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JacobiSN




Mathematica Notation

Traditional Notation









Elliptic Functions > JacobiSN[z,m] > Series representations > q-series





http://functions.wolfram.com/09.36.06.0003.01









  


  








Input Form





Log[JacobiSN[z, m]] == 2 Log[EllipticTheta[3, 0, EllipticNomeQ[m]]] + Log[Sin[(Pi z)/(2 EllipticK[m])]] - 4 Sum[(EllipticNomeQ[m]^r/(r (1 + EllipticNomeQ[m]^r))) Sin[(r Pi z)/(2 EllipticK[m])]^2, {r, 1, Infinity}]








Standard Form





Cell[BoxData[RowBox[List[RowBox[List["Log", "[", RowBox[List["JacobiSN", "[", RowBox[List["z", ",", "m"]], "]"]], "]"]], "\[Equal]", RowBox[List[RowBox[List["2", RowBox[List["Log", "[", RowBox[List["EllipticTheta", "[", RowBox[List["3", ",", "0", ",", RowBox[List["EllipticNomeQ", "[", "m", "]"]]]], "]"]], "]"]]]], "+", RowBox[List["Log", "[", RowBox[List["Sin", "[", FractionBox[RowBox[List["\[Pi]", " ", "z"]], RowBox[List["2", RowBox[List["EllipticK", "[", "m", "]"]]]]], "]"]], "]"]], "-", RowBox[List["4", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["r", "=", "1"]], "\[Infinity]"], RowBox[List[FractionBox[SuperscriptBox[RowBox[List["EllipticNomeQ", "[", "m", "]"]], "r"], RowBox[List["r", " ", RowBox[List["(", RowBox[List["1", "+", SuperscriptBox[RowBox[List["EllipticNomeQ", "[", "m", "]"]], "r"]]], ")"]]]]], " ", SuperscriptBox[RowBox[List["Sin", "[", FractionBox[RowBox[List["r", " ", "\[Pi]", " ", "z"]], RowBox[List["2", RowBox[List["EllipticK", "[", "m", "]"]]]]], "]"]], "2"]]]]]]]]]]]]]








MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> sn </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> z </mi> <mo> &#10072; </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <msub> <mi> &#977; </mi> <mn> 3 </mn> </msub> <mo> ( </mo> <mrow> <mn> 0 </mn> <mo> , </mo> <mrow> <semantics> <mi> q </mi> <annotation-xml encoding='MathML-Content'> <ci> EllipticNomeQ </ci> </annotation-xml> </semantics> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mi> K </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> r </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mrow> <mfrac> <mrow> <msup> <mrow> <semantics> <mi> q </mi> <annotation-xml encoding='MathML-Content'> <ci> EllipticNomeQ </ci> </annotation-xml> </semantics> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> <mi> r </mi> </msup> <mtext> </mtext> </mrow> <mrow> <mi> r </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <msup> <mrow> <semantics> <mi> q </mi> <annotation-xml encoding='MathML-Content'> <ci> EllipticNomeQ </ci> </annotation-xml> </semantics> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> <mi> r </mi> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <msup> <mi> sin </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mfrac> <mrow> <mi> r </mi> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mi> K </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ln /> <apply> <ci> JacobiSN </ci> <ci> z </ci> <ci> m </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ln /> <apply> <ci> EllipticTheta </ci> <cn type='integer'> 3 </cn> <cn type='integer'> 0 </cn> <apply> <ci> EllipticNomeQ </ci> <ci> m </ci> </apply> </apply> </apply> </apply> <apply> <ln /> <apply> <sin /> <apply> <times /> <pi /> <ci> z </ci> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ci> EllipticK </ci> <ci> m </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <sum /> <bvar> <ci> r </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <times /> <apply> <power /> <apply> <ci> EllipticNomeQ </ci> <ci> m </ci> </apply> <ci> r </ci> </apply> <apply> <power /> <apply> <times /> <ci> r </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <ci> EllipticNomeQ </ci> <ci> m </ci> </apply> <ci> r </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <sin /> <apply> <times /> <ci> r </ci> <pi /> <ci> z </ci> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ci> EllipticK </ci> <ci> m </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Log", "[", RowBox[List["JacobiSN", "[", RowBox[List["z_", ",", "m_"]], "]"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["2", " ", RowBox[List["Log", "[", RowBox[List["EllipticTheta", "[", RowBox[List["3", ",", "0", ",", RowBox[List["EllipticNomeQ", "[", "m", "]"]]]], "]"]], "]"]]]], "+", RowBox[List["Log", "[", RowBox[List["Sin", "[", FractionBox[RowBox[List["\[Pi]", " ", "z"]], RowBox[List["2", " ", RowBox[List["EllipticK", "[", "m", "]"]]]]], "]"]], "]"]], "-", RowBox[List["4", " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["r", "=", "1"]], "\[Infinity]"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["EllipticNomeQ", "[", "m", "]"]], "r"], " ", SuperscriptBox[RowBox[List["Sin", "[", FractionBox[RowBox[List["r", " ", "\[Pi]", " ", "z"]], RowBox[List["2", " ", RowBox[List["EllipticK", "[", "m", "]"]]]]], "]"]], "2"]]], RowBox[List["r", " ", RowBox[List["(", RowBox[List["1", "+", SuperscriptBox[RowBox[List["EllipticNomeQ", "[", "m", "]"]], "r"]]], ")"]]]]]]]]]]]]]]]


Date Added to functions.wolfram.com (modification date)





2001-10-29