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http://functions.wolfram.com/09.03.03.0011.01
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EllipticTheta[3, Pi (1/2 + m), q] ==
Sqrt[2/Pi] (1 - InverseEllipticNomeQ[q])^(1/4)
Sqrt[EllipticK[InverseEllipticNomeQ[q]]] /; Element[m, Integers]
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["EllipticTheta", "[", RowBox[List["3", ",", RowBox[List["\[Pi]", RowBox[List["(", RowBox[List[FractionBox["1", "2"], "+", "m"]], ")"]]]], ",", "q"]], "]"]], "\[Equal]", RowBox[List[SqrtBox[FractionBox["2", "\[Pi]"]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", RowBox[List["InverseEllipticNomeQ", "[", "q", "]"]]]], ")"]], RowBox[List["1", "/", "4"]]], SqrtBox[RowBox[List["EllipticK", "[", RowBox[List["InverseEllipticNomeQ", "[", "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> <msub> <mi> ϑ </mi> <mn> 3 </mn> </msub> <mo> ( </mo> <mrow> <mrow> <mi> π </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> + </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> , </mo> <mi> q </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <msqrt> <mfrac> <mn> 2 </mn> <mi> π </mi> </mfrac> </msqrt> <mo> ⁢ </mo> <mroot> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <msup> <semantics> <mi> q </mi> <annotation-xml encoding='MathML-Content'> <ci> EllipticNomeQ </ci> </annotation-xml> </semantics> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> q </mi> <mo> ) </mo> </mrow> </mrow> <mn> 4 </mn> </mroot> <mo> ⁢ </mo> <msqrt> <mrow> <mi> K </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <msup> <semantics> <mi> q </mi> <annotation-xml encoding='MathML-Content'> <ci> EllipticNomeQ </ci> </annotation-xml> </semantics> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> q </mi> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> </msqrt> </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> <ci> EllipticTheta </ci> <cn type='integer'> 3 </cn> <apply> <pi /> <apply> <plus /> <cn type='rational'> 1 <sep /> 2 </cn> <ci> m </ci> </apply> </apply> <ci> q </ci> </apply> <apply> <times /> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <pi /> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> InverseEllipticNomeQ </ci> <ci> q </ci> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 4 </cn> </apply> <apply> <power /> <apply> <ci> EllipticK </ci> <apply> <ci> InverseEllipticNomeQ </ci> <ci> q </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <in /> <ci> m </ci> <integers /> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["EllipticTheta", "[", RowBox[List["3", ",", RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List[FractionBox["1", "2"], "+", "m_"]], ")"]]]], ",", "q_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SqrtBox[FractionBox["2", "\[Pi]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", RowBox[List["InverseEllipticNomeQ", "[", "q", "]"]]]], ")"]], RowBox[List["1", "/", "4"]]], " ", SqrtBox[RowBox[List["EllipticK", "[", RowBox[List["InverseEllipticNomeQ", "[", "q", "]"]], "]"]]]]], "/;", RowBox[List["m", "\[Element]", "Integers"]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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