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variants of this functions
Zeta




Mathematica Notation

Traditional Notation









Zeta Functions and Polylogarithms > Zeta[s,a] > Differentiation > Low-order differentiation > With respect to s > For zeta(s,a) > Derivatives at negarive integer points





http://functions.wolfram.com/10.02.20.0024.01









  


  








Input Form





Derivative[1, 0][Zeta][-2, a] == (1/(12 Pi^2)) (2 (-9 + a (18 + 3 a (-6 + EulerGamma) + EulerGamma + 2 a^2 EulerGamma)) Pi^2 - 3 Zeta[3] + Pi^2 ((1 + 6 a (1 + a)) (-3 + 2 EulerGamma) Floor[-Re[a]] + 3 (1 + 2 a) (-3 + 2 EulerGamma) Floor[-Re[a]]^2 + (-6 + 4 EulerGamma) Floor[-Re[a]]^3 + 6 ((-1 + a) (Log[2] + 2 (-1 + a) Log[-1 + a] - 4 Log[Glaisher] + Log[Pi] - a Log[2 Pi]) + 4 PolyGamma[-3, -1 + a] + 2 Sum[(1/2) (a + k)^2 (2 Log[a + k] - Log[(a + k)^2]), {k, 0, Floor[-Re[a]]}] + (-3 + 2 EulerGamma) Zeta[-2, a + Max[0, Floor[1 - Re[a]]]])))








Standard Form





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MathML Form







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<apply> <ln /> <apply> <plus /> <ci> a </ci> <ci> k </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ln /> <apply> <power /> <apply> <plus /> <ci> a </ci> <ci> k </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <plus /> <cn type='integer'> -3 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <eulergamma /> </apply> </apply> <apply> <ci> Zeta </ci> <cn type='integer'> -2 </cn> <apply> <plus /> <ci> a </ci> <apply> <max /> <cn type='integer'> 0 </cn> <apply> <floor /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <real /> <ci> a </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





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Date Added to functions.wolfram.com (modification date)





2007-05-02