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http://functions.wolfram.com/10.04.04.0011.01
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Limit[RiemannSiegelZ[x + \[Epsilon]], \[Epsilon] -> Plus[0]] ==
RiemannSiegelZ[x] E^(I Pi Floor[1/4 - (I x)/2]) /; I x > 1/2
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Limit", "[", RowBox[List[RowBox[List["RiemannSiegelZ", "[", RowBox[List["x", "+", "\[Epsilon]"]], "]"]], ",", RowBox[List["\[Epsilon]", "\[Rule]", RowBox[List["+", "0"]]]]]], "]"]], "\[Equal]", " ", RowBox[List[RowBox[List["RiemannSiegelZ", "[", "x", "]"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "4"], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "x"]], "2"]]], "]"]]]]]]]]], "/;", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "x"]], ">", FractionBox["1", "2"]]]]]]]
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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <munder> <mi> lim </mi> <mrow> <mi> ϵ </mi> <semantics> <mo> → </mo> <annotation encoding='Mathematica'> "\[Rule]" </annotation> </semantics> <mrow> <mo> + </mo> <mn> 0 </mn> </mrow> </mrow> </munder> <mo> ⁢ </mo> <mtext>   </mtext> <mrow> <semantics> <mi> Z </mi> <annotation encoding='Mathematica'> TagBox["Z", RiemannSiegelZ] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> x </mi> <mo> + </mo> <mi> ϵ </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <semantics> <mi> Z </mi> <annotation encoding='Mathematica'> TagBox["Z", RiemannSiegelZ] </annotation> </semantics> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> exp </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <mrow> <mo> ⌊ </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> <mo> - </mo> <mfrac> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> x </mi> </mrow> <mn> 2 </mn> </mfrac> </mrow> <mo> ⌋ </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> x </mi> </mrow> <mo> > </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <limit /> <bvar> <ci> ϵ </ci> </bvar> <condition> <apply> <tendsto /> <ci> ϵ </ci> <apply> <plus /> <cn type='integer'> 0 </cn> </apply> </apply> </condition> <apply> <ci> RiemannSiegelZ </ci> <apply> <plus /> <ci> x </ci> <ci> ϵ </ci> </apply> </apply> </apply> <apply> <times /> <apply> <ci> RiemannSiegelZ </ci> <ci> x </ci> </apply> <apply> <exp /> <apply> <times /> <imaginaryi /> <pi /> <apply> <floor /> <apply> <plus /> <cn type='rational'> 1 <sep /> 4 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <ci> x </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <gt /> <apply> <times /> <imaginaryi /> <ci> x </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Limit", "[", RowBox[List[RowBox[List["RiemannSiegelZ", "[", RowBox[List["x_", "+", "\[Epsilon]_"]], "]"]], ",", RowBox[List["\[Epsilon]_", "\[Rule]", RowBox[List["+", "0"]]]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["RiemannSiegelZ", "[", "x", "]"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "4"], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "x"]], "2"]]], "]"]]]]]]], "/;", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "x"]], ">", FractionBox["1", "2"]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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