|
|
| Line 1: |
Line 1: |
| In [[mathematics]], a '''Barnes zeta function''' is a generalization of the [[Riemann zeta function]] introduced by {{harvs|txt|authorlink=E. W. Barnes|first=E. W. |last=Barnes|year=1901}}. It is further generalized by the [[Shintani zeta function]].
| | I like Urban exploration. <br>I also try to learn Spanish in my free time.<br><br>Feel free to visit my web site [http://ikzouhetnietweten.nl/index.php?mod=users&action=view&id=181498 Size guaide mountain bike sizing.] |
| | |
| ==Definition==
| |
| The Barnes zeta function is defined by
| |
| | |
| : <math>\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}</math>
| |
| where ''w'' and ''a''<sub>''j''</sub> have positive real part and ''s'' has real part greater than ''N''.
| |
| | |
| It has a [[meromorphic continuation]] to all complex ''s'', whose only [[Mathematical singularity|singularities]] are simple poles at ''s'' = 1, 2, ..., ''N''. For ''N'' = ''w'' = ''a''<sub>1</sub> = 1 it is the Riemann zeta function.
| |
| | |
| ==References== | |
| *{{Citation | last1=Barnes | first1=E. W. | title=The Theory of the Double Gamma Function. [Abstract] | jstor=116064 | publisher=The Royal Society | year=1899 | journal=Proceedings of the Royal Society of London | issn=0370-1662 | volume=66 | pages=265–268}}
| |
| *{{Citation | last1=Barnes | first1=E. W. | title=The Theory of the Double Gamma Function | jstor=90809 | publisher=The Royal Society | year=1901 | journal=Philosophical Transactions of the Royal Society of London. Series A, Containing Papers of a Mathematical or Physical Character | issn=0264-3952 | volume=196 | pages=265–387}}
| |
| *{{citation|first=E. W. |last=Barnes|title=On the theory of the multiple gamma function|journal= Trans. Cambridge Philos. Soc. |volume=19 |year=1904|pages=374–425}}
| |
| *{{Citation | last1=Friedman | first1=Eduardo | last2=Ruijsenaars | first2=Simon | title=Shintani–Barnes zeta and gamma functions | doi=10.1016/j.aim.2003.07.020 | mr=2078341 | year=2004 | journal=Advances in Mathematics | issn=0001-8708 | volume=187 | issue=2 | pages=362–395}}
| |
| *{{Citation | last1=Ruijsenaars | first1=S. N. M. | title=On Barnes' multiple zeta and gamma functions | doi=10.1006/aima.2000.1946 | mr=1800255 | year=2000 | journal=Advances in Mathematics | issn=0001-8708 | volume=156 | issue=1 | pages=107–132}}
| |
| | |
| [[Category:Zeta and L-functions]]
| |
I like Urban exploration.
I also try to learn Spanish in my free time.
Feel free to visit my web site Size guaide mountain bike sizing.