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variants of this functions
Zeta






Mathematica Notation

Traditional Notation









Zeta Functions and Polylogarithms > Zeta[s,a] > Representations through equivalent functions > Interrelations





http://functions.wolfram.com/10.02.27.0004.01









  


  










Input Form





Zeta[s, a] == (1 + UnitStep[-Re[a]] (E^((2 UnitStep[Im[a]] - 1) Pi I s) - 1)) ZetaClassical[s, a] + UnitStep[-Re[a]] (1 - E^((2 UnitStep[Im[a]] - 1) Pi I s)) (ZetaClassical[s, a + Floor[-Re[a]] + 1] + (UnitStep[Im[a]] (1 + Floor[-Re[a]] + Floor[Re[a]]))/ ((Floor[-Re[a]] + a)^2)^(s/2))










Standard Form





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MathML Form







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</mo> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <msup> <mi> &#8519; </mi> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <semantics> <mi> &#952; </mi> <annotation-xml encoding='MathML-Content'> <ci> UnitStep </ci> </annotation-xml> </semantics> <mo> ( </mo> <mrow> <mi> Im </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> s </mi> </mrow> </msup> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <semantics> <mi> &#952; </mi> <annotation-xml encoding='MathML-Content'> <ci> UnitStep </ci> </annotation-xml> </semantics> <mo> ( </mo> <mrow> <mo> - </mo> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mover> <mi> &#950; </mi> <mo> ^ </mo> </mover> <mo> ( </mo> <mrow> <mi> s </mi> <mo> , </mo> <mi> a </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> Zeta </ci> <ci> s </ci> <ci> a </ci> </apply> <apply> <plus /> <apply> <times /> <apply> <plus /> <apply> <times /> <apply> <plus /> <apply> <floor /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <real /> <ci> a </ci> </apply> </apply> </apply> <apply> <floor /> <apply> <real /> <ci> a </ci> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <ci> UnitStep </ci> <apply> <imaginary /> <ci> a </ci> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <power /> <apply> <plus /> <ci> a </ci> <apply> <floor /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <real /> <ci> a </ci> </apply> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <ci> s </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <apply> <ci> OverHat </ci> <ci> &#950; 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</ci> </apply> <ci> s </ci> <ci> a </ci> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Zeta", "[", RowBox[List["s_", ",", "a_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["1", "+", RowBox[List[RowBox[List["UnitStep", "[", RowBox[List["-", RowBox[List["Re", "[", "a", "]"]]]], "]"]], " ", RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", RowBox[List["UnitStep", "[", RowBox[List["Im", "[", "a", "]"]], "]"]]]], "-", "1"]], ")"]], " ", "\[Pi]", " ", "\[ImaginaryI]", " ", "s"]]], "-", "1"]], ")"]]]]]], ")"]], " ", RowBox[List["ZetaClassical", "[", RowBox[List["s", ",", "a"]], "]"]]]], "+", RowBox[List[RowBox[List["UnitStep", "[", RowBox[List["-", RowBox[List["Re", "[", "a", "]"]]]], "]"]], " ", RowBox[List["(", RowBox[List["1", "-", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", RowBox[List["UnitStep", "[", RowBox[List["Im", "[", "a", "]"]], "]"]]]], "-", "1"]], ")"]], " ", "\[Pi]", " ", "\[ImaginaryI]", " ", "s"]]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["ZetaClassical", "[", RowBox[List["s", ",", RowBox[List["a", "+", RowBox[List["Floor", "[", RowBox[List["-", RowBox[List["Re", "[", "a", "]"]]]], "]"]], "+", "1"]]]], "]"]], "+", FractionBox[RowBox[List[RowBox[List["UnitStep", "[", RowBox[List["Im", "[", "a", "]"]], "]"]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["Floor", "[", RowBox[List["-", RowBox[List["Re", "[", "a", "]"]]]], "]"]], "+", RowBox[List["Floor", "[", RowBox[List["Re", "[", "a", "]"]], "]"]]]], ")"]]]], SuperscriptBox[RowBox[List["(", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["Floor", "[", RowBox[List["-", RowBox[List["Re", "[", "a", "]"]]]], "]"]], "+", "a"]], ")"]], "2"], ")"]], RowBox[List["s", "/", "2"]]]]]], ")"]]]]]]]]]]










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





2007-05-02