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JacobiNC




Mathematica Notation

Traditional Notation









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





http://functions.wolfram.com/09.31.06.0002.01









  


  








Input Form





JacobiNC[z, m] == (Pi/(2 Sqrt[1 - m] EllipticK[m])) Sec[(Pi z)/(2 EllipticK[m])] - ((2 Pi)/(Sqrt[1 - m] EllipticK[m])) Sum[(((-1)^k EllipticNomeQ[m]^(2 k + 1))/(EllipticNomeQ[m]^(2 k + 1) + 1)) Cos[((2 k + 1) Pi z)/(2 EllipticK[m])], {k, 0, Infinity}]








Standard Form





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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> nc </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> z </mi> <mo> &#10072; </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mfrac> <mrow> <mi> &#960; </mi> <mtext> </mtext> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> m </mi> </mrow> </msqrt> <mo> &#8290; </mo> <mrow> <mi> K </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mi> sec </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> </mrow> <mo> - </mo> <mrow> <mfrac> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> <mtext> </mtext> </mrow> <mrow> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> m </mi> </mrow> </msqrt> <mo> &#8290; </mo> <mrow> <mi> K </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mrow> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msup> <mo> &#8290; </mo> <msup> <mrow> <mi> q </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> k </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </msup> </mrow> <mrow> <msup> <mrow> <mi> q </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> k </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </msup> <mo> + </mo> <mn> 1 </mn> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> k </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <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> <ci> JacobiNC </ci> <ci> z </ci> <ci> m </ci> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <pi /> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <ci> EllipticK </ci> <ci> m </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <sec /> <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 /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> <apply> <power /> <apply> <times /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <ci> EllipticK </ci> <ci> m </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> <apply> <power /> <apply> <ci> q </ci> <ci> m </ci> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <power /> <apply> <ci> q </ci> <ci> m </ci> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <cos /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> <cn type='integer'> 1 </cn> </apply> <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> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





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





2001-10-29