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http://functions.wolfram.com/09.06.27.0026.01
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EllipticThetaPrime[2, z, q] ==
(q^(1/4 + m + m^2) ((-I) (1 + 2 m) EllipticTheta[4,
z + (1/2) (1 + 2 m) (Pi + I Log[q]), q] + EllipticThetaPrime[4,
z + (1/2) (1 + 2 m) (Pi + I Log[q]), q]))/E^(I (1 + 2 m) z) /;
Element[m, Integers]
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["EllipticThetaPrime", "[", RowBox[List["2", ",", "z", ",", "q"]], "]"]], "\[Equal]", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "m"]]]], ")"]], " ", "z"]]], " ", SuperscriptBox["q", RowBox[List[FractionBox["1", "4"], "+", "m", "+", SuperscriptBox["m", "2"]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "m"]]]], ")"]], " ", RowBox[List["EllipticTheta", "[", RowBox[List["4", ",", RowBox[List["z", "+", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "m"]]]], ")"]], " ", RowBox[List["(", RowBox[List["\[Pi]", "+", RowBox[List["\[ImaginaryI]", " ", RowBox[List["Log", "[", "q", "]"]]]]]], ")"]]]]]], ",", "q"]], "]"]]]], "+", RowBox[List["EllipticThetaPrime", "[", RowBox[List["4", ",", RowBox[List["z", "+", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "m"]]]], ")"]], " ", RowBox[List["(", RowBox[List["\[Pi]", "+", RowBox[List["\[ImaginaryI]", " ", RowBox[List["Log", "[", "q", "]"]]]]]], ")"]]]]]], ",", "q"]], "]"]]]], ")"]]]]]], "/;", RowBox[List["m", "\[Element]", "Integers"]]]]]]
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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> 2 </mn> <mo> ′ </mo> </msubsup> <mo> ( </mo> <mrow> <mi> z </mi> <mo> , </mo> <mi> q </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <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> m </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mi> z </mi> </mrow> </msup> <mo> ⁢ </mo> <msup> <mi> q </mi> <mrow> <msup> <mi> m </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mi> m </mi> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msubsup> <mi> ϑ </mi> <mn> 4 </mn> <mo> ′ </mo> </msubsup> <mo> ( </mo> <mrow> <mrow> <mi> z </mi> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> m </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> q </mi> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mi> π </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> , </mo> <mi> q </mi> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> m </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msub> <mi> ϑ </mi> <mn> 4 </mn> </msub> <mo> ( </mo> <mrow> <mrow> <mi> z </mi> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> m </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> q </mi> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mi> π </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> , </mo> <mi> q </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> m </mi> <mo> ∈ </mo> <semantics> <mi> ℤ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] </annotation> </semantics> </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'> 2 </cn> </apply> </apply> <ci> z </ci> <ci> q </ci> </apply> <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> m </ci> </apply> <cn type='integer'> 1 </cn> </apply> <ci> z </ci> </apply> </apply> <apply> <power /> <ci> q </ci> <apply> <plus /> <apply> <power /> <ci> m </ci> <cn type='integer'> 2 </cn> </apply> <ci> m </ci> <cn type='rational'> 1 <sep /> 4 </cn> </apply> </apply> <apply> <plus /> <apply> <apply> <partialdiff /> <list> <cn type='integer'> 1 </cn> </list> <apply> <ci> Subscript </ci> <ci> ϑ </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> m </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <apply> <times /> <imaginaryi /> <apply> <ln /> <ci> q </ci> </apply> </apply> <pi /> </apply> </apply> </apply> <ci> q </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> m </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <ci> EllipticTheta </ci> <cn type='integer'> 4 </cn> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> m </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <apply> <times /> <imaginaryi /> <apply> <ln /> <ci> q </ci> </apply> </apply> <pi /> </apply> </apply> </apply> <ci> q </ci> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <in /> <ci> m </ci> <integers /> </apply> </apply> </annotation-xml> </semantics> </math>
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Date Added to functions.wolfram.com (modification date)
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