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http://functions.wolfram.com/09.02.16.0007.01
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EllipticTheta[2, z, q] == (((-1)^(1/4) E^(z^2/Log[q]) Sqrt[Pi])/
Sqrt[(-I) Log[q]]) EllipticTheta[4, (I Pi z)/Log[q], E^(Pi^2/Log[q])]
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Cell[BoxData[RowBox[List[RowBox[List["EllipticTheta", "[", RowBox[List["2", ",", "z", ",", "q"]], "]"]], "\[Equal]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]], " ", SuperscriptBox["\[ExponentialE]", FractionBox[SuperscriptBox["z", "2"], RowBox[List["Log", "[", "q", "]"]]]], " ", SqrtBox["\[Pi]"]]], SqrtBox[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", RowBox[List["Log", "[", "q", "]"]]]]]], RowBox[List["EllipticTheta", "[", RowBox[List["4", ",", FractionBox[RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", "z"]], RowBox[List["Log", "[", "q", "]"]]], ",", SuperscriptBox["\[ExponentialE]", FractionBox[SuperscriptBox["\[Pi]", "2"], RowBox[List["Log", "[", "q", "]"]]]]]], "]"]]]]]]]]
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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> <msub> <mi> ϑ </mi> <mn> 2 </mn> </msub> <mo> ( </mo> <mrow> <mi> z </mi> <mo> , </mo> <mi> q </mi> </mrow> <mo> ) </mo> </mrow> <mo>  </mo> <mrow> <mfrac> <mrow> <mroot> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mn> 4 </mn> </mroot> <mo> ⁢ </mo> <mtext> </mtext> <msqrt> <mi> π </mi> </msqrt> <mo> ⁢ </mo> <msup> <mi> ⅇ </mi> <mfrac> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> q </mi> <mo> ) </mo> </mrow> </mfrac> </msup> </mrow> <msqrt> <mrow> <mrow> <mo> - </mo> <mi> ⅈ </mi> </mrow> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> q </mi> <mo> ) </mo> </mrow> </mrow> </msqrt> </mfrac> <mo> ⁢ </mo> <mrow> <msub> <mi> ϑ </mi> <mn> 4 </mn> </msub> <mo> ( </mo> <mrow> <mfrac> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> q </mi> <mo> ) </mo> </mrow> </mfrac> <mo> , </mo> <msup> <mi> ⅇ </mi> <mfrac> <msup> <mi> π </mi> <mn> 2 </mn> </msup> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> q </mi> <mo> ) </mo> </mrow> </mfrac> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> EllipticTheta </ci> <cn type='integer'> 2 </cn> <ci> z </ci> <ci> q </ci> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 4 </cn> </apply> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <ln /> <ci> q </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> <apply> <ln /> <ci> q </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> EllipticTheta </ci> <cn type='integer'> 4 </cn> <apply> <times /> <imaginaryi /> <pi /> <ci> z </ci> <apply> <power /> <apply> <ln /> <ci> q </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <power /> <pi /> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <ln /> <ci> q </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["EllipticTheta", "[", RowBox[List["2", ",", "z_", ",", "q_"]], "]"]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]], " ", SuperscriptBox["\[ExponentialE]", FractionBox[SuperscriptBox["z", "2"], RowBox[List["Log", "[", "q", "]"]]]], " ", SqrtBox["\[Pi]"]]], ")"]], " ", RowBox[List["EllipticTheta", "[", RowBox[List["4", ",", FractionBox[RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", "z"]], RowBox[List["Log", "[", "q", "]"]]], ",", SuperscriptBox["\[ExponentialE]", FractionBox[SuperscriptBox["\[Pi]", "2"], RowBox[List["Log", "[", "q", "]"]]]]]], "]"]]]], SqrtBox[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", RowBox[List["Log", "[", "q", "]"]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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