple zeta and gamma functions", Advances in Mathematics 156 (1): 107–132, doi:10.1006/aima.2000.1946, ISSN 0001-8708, MR1800255 

Wikimedia Foundation. 2010.

Игры ⚽ Поможем написать реферат

Look at other dictionaries:

  • Barnes G-function — In mathematics, the Barnes G function (typically denoted 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… …   Wikipedia

  • Multiple zeta function — For a different but related multiple zeta function, see Barnes zeta function. In mathematics, the multiple zeta functions generalisations of the Riemann zeta function, defined by and converge when Re(s1)+...+Re(si) > i for all i.… …   Wikipedia

  • Riemann zeta function — ζ(s) in the complex plane. The color of a point s encodes the value of ζ(s): dark colors denote values close to zero and hue encodes the value s argument. The white spot at s = 1 is the pole of the zeta function; the black spots on the… …   Wikipedia

  • Barnes'sche G-Funktion — Die Barnessche G Funktion, typischerweise mit G(z) bezeichnet, ist eine Funktion, die eine Erweiterung der Superfakultäten auf die komplexen Zahlen darstellt. Sie steht in Beziehung zur Gammafunktion, der K Funktion und der Konstanten von… …   Deutsch Wikipedia

  • Multiple gamma function — For derivatives of the log of the gamma function, see polygamma function. In mathematics, the multiple gamma function ΓN is a generalization of the Euler Gamma function and the Barnes G function. The double gamma function was studied Barnes… …   Wikipedia

  • Gamma function — For the gamma function of ordinals, see Veblen function. The gamma function along part of the real axis In mathematics, the gamma function (represented by the capital Greek letter Γ) is an extension of the factorial function, with its… …   Wikipedia

  • Z function — In mathematics, the Z function is a function used for studying the Riemann zeta function along the critical line where the real part of the argument is one half. It is also called the Riemann Siegel Z function, the Riemann Siegel zeta function,… …   Wikipedia

  • Digamma function — For Barnes s gamma function of 2 variables, see double gamma function. Digamma function ψ(s) in the complex plane. The color of a point s encodes the value of ψ(s). Strong colors denote values close to zero and hue encodes the value s argument …   Wikipedia

  • K-function — In mathematics, the K function, typically denoted K ( z ), 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:K(z)=(2pi)^{( z… …   Wikipedia

  • Дзета-функции — Эта страница информационный список. См. также основную статью: Дзета функция Римана В математике дзета функция обычно это функция родственная или аналогичная дзета функции Римана …   Википедия

Share the article and excerpts

Direct link
… Do a right-click on the link above
and select “Copy Link”