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RiemannSiegelZ






Mathematica Notation

Traditional Notation









Zeta Functions and Polylogarithms > RiemannSiegelZ[z] > Series representations > Generalized power series > Expansions at z==i/2+2i k





http://functions.wolfram.com/10.04.06.0006.01









  


  










Input Form





RiemannSiegelZ[z] \[Proportional] ((-1)^k 2^(-(1/2) - 2 k) Pi^(1/4 - k) Sqrt[I (-(I/2) - 2 I k + z)] (2 k)! Product[2/Sqrt[1 + 4 j + 2 I z], {j, 0, k - 1}] Zeta[1 + 2 k] (1 - (1/(4 Zeta[1 + 2 k])) (I (-(I/2) - 2 I k + z) ((EulerGamma - 2 Log[4 Pi] - PolyGamma[1/2 + k] + 4 PolyGamma[1 + 2 k]) Zeta[1 + 2 k] + 4 Derivative[1][Zeta][1 + 2 k])) - ((-(I/2) - 2 I k + z)^2 (Zeta[1 + 2 k] (EulerGamma^2 - Pi^2 + 4 EulerGamma Log[Pi] + 4 Log[Pi]^2 - 8 EulerGamma Log[2 Pi] - 16 Log[Pi] Log[2 Pi] + 16 Log[2 Pi]^2 + PolyGamma[1/2 + k]^2 + PolyGamma[1/2 + k] (-2 EulerGamma + Log[256] + 4 Log[Pi] - 8 PolyGamma[1 + 2 k]) + 8 (EulerGamma - Log[16] - 2 Log[Pi]) PolyGamma[1 + 2 k] + 16 PolyGamma[1 + 2 k]^2 + 16 PolyGamma[1, 1 + 2 k] - 2 Zeta[2, 1/2 + k]) + 8 (EulerGamma - Log[16] - 2 Log[Pi] - PolyGamma[1/2 + k] + 4 PolyGamma[1 + 2 k]) Derivative[1][Zeta][1 + 2 k] + 16 Derivative[2][Zeta][1 + 2 k]))/(32 Zeta[1 + 2 k])))/ E^((1/2) LogGamma[1/2 + k]) + O[(-(I/2) - 2 I k + z)^3] /; (z -> I/2 + 2 I k) && Element[k, Integers] && k > 0










Standard Form





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MathML Form







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encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mn> 256 </mn> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> &#960; </mi> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mn> 8 </mn> <mo> &#8290; </mo> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> k </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <semantics> <mi> &#8509; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubledGamma]&quot;, 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<apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <ci> k </ci> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <imaginaryi /> <apply> <power /> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <ci> Zeta </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <ci> PolyGamma </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> 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Rule Form





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





2001-10-29





© 1998- Wolfram Research, Inc.