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http://functions.wolfram.com/09.07.27.0001.02
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EllipticThetaPrime[3, z, E^(I Pi \[Tau])] ==
EllipticThetaPrime[3, z - Pi \[Tau], E^(I Pi \[Tau])]/
E^(I ((-Pi) \[Tau] + 2 z)) -
(2 I EllipticTheta[3, z - Pi \[Tau], E^(I Pi \[Tau])])/
E^(I ((-Pi) \[Tau] + 2 z)) /; Im[\[Tau]] > 0
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["EllipticThetaPrime", "[", RowBox[List["3", ",", "z", ",", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", "\[Tau]"]]]]], "]"]], "\[Equal]", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[Pi]"]], " ", "\[Tau]"]], "+", RowBox[List["2", " ", "z"]]]], ")"]]]]], " ", RowBox[List["EllipticThetaPrime", "[", RowBox[List["3", ",", RowBox[List["z", "-", RowBox[List["\[Pi]", " ", "\[Tau]"]]]], ",", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", "\[Tau]"]]]]], "]"]]]], "-", RowBox[List["2", " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[Pi]"]], " ", "\[Tau]"]], "+", RowBox[List["2", " ", "z"]]]], ")"]]]]], " ", "\[ImaginaryI]", " ", RowBox[List["EllipticTheta", "[", RowBox[List["3", ",", RowBox[List["z", "-", RowBox[List["\[Pi]", " ", "\[Tau]"]]]], ",", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", "\[Tau]"]]]]], "]"]]]]]]]], "/;", RowBox[List[RowBox[List["Im", "[", "\[Tau]", "]"]], ">", "0"]]]]]]
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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> <msubsup> <mi> ϑ </mi> <mn> 3 </mn> <mo> ′ </mo> </msubsup> <mo> ( </mo> <mrow> <mi> z </mi> <mo> , </mo> <msup> <mi> ⅇ </mi> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <mi> τ </mi> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <msup> <mi> ⅇ </mi> <mrow> <mrow> <mo> - </mo> <mi> ⅈ </mi> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> - </mo> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mi> τ </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </msup> <mo> ⁢ </mo> <mrow> <msubsup> <mi> ϑ </mi> <mn> 3 </mn> <mo> ′ </mo> </msubsup> <mo> ( </mo> <mrow> <mrow> <mi> z </mi> <mo> - </mo> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mi> τ </mi> </mrow> </mrow> <mo> , </mo> <msup> <mi> ⅇ </mi> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <mi> τ </mi> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msup> <mi> ⅇ </mi> <mrow> <mrow> <mo> - </mo> <mi> ⅈ </mi> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> - </mo> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mi> τ </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </msup> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <mrow> <msub> <mi> ϑ </mi> <mn> 3 </mn> </msub> <mo> ( </mo> <mrow> <mrow> <mi> z </mi> <mo> - </mo> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mi> τ </mi> </mrow> </mrow> <mo> , </mo> <msup> <mi> ⅇ </mi> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <mi> τ </mi> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> Im </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> τ </mi> <mo> ) </mo> </mrow> <mo> > </mo> <mn> 0 </mn> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <apply> <partialdiff /> <list> <cn type='integer'> 1 </cn> </list> <apply> <ci> Subscript </ci> <ci> ϑ </ci> <cn type='integer'> 3 </cn> </apply> </apply> <ci> z </ci> <apply> <power /> <exponentiale /> <apply> <times /> <imaginaryi /> <pi /> <ci> τ </ci> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <pi /> <ci> τ </ci> </apply> </apply> </apply> </apply> </apply> <apply> <apply> <partialdiff /> <list> <cn type='integer'> 1 </cn> </list> <apply> <ci> Subscript </ci> <ci> ϑ </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <pi /> <ci> τ </ci> </apply> </apply> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <imaginaryi /> <pi /> <ci> τ </ci> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <pi /> <ci> τ </ci> </apply> </apply> </apply> </apply> </apply> <imaginaryi /> <apply> <ci> EllipticTheta </ci> <cn type='integer'> 3 </cn> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <pi /> <ci> τ </ci> </apply> </apply> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <imaginaryi /> <pi /> <ci> τ </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <gt /> <apply> <imaginary /> <ci> τ </ci> </apply> <cn type='integer'> 0 </cn> </apply> </apply> </annotation-xml> </semantics> </math>
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Date Added to functions.wolfram.com (modification date)
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){kind=small}
