EllipticTheta[3, 0, q] == Sqrt[2/Pi] Sqrt[EllipticK[InverseEllipticNomeQ[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> &#977; </mi> <mn> 3 </mn> </msub> <mo> ( </mo> <mrow> <mn> 0 </mn> <mo> , </mo> <mi> q </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <msqrt> <mfrac> <mn> 2 </mn> <mi> &#960; </mi> </mfrac> </msqrt> <mo> &#8290; </mo> <msqrt> <mrow> <mi> K </mi> <mo> &#8289; </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> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> EllipticTheta </ci> <cn type='integer'> 3 </cn> <cn type='integer'> 0 </cn> <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> <ci> EllipticK </ci> <apply> <ci> InverseEllipticNomeQ </ci> <ci> q </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["EllipticTheta", "[", RowBox[List["3", ",", "0", ",", "q_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[SqrtBox[FractionBox["2", "\[Pi]"]], " ", SqrtBox[RowBox[List["EllipticK", "[", RowBox[List["InverseEllipticNomeQ", "[", "q", "]"]], "]"]]]]]]]]]

2001-10-29