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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/10.02.20.0027.01









  


  








Input Form





Derivative[1, 0][Zeta][-4, a] == 24 PolyGamma[-5, -1 + a] - (1/180) (-1 + a) (-386 + 375 a (3 + (-3 + a) a)) + (1/30) a (-1 + a^2 (10 + 3 a (-5 + 2 a))) EulerGamma + (-1 + a)^4 Log[-1 + a] - 4 (-1 + a)^3 Log[Glaisher] - (1/2) (-1 + a)^4 Log[2 Pi] + (1/(12 Pi^4)) (9 (-2 (-1 + a)^2 Pi^2 Zeta[3] + Zeta[5]) + (-25 + 12 EulerGamma) Pi^4 Zeta[-4, a]) + 4 (-1 + a) Derivative[1][Zeta][-3] + Sum[(1/2) (a + k)^4 (2 Log[a + k] - Log[(a + k)^2]), {k, 0, Floor[1 - Re[a]]}]








Standard Form





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MathML Form







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</mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mi> k </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mi> log </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mi> k </mi> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <apply> <partialdiff /> <list> <cn type='integer'> 1 </cn> <cn type='integer'> 0 </cn> </list> <ci> Zeta </ci> </apply> <cn type='integer'> -4 </cn> <ci> a </ci> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 24 </cn> <apply> <ci> PolyGamma </ci> <cn type='integer'> -5 </cn> <apply> <plus /> <ci> a </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <ln /> <apply> <plus /> <ci> a </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> a </ci> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <ln /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> a </ci> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> 4 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <ln /> <ci> Glaisher </ci> </apply> <apply> <power /> <apply> <plus /> <ci> a </ci> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> 3 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='rational'> 1 <sep /> 180 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 375 </cn> <ci> a </ci> <apply> <plus /> <apply> <times /> <apply> <plus /> <ci> a </ci> <cn type='integer'> -3 </cn> </apply> <ci> a </ci> </apply> <cn type='integer'> 3 </cn> </apply> </apply> <cn type='integer'> -386 </cn> </apply> <apply> <plus /> <ci> a </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <ci> D </ci> <apply> <ci> Zeta </ci> <cn type='integer'> -3 </cn> </apply> <cn type='integer'> -3 </cn> </apply> <apply> <plus /> <ci> a </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 9 </cn> <apply> <plus /> <apply> <ci> Zeta </ci> <cn type='integer'> 5 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <plus /> <ci> a </ci> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <pi /> <cn type='integer'> 2 </cn> </apply> <apply> <ci> Zeta </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <plus /> <cn type='integer'> -25 </cn> <apply> <times /> <cn type='integer'> 12 </cn> <eulergamma /> </apply> </apply> <apply> <power /> <pi /> <cn type='integer'> 4 </cn> </apply> <apply> <ci> Zeta </ci> <cn type='integer'> -4 </cn> <ci> a </ci> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 12 </cn> <apply> <power /> <pi /> <cn type='integer'> 4 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 30 </cn> <ci> a </ci> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 3 </cn> <ci> a </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> a </ci> </apply> <cn type='integer'> -5 </cn> </apply> </apply> <cn type='integer'> 10 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> <eulergamma /> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <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> </uplimit> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <power /> <apply> <plus /> <ci> a </ci> <ci> k </ci> </apply> <cn type='integer'> 4 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ln /> <apply> <plus /> <ci> a </ci> <ci> k </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> log </ci> <apply> <power /> <apply> <plus /> <ci> a </ci> <ci> k </ci> </apply> <cn type='integer'> 2 </cn> </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